In modern optical research, the electromagnetic theory of light propagation, interference, and diffraction was systematically described in M. Born and E. Wolf's Principles of Optics and Joseph W. Goodman's Introduction to Fourier Optics. The coherent optical imaging theories in the two classic works are widely cited by contemporary scientists and technological workers. However, both theories are obtained in approximate conditions. The calculation formulas given in Principles of Optics are derived based on assuming the existence of an "isoplanatic region" in the image plane. The derived formula introduces a pupil function that is only related to the optical system aberration and the exit pupil does not provide a specific expression for the pupil function, which cannot be employed for practical calculations. The calculation formula given in Introduction to Fourier Optics can only calculate the amplitude distribution of the image field when the object size is less than 1/4 of the diameter of the incident pupil.

From a mathematical perspective, the formulas derived from the two optical masterpieces have the same form, and coherent optical imaging systems are both linear space-invariant systems. The physical meaning of the transfer function defined by the outgoing pupil is a filter for the ideal image spectrum.

With the advancing technology, the above approximate theories are gradually unable to meet practical needs. For example, experimental observations indicate that the imaging quality varies in different regions of the image plane, and the imaging system illuminated by coherent light is not a linear space-invariant system. Additionally, in modern optical detection research, the amplitude and phase of the image field are equally important physical quantities, and the theory that can only calculate the amplitude distribution of the image field cannot meet the requirements. Therefore, it is necessary to study theories that can accurately calculate the amplitude and phase distribution of image light fields.

Based on Fresnel diffraction integration, the spatial tracing of the optical wave field during the imaging of the lens imaging system is carried out, and the expression that can calculate the amplitude and phase distribution of the image light field is derived. Based on the derived formula, the shortcomings of the coherent optical imaging theory in the above-mentioned two optical masterpieces are first studied.

Considering currently no reports of quantitative numerical calculations on "ringing oscillation" in coherent optical imaging, ringing oscillation is an interference that must be eliminated for the image field of digital holographic detection. To experimentally prove the formula derived by the authors and provide a theoretical basis for eliminating ringing interference, we design a microscopic digital holography system. By adopting a USAF1951 resolution plate as the object, the intensity distribution of the image field is calculated using the calculation formula given in Introduction to Fourier Optics and the formula derived by us and compared with experimental measurements. In comparative studies, special attention should be paid to whether the theoretical distribution calculation of ringing and oscillation fringes is consistent with experimental measurements.

Based on the derived formula [Eq. (6)], the coherent optical imaging system is no longer a linear space-invariant system. The research results on the coherent optical illumination imaging formula in Principles of Optics indicate that for actual optical systems, there is no "isoplanatic region" in the image field, and there is no pupil function that is independent of the object field but only related to the exit pupil and aberration of the imaging system. The calculation formula given in Introduction to Fourier Optics can only approximate the amplitude distribution of the image light field when the object size is smaller than the diameter of the incident pupil by 1/4.

The comparison between theoretical calculations and experimental measurements shows that the derived Eq. (6) can more accurately calculate the distribution of ringing and oscillating fringes appearing in the image light field (Figs. 8-10).

Formulas that can calculate the amplitude and phase of the image light field are derived by spatial tracking of the coherent optical imaging process, and the shortcomings of classical coherent optical imaging theory are discussed. An imaging experiment with a rectangular transparent hole as the object is designed to prove the correctness of the derived formula. Meanwhile, the classical imaging calculation formula and the derived formula are utilized to simulate and calculate the experimental measurement image. The research results indicate that the derived formula can not only accurately calculate the intensity image of the image light field, but also more accurately calculate the distribution of ringing oscillations.