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Really confused about under/over-correction

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#1 flywing1

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Posted 03 December 2021 - 07:11 PM

Sorry about double posting, just realized ATM might be a better place for my questions than the refractors forum.

 

My current understanding is that this lens is Undercorrected with a Positive SA, which creates an airy disk (simulated probably, left to right=intra-focus to extra-focus) as shown. Is this correct? Because I have seen multiple posts on CN and even academic pdfs either calling this overcorrected or saying that a negative SA is associated with undercorrected lens.  For example, from telescope optics site (https://www.telescop...aberrations.htm ): “spherical aberration is " overcorrected", i.e. with marginal rays crossing before paraxial rays” (sorry if I misunderstood, but this should be Undercorrected, right?)

These two images are from Edmund Optics and wiki:
https://www.edmundop...n-plates-en.pdf
https://en.wikipedia...ical_aberration

Attached Thumbnails

  • undercorrect.JPG
  • undercorrectAirydisk (Medium).JPG

Edited by flywing1, 03 December 2021 - 07:12 PM.


#2 KBHornblower

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Posted 03 December 2021 - 08:00 PM

The sketch of the rays is exaggerated but if I am not mistaken it is true for a lens with spherical surfaces.  The peripheral rays focus closer to the lens.  This is uncorrected.  Suppose we figure the lens to move the peripheral rays part way to the paraxial focus.  That would be undercorrection.  We work some more and get all the rays to the same focus.  That is full correction.  Suppose we overdo it and move the peripheral focus beyond the paraxial focus.  That would be overcorrection.

 

The images are simulations of how the spherical aberration degrades the image at various positions inside and outside of the point of best focus.  If we have a fully corrected lens, the patterns will be identical inside and outside, and the best focus will be sharper, with an Airy disk and perhaps one or two faint rings.  In this one the rings are brighter and more extensive.



#3 jimhoward999

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Posted 03 December 2021 - 08:26 PM

You diagram depicts a lens with under-corrected spherical aberration.

 

The way I learned it, the longitudinal spherical aberration is L'-l' where L' is the real focal distance and l' is the paraxial one.    So if the marginal rays focus short, then L' is smaller than l' and you have negative spherical aberration.  That is the opposite of what you have written and also the opposite of some things I have seen on the internet.

 

I went back and pulled my college notes from Kingslake's lens design class, and verified that at least according to his sign conventions, longitudinal spherical aberration is as I have described above L' - l'.  That means under corrected spherical aberration , as found in a simple positive lens, is negative.


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#4 Pinbout

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Posted 03 December 2021 - 10:02 PM

 

Undercorrected with a Positive SA,

In units of wavefronts… sometimes using this symbol ~

 

fracs typically have a spherical wavefront, having a 0 conic. 

parabolic mirrors have -1 conic

 

in either situation any SA would be added to the conic.

a spherical mirror is a undercorrected parabolic. the amount of SA in wavefronts is +1~

 

an undercorrected spherical wavefront frac could be  0+ (1/4~) as a minimum acceptable figured.

an overcorrected frac could be 0+ (-1/4~) as a minimum acceptable figure 

 

If you look at the sections of a mirror and their conics it will illustrate why under is + and over is - 

those illustration are beyond my skills at the moment lol.gif


Edited by Pinbout, 03 December 2021 - 10:03 PM.

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#5 dan chaffee

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Posted 03 December 2021 - 10:14 PM

Both undercorrection or overcorrection for any optic are considered third order

spherical aberration. The majority of achromatic lenses are designed to be

fully corrected with spherical surfaces. If they test with spherical aberration, it means

either it is too close to the target or the space between lenses is not optimal or one or more surfaces is

not truly spherical, or any of the three. Apochromatic refractors and some achromat designs

are sometimes designed to be aspherized for best performance, especially fast systems,

but no design I'm aware of is immune to the possibility of spherical aberration from

fabrication/spacing errors.

Undercorrection: outer portion of optic focuses closer than inner radii.

Overcorrection: outer portion of optic focuses further than inner radii.


Edited by dan chaffee, 04 December 2021 - 02:14 AM.

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#6 davidc135

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Posted 04 December 2021 - 04:19 AM

As Jimhoward says above under-correction has negative S.A. 'Telescope-Optics.net' states this in section 4.1.1.  David



#7 Pinbout

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Posted 04 December 2021 - 08:08 AM

As Jimhoward says above under-correction has negative S.A. 'Telescope-Optics.net' states this in section 4.1.1.  David

Yes, he does ,

 

https://www.telescop.../spherical1.htm



#8 Asbytec

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Posted 04 December 2021 - 08:53 AM

 

My current understanding is that this lens is Undercorrected with a Positive SA, which creates an airy disk (simulated probably, left to right=intra-focus to extra-focus) as shown. Is this correct? 

The Airy disc is only seen in focus. It is the result at best focus of the diffraction interference we see when out of focus. 

 

The sign, as I recall from a discussion, has to do with the distance the actual wave front has to travel relative to a reference sphere centered on best(?) focus. Every point on a perfect reference sphere will have the same path length (radii) to the focal point. If the actual wavefront is inside the reference sphere, then it has less distance (or optical path length) to the point of best focus so the sign is negative. An under corrected wave is steeper than a reference sphere at the marginal zones, which is why it comes to focus before the point of best diffraction focus. So, if my understanding is correct, then under correction has a negative sign.

 

For example, a meniscus lens (MCT) creates a wavefront that is convex toward the primary, so the marginal zones are further from the primary mirror and the point of best focus than the paraxial zones. They have further to travel to get to the point of best focus. That is a form of over correction and of opposite sign (+) to the under correction (-) of a spherical primary mirror. I throw that out there for the sake of discussion, please correct me if I am wrong.

 

According to the link above, "With over-corrected (positive) spherical aberration, marginal rays focus farther away than paraxial rays. In either case, geometrical structure of the defocused zone remains identical in regard to the paraxial focus." 


Edited by Asbytec, 04 December 2021 - 08:57 AM.

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#9 MKV

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Posted 04 December 2021 - 09:36 AM

You diagram depicts a lens with under-corrected spherical aberration.

 

The way I learned it, the longitudinal spherical aberration is L'-l' where L' is the real focal distance and l' is the paraxial one.    So if the marginal rays focus short, then L' is smaller than l' and you have negative spherical aberration.  That is the opposite of what you have written and also the opposite of some things I have seen on the internet.

 

I went back and pulled my college notes from Kingslake's lens design class, and verified that at least according to his sign conventions, longitudinal spherical aberration is as I have described above L' - l'.  That means under corrected spherical aberration , as found in a simple positive lens, is negative.

Spot on! If marginal focus is shorter than the paraxial focus, the spherical error is negative (meaning shorter). A sphere is not an undercorrected parabola. An optical reference focal length is will always 1/2 of the vertex radius of curvature R/2, regardless of the conic constant. Hence, all conics are referenced relative to the the sphere. 

 

The confusion comes from the the way optics are analyzed graphically. The convention is that the light propagates form LEFT to RIGHT, and this is counted as positive propagation. On reflection, the direction reverses and the reflected light propagates in the NEGATIVE direction (RIGHT to LEFT)

 

Another source of confusion is the way amateurs make aspheric surfaces. They "deepen" the center, looking for the best fitted parabola between two spheres, but in reality prolate ellipsoids, paraboloids and hyperboloids are shallower than the sphere.  A deeper surface will by definition have negative SA, but paraboloids actually have positive SA which is written as negative because of the convention of how it is graphically represented.


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#10 flywing1

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Posted 04 December 2021 - 11:04 AM

Thank you all for the help! Now I am clear that the lens I showed is Undercorrected with a Negative SA, and the wiki page probably has the incorrect information when it says "Positive spherical aberration means peripheral rays are bent too much." Will read more books, rely less on wiki. lol.gif



#11 MKV

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Posted 04 December 2021 - 10:19 PM

Thank you all for the help! Now I am clear that the lens I showed is Undercorrected with a Negative SA, and the wiki page probably has the incorrect information when it says "Positive spherical aberration means peripheral rays are bent too much." Will read more books, rely less on wiki. 

I think you should forget about the positive/negative labels and concentrate on whether the marginal (peripheral) rays fall short (undercorrected) or long (overcorrected) when the mirror is tested at its radius of curvature R with a knife-edge test, which is a null (zero or no aberration) test for a sphere.

 

For a parabola the peripheral or marginal rays will focus farther from the mirror (overcorrection) by an amount equal to KYm²/2R, where K = conic constant (-1 for parabola) and Ym is the peripheral/marginal ray aperture radius. Hence, empirically the marginal focus will be >0 (positive) relative to that of a sphere but the negative sign of the conic constant makes it look negative, which does not determine determine whether the SA is over- or under-corrected.

 

Here's an illustration that shows clearly marginal overcorrection.

 

LA_TA.jpg

source: Warren Smith's Modern Optical Engineering, p. 292. McGraw-Hill 1966 edition

 

 


Edited by MKV, 05 December 2021 - 09:10 AM.

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