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To determine the system MTF, a sinusoidal pattern with perfect contrast is used as the object. Image contrast is defined as:

*I _{max }*and

*f _{c}*. The cutoff frequency, which is the highest frequency the system can resolve, is given by:

*Figure 4: Examples of MTF vs defocus position and MTF vs field charts.*

*Figure 5: On-axis CTF. The target is a square-wave pattern instead of a sine-wave pattern. *

The optical designer doesn’t necessarily aim to achieve the diffraction limit performance. The desired MTF curve is based on design requirements. The lens specifications are usually of the form of an MTF value for specific frequencies. The MTF specification may come, for instance, from the sensor pixel size and the Nyquist theorem, which describes the maximum spatial frequency that the sensor can resolve.

For example, assume a digital sensor with a pixel size of 7.4 μm x 7.4 μm. According to the Nyquist theorem, the highest frequency that can be resolved is 1 cycle/(2 x 7.4 um) = 0.0676 cycles/μm ≈ 68 cycles/mm. Based on the sensor characteristics, a typical performance requirement for the lens could be:

- MTF > 50% at 17 cycles/mm, and
- MTF > 25 % at 68 cycles/mm.

Figure 6 shows the MTF chart from a lens that would meet these requirements for all fields. It is important to know the sensor characteristics during the design process since in many cases the Nyquist frequency of the sensor is only a small fraction of the diffraction cutoff frequency *f _{c.}*

*Figure 6: System that meets requirements of MTF > 50% at 17 cycles/mm and MTF > 25% at 68 cycles/mm.*

Experimentally, there are different ways to measure the frequency response of an optical system. Two examples are the slanted-edge target and the three-bar target. These targets are easy to use since they just need to be imaged. The three-bar target is a common method to measure resolution, it consists of three parallel bars of certain width and separation that corresponds to a specific spatial frequency. To measure system resolution, multiple three-bar patterns at specific frequencies are arranged in a single chart, and the resulting image is inspected to determine the smallest features that can be resolved. A popular target with these characteristics is the USAF 1951 chart shown in Figure 7. The measured resolution from a three-bar target might differ from the predicted CTF, since the CTF assumes an infinitely long pattern of bars. In a pattern with a finite number of bars, there can be end effects that reduce the contrast of the bars at the ends of the pattern. For this reason, the measured contrast might be less than the predicted by the CTF.

Another method to experimentally measure the MTF is the slanted-edge target. This method is different in that frequency information can be obtained from a single target. The edge must be at a specified small angle with respect to the pixel array of the sensor, as shown in Figure 7. When the input is a step function instead of a point source, the irradiance distribution at the image is an edge spread function. The MTF can then be determined from the Fourier transform of the derivative of the edge spread function.

*Figure 7: Examples of targets utilized to measure MTF experimentally.*

where is a 2D spatial-position vector.

The incoherent transfer function of a linear, shift-invariant system is given by the Fourier transform of the incoherent point spread function:

The Modulation Transfer Function (MTF) is the modulus of the optical transfer function:

where is the phase. The MTF can only take positive values, but the OTF can be negative when there is phase reversal. For the image of the bar pattern, this would result in contrast reversal, meaning that white features become dark and dark features become light. In an MTF plot, the behavior is that the MTF curve reaches zero and then “bounces”. It is important to be aware of this behavior. After the bounce, the MTF technically increases even though it reached zero at a lower frequency. Another possible impact is that if the MTF is used in the error function during optimization, phase reversal may produce local minima.

Other common system specifications are Strehl ratio and RMS wavefront error. For small aberrations, specifying the Strehl ratio is equivalent to specifying the RMS wavefront error. The Strehl ratio is defined as the ratio of the peak intensity of the measured PSF to the peak intensity of the perfect PSF. The MTF is related to the inverse Fourier transform of the incoherent PSF. As a result, it is possible to express the Strehl ratio as the integral under the entire MTF curve, including frequencies beyond the Nyquist frequency (which are irrelevant). For this reason, in situations where Nyquist is well below the diffraction cutoff, it makes more sense to specify MTF at frequencies between zero and the Nyquist frequency, rather than specifying the Strehl ratio or RMS wavefront error.

- Barrett, H. H., & Myers, K. J. (2004).
*Foundations of image science*. John Wiley & Sons. - Bentley, J., & Olson, C. (2012).
*Field guide to lens design*. SPIE Press. - Goodman, J. W. (2005).
*Introduction to Fourier optics*. Roberts and Company Publishers. - Hecht, E. (2002).
*Optics*. Pearson Education, Inc. - Mahajan, V. N. (1982). Strehl ratio for primary aberrations: some analytical results for circular and annular pupils.
*JOSA, 72*(9), 1258-1266. - Rogers, J. R. (2008, September). Three-bar resolution versus MTF: how different can they be anyway?. In
*An Optical Believe It or Not: Key Lessons Learned*(Vol. 7071, pp. 38-45). SPIE. - Sasian, J. (2019).
*Introduction to lens design*. Cambridge University Press. - Smith, W. J. (2008).
*Modern optical engineering*. McGraw-Hill. - Zhang, X., Kashti, T., Kella, D., Frank, T., Shaked, D., Ulichney, R., Fischer, M., & Allebach, J. P. (2012, January). Measuring the modulation transfer function of image capture devices: what do the numbers really mean?.
*In Image Quality and System Performance IX*(Vol. 8293, pp. 64-74). SPIE.

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