The meeting on Modulation Transfer Functions and their Application, held in Boston, in March of 1968, was one of the most useful and significant meetings of its type. In summary, more useful, factual information was transmitted than at any previous meeting of its type. There appeared more examples of work "done" than "to be done".

It is the purpose of this report to present the case for the use of a coherent transfer function in optics. At present both a coherent and a non-coherent trans-fer function are defined by various authors. Additionally,transfer functions for partially coherent light can be defined. It is proposed in this paper that a transfer function operating on amplitude be used.

The Modulation Transfer Function for a diffraction limited lens has been calculated for the cases of coherent, partially coherent, and incoherent illumination. These treatments have been based on mathematical transforms that are not necessarily well understood by the practicing optical engineer. Thus, while the results may be used daily, there is not a clear understanding of why the results are valid.

A general description of optical instruments is presented, which can be used to treat geometrical optics, Fourier optics, and the wave theory in a unified manner. It is shown that a joint treatment of these various branches of optics clarifies many results that quite often remain intuitively obscure to the student; e.g., the Legendre transformations in eikonal theory attain an immediate significance in the context of the wave theory. Students of a course along these lines must have a moderate knowledge of Fourier integrals. In the final section it is shown that diffraction theory can be described in terms of a "sum over all rays" concept.

Fourier analysis can be applied to an anisoplanatic optical system, if we define the modulation transfer function of such a system as the four-dimensional Fourier transform of its space-variant point spread function. We show that this approach can be used to estimate the band-width of the output image, and thereby provides a way of determin-ing the sampling rates required for computer simulation of such systems. We also show that the computer calculation of the image is less time-consuming to do in the spatial domain than in the frequency domain.

The calculation of the optical transfer function from lens design data by digital computer has become common today. The optical transfer function describes the imaging of a spatial sine wave by the optical system. The amplitude of a sine wave is described by the modulation, M, given by the equation

We are undertaking a study for the Air Force to determine the state of the art in the measurement of the OTF of lenses. Measurement of OTF has received a great deal of attention, but has it become a practical tool for industry? If the measurement of OTF is a well established procedure for evaluating lenses then perhaps the Air Force should include this measurement in its military standard procedures.

A recent advance makes feasible the evaluation of the physical optical transfer function at points which cover the entire frequency plane, or a selected portion of it, at any desired density. The vector frequencies at which the transfer function is evaluated are independent of the wavelength, the numerical aperture of the imagery, and the element size of the grid used to determine the locations of the points at which the pupil function is evaluated. Accurate results are achieved by interpolating in the piece-wise nearly linear phase of the pupil function. A table of maximum, minimum, and average response at each of a set of frequencies as a function of azimuth serves as a summary of the more detailed results. virtually, any optical system of interest regardless of its numerical aperture and/or the size of its aberrations can be evaluated by using this method with a 20 x 20 grid.

Our objective in this paper is to present a practical system for the calculation of real optical system transfer functions under various conditions of the lens and light source. The techniques and examples of results which we shall include in the presentation represent a currently operating data manipulation system which combines lens design data with measurements of surface or system errors to develop the OTF on a two dimensional basis.

This paper concerns a simple and inexpensive method for computing the polychromatic MTF of an optical system whose performance is determined essentially by longitudinal chromatic aberration. This method employs the results of wave optics computation and therefore retains high accuracy regardless of whether image quality is governed primarily by diffraction, geometric aberrations, or any combination thereof. We believe the analysis methods outlined in this paper will provide a useful engineering tool to the systems engineer who wishes some reasonably accurate insight into the effect of chromatic aberration on MTF tradeoffs. For lenses intended for high-acuity aerial photography, as a general rule, chromatic aberration will seriously limit MTF performance for focal lengths longer than 6 inches.

The optical transfer function (OTF), obtained with the Perkin-Elmer edge gradient analysis (EGA) measurement technique, is a complex function consisting of the modulation transfer function (MTF) and the phase transfer function (91F). Previous experiments demonstrated the EGA technique to be accurate for measuring the modulation transfer function. An experiment was conducted to test the Perkin-Elmer automated EGA technique as to its accuracy in measuring the phase transfer function. Experimental equipment was designed to pro-duce photographic edges with a theoretically known phase component. The experimental apparatus provided sinusoidal image motion to an edge test target while it was photographed with a high quality optical system. The resulting photographic edges were degraded by a known amount of sinusoidal image motion during exposure. These edges were measured using the EGA procedure. The experimental values for all OTF effects other than that due to sinusoidal image motion were then removed. The resultant OTF's were compared to the theoretically derived values for the appropriate sinusoidal image motion. The results demonstrate that the EGA technique is an accurate method of measuring both the modulation transfer function and the2hasetran'sfer function over the spatial frequency region tested.

The concept of a unique modulation transfer function for a photographic system is often inadequate because of the nonlinearities that exist in the system. Examples of the nonlinearities are the spatial interactions in the development process and the relation between exposure and the transmittance of the developed image. The effective MTF of the system may vary with the exposure level and with the nature of the input exposure patterns. Sinus-oidal exposure patterns, for example, can yield a different effective MTF than white noise patterns. In a negative-positive system, the use of low-gamma negative material with a high-gamma positive material may give better sharpness than the converse even though cascading of the component modulation transfer functions indicates no difference in system MTF. The distorted waveform generated by the nonlinearity of a high-gamma negative contains high-frequency harmonics that can overtax the resolution capabilities of the printing components. Since linear analysis is inadequate for these problems, certain methods of non-linear analysis are being applied. In one of them, a second degree prediction formula is used in conjunction with separate optical and chemical spread functions.

The Modulation Transfer Function is based on Fourier techniques which imply the validity of the principle of superposition. This is clearly violated in the nonlinear film development process. It becomes necessary to face up to this question if one is to know how much confidence to place in the results of such analysis. A pseudotransfer function is therefore defined to represent an artificial linear filter which, if it replaced the actual photographic process, would result in precisely the same image as would be produced by the real system. The results of a computer program are presented to illustrate how such an approach may be used. Some of the conditions under which the atmospheric transfer function remains undefined are also reviewed.

Many methods have evolved over the past fifteen years for the measurement of optical and modulation transfer functions. Sufficient progress has been made to regard this as a most powerful criteria for the design and assessment of optical systems.

After a short summary, the used measurement methods of modulation transfer function in the Physikalisch-Technische Bundesanstalt (PTB) are described. The experience and the difference of the methods regarding measuring time, their sources of error as well as the attainable accuracy are discussed. Furtheron a simple program to calculate the otf from lens data under consideration of the dif-fraction is explained and the results of the calculated otf compared with the experimental otf-values are given and the differences between them are shown. Finally, the advantage to use the otf in order to derive quality criteria of the image performance is emphasized in spite of some shortcomings resulting from the difficulties of the mostly used lenses and measuring devices.

An outline is given of the factors which can lead to errors in the experimental measurement of 0.t.f. and therefore to disagreement between the results obtained by different laboratories. Some of the subsidiary checks which can be carried out on 0.t.f. equipment to eliminate or reduce these errors are indicated. Methods of establishing the overall validity and accuracy of such equipment are described and their relative merits discussed. This covers the work done by the 'Sira' group sponsored project on simple standard test lenses and includes a discussion of design requirements, the limitations on the applicability of this method and proposals for extending the work.

A production unit of a new, completely self-contained device for measuring optical transfer functions will be demonstrated and explained in detail. Some of the remarkable features are: A simple photocell is employed instead of a photo multiplier because of the high efficiency of the optical track. It has a unique attachment for exact spatial zero frequency measurements. It has a built-in oscilloscope and a recorder capable of taking a complete diagram every 20 seconds. It works with one single electrical frequency of 400 cps and therefore eliminates the higher harmonics without the necessity of a sinusoidal target. The instrument can be operated by a technician without any special optical knowledge.

This paper will describe the instrumental details of a modulation transfer function analyzer capable of presenting automatically the MTF curves in the frequency range of : a) 0.4 cycles/mm to 10 cycles/ mm; b) 4 cycles/mm to 100 cycles/mm; and c) 24 cycles/mm to 600 cycles/mm. The instrument is designed to produce an MTF curve,after the optical element has been set in place, in approximately 5 seconds, with an accuracy of ±5 percent of the indicated response value. This instrument has been used extensively for testing of lens systems, fiber optics components, and such electro-optical systems as image converters and image intensifiers. Pertinent details of the instrument setup for these various applications as well as data on numerous devices will be presented. Instrument accuracies and error analysis will be discussed along with details of y-axis normalization.

The modulation transfer function of optical systems is computed under strict consideration of diffraction at a given spectral distribution of light source and radiation detector. The consideration is based on the precondition that the monochromatic aberrations of the optical system are negligibly small and that only the chromatic aberra-tions are effective. The modulation transfer function is computed for different states of correction with regard to the correction of the chromatic aberrations (over and under-correction, secondary spectrum) and the corresponding results discussed in order to obtain rules for optimum correction of chromatic aberration.

This paper describes an application of the Modulation Transfer Function (MTF) to problems encountered in the fabrication of an unusual 45° field eyepiece, consisting of three strongly aspheric plastic lenses and one conventional glass eyelens; eyepiece performance specifications are given in terms of MTF. The optical design problem combines high performance and extreme lightweight with the ability to mass produce at very low cost by injection molding. Since surfaces are general aspherics, they cannot be evaluated by normal optical techniques. Material shrinkage during molding further complicates the problem by requiring the mold to differ from the desired lens. Molds are compensated by directly measuring the contour of the lens and comparing the results to the design requirements. This process is iterative. To minimize iterations, a program was developed to fit measured lens data to aspheric equations of the form used in the optical design, to allow evaluation of system performance by direct calculation of MTF. In this way, not only could system performance be evaluated, but the affect of individual lens surfaces on the total performance could be studied as well. The study of the affect of individual lens surfaces was carried out by replacing the equation developed from fitted data for a given surface with the nominal design equa-tion and calculating the MTF. We, thus, were able to determine the effect of improving this surface without the need of performing the actual operation. Through this technique, the surfaces which were most sensitive to errors were located and correction efforts concentrated on those which gave the greatest optical performance improvement in terms of MTF. This utilization of MTF allowed system performance to be optimized with a considerable savings in time and effort. Computer predictions of MTF performance compred favorably with directly measured results.

The significance of modulation transfer function for optical components such as applied to the image intensifier and low light level television systems is well recognized. Photographic lenses are generally designed for high resolution, and for high contrast targets. These lenses usually do not have high modulation transfer characteristics at low spatial frequencies where performances of the image intensifier and television tubes are critical. The vital requirements for the low light level optical components are