Transmission through mismatched layer in glass of marginal ray vs. numerical aperture

[hans]

Similar to the plots on Wikipedia (Fresnel equations) but showing intensity transmission from glass to an intermediate mismatched layer (air, water, glycerol) then back to glass for randomly polarized light. (Average of the p- and s-polarized cases.) Using NA on the horizontal axis compresses the right side of the graph due to the sine relationship and makes the falloff look sharper. The falloff is pretty sharp in terms of NA, for example at 0.90 with a dry condenser transmission is still almost 80% for the marginal ray through the air layer.

layer-transmissivity.png (60.13 KiB) Viewed 2826 times

gnuplot commands:

Code: Select all

sz(n, na) = sqrt(1 - (na/n)**2)
tap(n1, n2, na) = 2*n1*sz(n1, na)/(n2*sz(n1, na) + n1*sz(n2, na))
tas(n1, n2, na) = 2*n1*sz(n1, na)/(n1*sz(n1, na) + n2*sz(n2, na))
tj(n1, n2, na, ta) = ((n2*sz(n2, na))/(n1*sz(n1, na)))*ta**2
tjp(n1, n2, na) = tj(n1, n2, na, tap(n1, n2, na))
tjs(n1, n2, na) = tj(n1, n2, na, tas(n1, n2, na))
tjr(n1, n2, na) = (tjp(n1, n2, na) + tjs(n1, n2, na))/2
tjr_layer(n1, n2, na) = tjr(n1, n2, na)*tjr(n2, n1, na)

set xrange [0:1.5]
set yrange [0:1]
set samples 1500

set grid xtics mxtics ytics
set xtics 0.1
set ytics 0.1
set mxtics 2
set key above
set term pngcairo size 1024, 512
set output "layer-transmissivity.png"

plot \
	tjr_layer(1.52, 1.00, x), \
	tjr_layer(1.52, 1.33, x), \
	tjr_layer(1.52, 1.47, x)

[Hobbyst46]

Nice graphs. Hoping I understand them correctly:
Does the graph mean, that the total intensity of the light that reaches the specimen, at NA=0.9 of the condenser, is almost 80% of the intensity of light that passes through the condenser and reaches its top (front) lens ?

I wonder how do my recent experiments with darkfield, using different immersion media (the same three) correlate with the theory. Too late here right now for deep thinking...

[hans]

Does the graph mean, that the total intensity of the light that reaches the specimen, at NA=0.9 of the condenser, is almost 80% of the intensity of light that passes through the condenser and reaches its top (front) lens ?

No, unfortunately, total intensity would be quite a bit more difficult to calculate, I think. Probably requires a double integral of this sort of radial function over some aperture, and not obvious to me what coordinates would be appropriate. This plot is just showing transmission of a plane wave at angle corresponding to the maximum within given NA light cone. So it can give some sense of the reduced intensity contribution of rays approaching total internal reflection (which presumably also reduces effective resolution a bit) but not intensity or resolution of the final image in a straightforward way. I was mainly trying to get a sense of how much estimates based on a hard cutoff at NA equal to the refractive index might differ vs. taking into account partial reflection at angles close to but still below total reflection.

[Hobbyst46]

Does the graph mean, that the total intensity of the light that reaches the specimen, at NA=0.9 of the condenser, is almost 80% of the intensity of light that passes through the condenser and reaches its top (front) lens ?

No, unfortunately, total intensity would be quite a bit more difficult to calculate, I think. Probably requires a double integral of this sort of radial function over some aperture, and not obvious to me what coordinates would be appropriate. This plot is just showing transmission of a plane wave at angle corresponding to the maximum within given NA light cone. So it can give some sense of the reduced intensity contribution of rays approaching total internal reflection (which presumably also reduces effective resolution a bit) but not intensity or resolution of the final image in a straightforward way. I was mainly trying to get a sense of how much estimates based on a hard cutoff at NA equal to the refractive index might differ vs. taking into account partial reflection at angles close to but still below total reflection.

To create DF, I place the drop of liquid (W, G, O) on the front lens and raise the condenser. The condenser is defined as NA=1.4 (with oil), so it will have an NA of ~1.4 with glycerine as well, ~1.3 with water and ~1.0 with air. When the liquid touches the slide bottom, it is brightly lit, as if a flash, whatever the liquid. I see it through the thickness of the slide, so a squared transmission must be factored in. For NA<=1 vs NA<=1.3 the change in brightness is considerable.For higher NAs, vs the maximal NA, not so much.

[Placozoa]

This is confusing, and that wikipedia page goes on forever, I did skim through though.

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.

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.

All very confusing. :-/

[Hobbyst46]

This is confusing, and that wikipedia page goes on forever, I did skim through though.

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.

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.

All very confusing. :-/

Yes, transmission of light through a medium depends on the thickness of the medium, among other factors. This dependence was not addressed in the graphs. I think that the purpose of the OP was not to compare between two or more objectives, or between two or more condensers, but show the dependence of transmission on the surrounding media. And the factor studied is the NA. So, at least for a given condenser, the graphs are valid IMHO. The same goes for a given objective. And being able to switch the immersion medium is convenient and benefical.

[hans]

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.

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.

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.

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.

[hans]

When the liquid touches the slide bottom, it is brightly lit, as if a flash, whatever the liquid.

Is this referring to the edges of the slide lighting up due to some of the light being guided out via multiple internal reflections? Or the view through the eyepieces?

I see it through the thickness of the slide, so a squared transmission must be factored in. For NA<=1 vs NA<=1.3 the change in brightness is considerable.For higher NAs, vs the maximal NA, not so much.

I think intensity should go as NA**2 for low NA, but similar to the earlier remark about now knowing appropriate coordinate system, not obvious to me how it should scale at high NA. Proportional to area of spherical wavefront?

[Hobbyst46]

When the liquid touches the slide bottom, it is brightly lit, as if a flash, whatever the liquid.

Is this referring to the edges of the slide lighting up due to some of the light being guided out via multiple internal reflections? Or the view through the eyepieces?
Not through the eyepiece. Just looking at the slide from above. Due to light guided out.

I see it through the thickness of the slide, so a squared transmission must be factored in. For NA<=1 vs NA<=1.3 the change in brightness is considerable.For higher NAs, vs the maximal NA, not so much.

I think intensity should go as NA**2 for low NA, but similar to the earlier remark about now knowing appropriate coordinate system, not obvious to me how it should scale at high NA. Proportional to area of spherical wavefront?

Just saying that the total transmittance through several layers of glass is a multiplication of the individual transmittances.