Find pupil locations w/ paraxial ray tracing (thick lens, Geary CH 5)

[phillip_at_work]

Homework Statement

Given the triplet prescription as illustrated in Figure 5.17... find a) the size and location (relative to stop) of the entrance pupil using paraxial ray trace equations (PRTE); b) the starting heights and angles on surface 1 for the marginal and chief rays, again using PRTE [Problem statement attached]

Relevant Equations

y_f = y_i + u' * t
n' * u' = n * u - y * phi
(typical paraxial ray trace equations)

Per the description given in the book, one can trace rays FROM the physical aperture stop into object space to find the size and location of the entrance pupil (EP). Also, one can trace rays FROM the physical aperture stop into image space to find the size and location of the exit pupil (XP). In both cases, the EP and XP are images and the physical stop is the object to be imaged.

I implemented this ray trace using an Excel spread sheet (part a). This is attached in original Excel and also as a PDF.

To solve part b, I add the EP and XP into a second Excel ray trace. As I understand it, when I aim my system marginal ray at the rim of the EP, it should also travel through the rim of the physical stop and the rim of the XP as shown on slide 10-29 of Greivenkamp's thin lens EP/XP ray trace here: https://wp.optics.arizona.edu/jgrei...es/11/2018/12/201-202-10-Stops-and-Pupils.pdf

At the moment, I'm doing this using an object at an arbitrary distance (80mm) and of arbitrary height (10mm), since the pupil locations shouldn't depend on this (also per Greivenkamp).

However, aiming my marginal ray at the rim of the EP does not show that ray touching the rim of the stop and XP. So I suppose I've either traced the EP and XP incorrectly (part a) or I'm implementing the EP and XP incorrectly in the second ray trace (part b).

Any feedback is much appreciated.

 

[Drakkith]

To start, I think your distances from the stop to each optical element to the EP need to be negative in part A (or your refractive indices may need to be negative or something since your ray is going right to left instead of left to right). Then you seem to have all your refractive indices set to 1.51 instead of either 1.617 or 1.649 in part B. Just correcting those gives me a close match for the distance to the EP between your spreadsheet and this raytrace calculator. I'm getting a distance between the EP and the first element of -29mm (so EP is to the right of surface 1) and a distance from the last surface to the XP of -14 mm (xp is to left of last surface).

Try that and see if it makes sense.

I haven't done much optics since I had to drop out of the Optical Engineering program at the University of Arizona back in 2018, so take my suggestions with a healthy dose of skepticism.

Edit: Mixed up signs on the pupils, corrected them.

 

[Drakkith]

Also, the distance between the object and the EP needs to be the sum of the distance between the object and element 1 and the distance between element 1 and the EP. Since the EP is to the right of element 1, that's 80+29 = 109mm in my calculation. See the bottom left of 10-27 in the slides.

 

[phillip_at_work]

Many thanks for your reply. I apologize for the indices at 1.51. That was a carry over from an earlier version that I failed to revise.

You're onto something when you say the signs should be flipped. I suspect now that the curvature values must be negative when moving from right to left, which would make the power products flip signs.

When I make these changes, I get EP position -29.49mm with radius of 13.28mm. XP position of -14.06mm and radius of 12.4mm. Revising part b to reflect these values now shows the marginal ray contacting the rim of each pupil/stop correctly. Sweet!!

One followup question. Unless I'm mistaken how Greivenkamp is tracing rays on slides 26-27, he too seems to trace from left-to-right and right-to-left to get EP and XP location/size. But his final trace on slide 29 from object to image (all from left-to-right) shows the thin lens powers keep the same signs as slides 26-27. This is why I created my thick lens trace this way initially. I replicated Greivenkamp's example entirely and it works. Why do you suppose my signs needed to flip to get the right EP location, but his signs didn't?

I'm including my final spread sheets here for future readers.

Edit: typos

 

[Drakkith]

One followup question. Unless I'm mistaken how Greivenkamp is tracing rays on slides 26-27, he too seems to trace from left-to-right and right-to-left to get EP and XP location/size. But his final trace on slide 29 from object to image (all from left-to-right) shows the thin lens powers keep the same signs as slides 26-27. This is why I created my thick lens trace this way initially. I replicated Greivenkamp's example entirely and it works. Why do you suppose my signs needed to flip to get the right EP location, but his signs didn't?

Ah, that's because we're doing the raytrace wrong. According to my old class notes, when you perform a reverse ray trace you multiply each Y by the NEGATIVE of negative phi, or Y*-(-phi) = Y*phi. In your first spreadsheet, if I keep all the lens powers and distances and everything the same as you had them originally and just change then signs in the cell equations then I get an EP distance of about -28.49 mm. For example, cell K13 =M13+(-L9*L12). So no need to change the signs of any of the actual element properties.

 

[phillip_at_work]

I agree that phi must get another negative, but I'm wondering how to justify the change.

The sign convention for ray tracing left-to-right says that curvatures with the center of radius to the right are positive. Curvatures with the center to the left are negative. If we flip the direction of the ray trace, I think we must also flip the sign of the curvature. Because phi = (n-n')*c, that would give us the additional negative (e.g., cell L9 becomes negative). I could also be over-thinking this and I should just add another negative to phi!

Regardless of why phi gets another negative, I don't see the sign of phi change in Greivenkamp's as a function of ray trace direction. I'm wondering why I need to do that for a thick lens and not a thin lens.

 

[Drakkith]

I could also be over-thinking this and I should just add another negative to phi!

It comes from the raytrace equations. For a forward raytrace:
$y' = y+\omega ' \tau '$
$\omega ' = \omega - y\phi$
Where:
$\omega = nu$
$\phi = (n'-n)C = \frac{1}{f_e}$
$\tau '=\frac{t'}{n'}$

For a reverse raytrace, you simply reorder the equations
$y = y'-\omega ' \tau '$
$\omega = \omega '+y\phi$

Notice the $\phi$ terms in the equations. For a forward raytrace you are subtracting y times phi, but for a reverse raytrace you are adding them. But we don't have phi written down in our worksheet, we have negative phi. So to recover phi we have to multiply by negative 1, or the negative of negative phi. For a forward raytrace it's more convenient to put negative phi down and just add $-y\phi$ to $nu$ versus having to subtract a result of $y\phi$.

Regardless of why phi gets another negative, I don't see the sign of phi change in Greivenkamp's as a function of ray trace direction. I'm wondering why I need to do that for a thick lens and not a thin lens.

The equations hold for thin lenses or thick. There is no difference between a thin or thick lens raytrace using paraxial ray tracing.

 

[phillip_at_work]

Okay, now I see what you're saying. That's a very silly mistake indeed.

When I fix the order of the equations for power, everything is correct, without messing around with any extra signs.

Attaching final and corrected ray traces here for future readers. Many thanks for your assistance!