To improve the imaging properties of a defocused optical system, the use of shaded apertures is studied theoretically and experimentally. The study is based on the optical transfer function (OTF). The two shaded apertures studied are the type in which the amplitude transmittance decreases gradually from the center of the pupil toward its rim, TA, and the type in which the amplitude transmittance decreases from its rim toward the center, TB. For comparison, the effects achieved with a clear aperture, TC, are included. The results of the calculations show that near focus the OTF for TA has higher values in the low frequency region than has either TB or Tc. When the system is defocused, the shaded aperture of the type TA yields an improved defocused image that is faithful to the outline of the object. The quality of the defocused image obtained with TB is worsened. When the OTF is used as a means for judging the quality of the defocused image, the two necessary conditions on the functions appear to be that the OTF (1) must be a monotonically decreasing function and (2) must be nonnegative. These conditions are confirmed by experiment. Since the transmittance variation of the shaded apertures is achieved by absorption, the effects due to the resultant decreases in light level are also considered.
M. Mino and Y. Okano, "Improvement in the OTF of a Defocused Optical System Through the Use of Shaded Apertures," Appl. Opt. 10, 2219-2225 (1971)