## 2 Related Works  
Ray tracing is the practical tool of geometrical optics [10]. It is based on the following three postulates:  
1. The wave direction is given by the normal, to what is referred to as the equiphase planes.
2. Rays are assumed to progress in straight lines in a homogeneous medium.
3. Power is conserved inside a bundle of rays.
4. Reflection and refraction obey the refraction rule.  
It is obvious that geometric optics cannot offer a comprehensive interpretation of image formation for an object through a given optical instrument. For example, aberrations and diffraction could not be straightforwardly treated by geometrical optics. In this paper, we intend to bypass these constraints by offering the concept of a diffraction-based compound system.  
Diffraction was first observed by [PERSON], when illuminating a rectangular aperture in an opaque medium. He used a light source at his disposal at that time. He observed the image at some distance behind the screen [15]. He did not observe sharp borders.  
Figure 1: Different theoretical frameworks to treat light. Quantum mechanics is the most rigorous framework.  
He observed a gradual transition from light to shadow. [PERSON] built up [PERSON]'s observations and advanced wave optics [16]. [PERSON], who discovered interference [5], supported wave theory [6]. [PERSON] was convinced that the postulation of [PERSON] does not offer an explanation for the non-existence of propagating backwards waves and advanced his theory. He proposed a combination of the [PERSON] principle, with the \"envelope\" building [7]. [PERSON] gave a full mathematical formulation to the scalar diffraction theory, based on the works of [PERSON] and [PERSON] [7]. [PERSON] and [PERSON] further developed [PERSON]'s work and advanced what was later referred to as the \"Rayleigh-Sommerfeld diffraction theory\" [7].  
Let us go back to geometrical optics. There are recent researches using geometrical optics to handle new issues, such as paraxial-based optical predesign of systems, polarization, inverse problems, refractive index distributions, surface geometries from ray-tracing, and aberrations. A recent work highlighted the topological space of 2 \(\times\) 2 matrices. The authors used the classical versus quantum, as well as the wave models. They proposed 2 \(\times\) 2 matrices to treat the quantization process [17]. Based on geometrical optics, [PERSON] proposed a new type of atmospheric dispersion corrector, suitable for wide-field high-resolution imaging and spectroscopy on large telescopes [18]. [PERSON] et al. proposed a new optical design method of a floating type collimator for microscopic camera inspection [19]. An improved analytical theory of ophthalmic lens design, based on geometrical optics, was proposed recently [20]. To control the linear astigmatism aberration in a perturbed axially symmetric optical system and tolerancing was performed by using geometrical optics [21]. A focus-adjustable, head-mounted display with off-axis system was proposed [22]. The objective of this paper is not to be limited to a geometrical theory of diffraction as suggested by [PERSON] [1]. What it is proposed in this paper is a powerful tool in a similar way as done with the compound system in geometrical theory.  
In summary, the wave aspect is intrinsic to scalar theory of diffraction but not to geometrical theory. The addition of wave aspects, such as wavefront aberration, polarization, PSF, OTF, and MTF, into geometrical optics are neither natural nor straightforward in this theory. What is generally done is an addition of wave aspects to ray-tracing. In other words, the wave character is artificially added in modified geometrical optical approaches.