Placozoa wrote: ↑Tue Mar 30, 2021 9:22 am
This is confusing, and that wikipedia page goes on forever, I did skim through though.
Some more context probably would have helped. There have been some discussion recently about substituting fluids in a homogeneous immersion stack that would typically go condenser -> immersion oil -> slide -> index-matched mounding media -> cover glass -> immersion oil -> objective. As Hobbyst says this graph is just the Fresnel coefficient for intensity transmission, nothing more, but plotted in a way that is maybe more directly relevant to the case of fluid substitutions deviating from homogeneous. The differences from the graph on the Wikipedia page are:
- p- and s-polarized cases averaged to show transmission of randomly-polarized light.
- Horizontal axis distorted (not a simple rescaling) to show transmission of the marginal ray vs. NA. (Rather than transmission vs. angle of incidence.)
- Transmission through a mismatched layer (two interfaces) rather than a single interface.
Placozoa wrote: ↑Tue Mar 30, 2021 9:22 am
Are all N.A. 0.90 objective lenses ground the same? I somehow doubt it, and if not then these graphs cannot be right. I think, and I could certainly be wrong, that objectives vary widely in how many lenses they have, what coatings the lenses have, as well as what materials the lenses are made from. Each of these factors would make a mess of a simple relationship between what angle a portion of a photon leaves the sample at and how much of it is reflected.
Immersion condensers and objectives I am familiar with have flat, uncoated surfaces -- in the exact homogenous case coating would not be necessary and curvature of the surfaces would have no optical effect. From what I understand though, high-end, modern immersion objectives are not actually corrected for homogeneous immersion and take into account for small differences in RI and dispersion of the cover glass, immersion fluid, and glass the front element is made out of. I would assume these differences are small compared to substituting air, water, or glycerol in place of the normal immersion fluid.
Placozoa wrote: ↑Tue Mar 30, 2021 9:22 am
I suppose if you wanted to see by experiment you could use a stop to act as an iris in the objective to set the NA at a fixed amount and compare the image quality to an objective that achieves that same NA by having lenses ground to achieve the same NA naturally. The first objective should be better somehow since its intensity would fall off more sharply if this is true.
Not sure I follow the logic here.
Placozoa wrote: ↑Tue Mar 30, 2021 9:22 am
All very confusing. :-/
Agree, and there could easily be a mistake somewhere. There two simple cases you can verify on the graph: First, transmission of the marginal ray goes to zero for NA >= n of the mismatched layer, as expected due to total internal reflection. Second, at normal incidence there is no polarization dependence, the Fresnel coefficients simplify considerably (shown on the Wikipedia page), and the ~92% transmission of marginal ray at NA = 0 shown on the graph agrees with the well-known number of ~4% reflection from a single glass-air or air-glass interface.
Also, the function is built up incrementally in gunplot so if you have gnuplot you can play around with plotting the various intermediate functions. The naming follows the derivations in Born and Wolf, starting with sz which is the z (normal) component of unit propagation vector s expressed in terms of NA using the relation cos**2 + sin**2 == 1. sz is then substituted into the p- and s-polarized amplitude coefficients [cos(theta_i) -> sz(n1, NA), cos(theta_t) -> sz(n2, NA)] giving functions tap and tas. tj is intensity transmission in terms of amplitude transmission, taking into account the differing refractive index. (Relation between amplitude and intensity depends on refractive index.) Finally tjp and tjs are are the p- and s- intensity coefficients, tjr is the average corresponding to random polarization, and tjr_layer is transmission in then out of the mismatched layer.