Distinction between image magnification and irradiance magnification: a commentary

Image irradiance in the absence of aplanatism and isoplanatism

“…Taking L = 5.00 × 10 −03 W cm −2 sr −1 and θ = 12.9633 degrees (as indicated in figure 1(a) with up to four decimal places to retain computational accuracy), equation ( 1) yields E ≈ 7.90 × 10 −04 W cm −2 , which is the same as the axial image irradiance that is determined through nonsequential ray tracing (see figure 2). This result further highlights the observation expressed in the author's previous communication [1]. Why does equation ( 1) work for non-aplanatic lenses?…”

“…The applicability of equation ( 1) to non-aplanatic images was also pointed out in 1966 by Gilmore [8], who had highlighted that despite texts-including even a classic book by Born and Wolf [6]-stating the assumption of aplanatism to derive equation ( 1), many practical optical systems are not aplanatic, and the axial irradiance of the images that they form seems to be given by equation (1). Gilmore applied thermodynamic arguments to show that equation ( 1) would be at least valid for the case of non-aplanatic optical imaging systems that are forming the image of extended objects.…”

“…a system that is free of spherical aberration and coma such that it obeys the so-called 'sine condition'), as one can confirm through examining a sample of references [2][3][4][5][6]. However, suppose we apply equation (1) to compute the axial image irradiance for the non-aplanatic system in figure 1(a). Taking L = 5.00 × 10 −03 W cm −2 sr −1 and θ = 12.9633 degrees (as indicated in figure 1(a) with up to four decimal places to retain computational accuracy), equation ( 1) yields E ≈ 7.90 × 10 −04 W cm −2 , which is the same as the axial image irradiance that is determined through nonsequential ray tracing (see figure 2).…”

“…Now, if rays at any angle θ fill the entire exit pupil, then the irradiance at Q is the integral of equation ( 2) over the full pupil area (in this paper, the 'exit pupil' shall often be called 'the pupil' whenever there is no ambiguity within the context of the discussion). From radiometry [2][3][4][5], the integral of equation (2) over a full pupil yields equation (1). For the stigmatic lens in figure 1(a), since all axial rays at Q fill the pupil, it is clear that equation (1) applies for computing the axial image irradiance for that lens, despite the fact that the image is not aplanatic.…”

“…determine the axial image irradiance) in each system, using an expression in closed-form. In prior communication [1], the author showed that the axial image irradiance for a stigmatic (but non-aplanatic) singlet that is imaging an incoherent extended disk Lambertian source can be determined through nonsequential ray tracing as well as by applying a well-known formula from radiometry [2][3][4][5][6], which is expressed as…”