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wh48gs
Carpal Tunnel
Reged: 03/02/07

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: jpcannavo]
#5797219  04/13/13 05:39 PM

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Joe,
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Perhaps getting back to the spirit of the OPs question, and in addressing my own ignorance of the relevant formalisms, can some rough generalizations be made? That is, can (or perhaps not) some general conclusions be drawn regarding the relative impact on Strehl of nth order vs (n+1)th order aberrations. Similarly for nthary vs. (n+1)thary aberrations.
For spherical aberration and mirror surface the relation between higher and lower aberration as wavefront error at paraxial focus is given by the ratio of the respective two terms in the sagitta expansion series. It comes to (K+1)/32F^2, F being the focal ratio, which means that the higherorder term diminishes toward parabola.
For best focus location, where 4th order is 1/4 of the error at paraxial focus, and 6th order is 2.5 times smaller, the ratio would be (K+1)/20F^2.
This is for pure higher order spherical. If limited to spherical surfaces, the higherorder spherical cannot be corrected, only minimized by balancing it with lowerorder. In such case, the PV wavefront error of balanced aberration is 4.3 times smaller than that of pure higherorder spherical, and the RMS wavefront error nearly 6 times smaller.
The effect on Strehl depends on which scenario is played.
It is much more complicated for a lens.
Vla

jhayes_tucson
sage
Reged: 08/26/12
Loc: Bend, OR

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: wh48gs]
#5797476  04/13/13 08:59 PM Attachment (22 downloads)

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There are two different concepts, that shouldn't be confused. The one is based on using power series of trigonometric functions to simplify the calculation of trigonometric functions back in days when raytracing calculation was done by hand. For most surfaces back then using only the first two terms was sufficiently accurate approximation of trigonometric function. The end (used) term's power was the basis to name the corresponding aberration as "first order", "third order" and so on, as illustrated by the power series posted by Mike.
But this concept is now only of historical significance (think of how much a simple hand calculator would make life easier for the old timers). The concept of aberration order used these days is based on the power series describing conic sagitta. Again, in order to simplify calculation, the sagitta can be described by a power series, with each next term much smaller than the previous. The first term gives exact sagitta for paraboloid; for all other conics additional terms are needed to describe the surface, according to
z=(d^2/2R)+[(1+K)d^4/8R^3]+[(1+K)^2d^6/16R^5]+...
The second term is the 4th order term, with related aberrations being of 4th order, the next is 6th order, and so on. For other than axial (spherical) aberration, the exponents used are more complex, but it can be simplified to a simple rule that the aberration order is determined by a sum of two exponents: one determining the dependance of wavefront error on zonal height in the pupil, and the other dependance on the field angle.
Primary (or lowerorder) spherical changes with the 4th power of zonal height, with zero field angle dependance (4+0), which makes it 4th order
Primary coma changes with the third power of zonal height and in proportion with the field angle (3+1), which also makes it 4th order
Primary astigmatism changes with the square of zonal height and square of the image angle (2+2), so it's 4th order
Secondary (next higherorder spherical) changes with the 6th power of zonal height, with zero field angle dependance (6+0), which makes it 6th order, and so on
These sagittarelated aberrations are wavefront aberrations. Transverse aberrations have the exponent for zonal dependance lower by one, which makes transverse (or ray) primary spherical 3rd order aberration, transverse primary coma 2nd order, and so on.
Likewise, secondary spherical transverse aberration is 5th order, and so on.
Vla
I disagree; there is no name difference between transverse and wavefront aberrations. Piston (the constant term) is considered the zero order term and the next two terms (first and second order) of the expansion form the tilt (linear) and power (r^2) terms. So, the third order terms include spherical aberration, which varies as the fourth power of the radial coordinate in the pupil. So even though spherical aberration varies as r^4, it is called “thirdorder” SA. These third order properties were developed and named by Seidel and (to some extent) Petzval and are what allowed the early development of more sophisticated designs (like achromats.) This approach is still widely used to gain insight into aberration theory and optical properties; though, computer aided design has eliminated the need to consider low order aberrations during the design process. To cut through the equations, I've attached a page showing what the wavefront error looks like for pure 3rd, 5th, and 7th order spherical aberration. You can see that as the order gets higher, the wavefront gets steeper toward the edge of pupil. So, a turned edge on a mirror would be best represented with a very high order SA component.
When we reverse the process and try to fit a surface, it is far more useful to use what are called orthogonal data sets. Zernike polynomials are orthogonal (which ultimately means independent of each other) over an unobscured circular pupil and are commonly used for fitting sparse data to create a uniform surface. The whole topic of Zernikes gets complicated and doesn't address the question posed by the OP so I don't want to go into much more detail. I mention it only to make participants aware that other polynomials are normally used to create an artificially generated wavefront that is "close" to the actual shape of the wavefront.
John
PS The silly 200k limit on CN attachments makes my attachment a pretty poor picture!
Edited by jhayes_tucson (04/13/13 09:20 PM)

Loren Chang
member
Reged: 04/28/09

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: wh48gs]
#5797742  04/14/13 12:52 AM

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Loren,
Strictly talking, we should say 3rd order for the transverse (ray spot) aberration and 4th order for the wavefront. Next higher order are 5th and 6th, respectively, and so on.
Got it and Thank you, Vla.

Darren Drake
Carpal Tunnel
Reged: 10/09/02
Loc: Chicagoland

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: Loren Chang]
#5797915  04/14/13 07:55 AM

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I still don't get it. Thanks all for trying though. And I'm usually pretty good at this stuff...

jpcannavo
scholastic sledgehammer
Reged: 02/21/05
Loc: Long Island New York

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: Darren Drake]
#5797929  04/14/13 08:12 AM

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I still don't get it. Thanks all for trying though. And I'm usually pretty good at this stuff...
Daren
For me at least, there is (I think) an aspect to your question that i am struggling to pull out of the more formal account.
I have some qualitative sense of what spherical aberration, coma, astigmatism etc. mean at the eyepiece end of things. I am curious then to know if a qualitative visual distinction obtains for orders/levels of aberration as they are being discussed here. So, and apart from what obtains in the star test, are there some generalizations that hold for image quality?
Can we say, for example, that the effects of high order SA will always be lower than those due to low order SA. Or does the mere asking of said question entail some fundamental misunderstanding or over simplification?

Ajohn
sage
Reged: 12/03/07

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: Darren Drake]
#5797974  04/14/13 08:57 AM

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I'm not surprised Darren. I find the simplistic and probably rather dated view I expressed adequate  they indicate that more "terms" are needed to accurately represent them. Actually considering some of the more complicated responses it's not dated at all.
All of the aberrations can be easily shown pictorially and are down to finite apertures and curved surfaces. In real terms they are just the factors chosen to model optical systems  with ever increasing accuracy. ie Taking a ray intercept curve 3rd order is adequately modelled by a 3rd order polynomial, 5th order needs 5 terms and so on.Taking the Zernike attachment posted 3rd order is a relatively shallow dish with little in the line of an abrupt slope  by the time you get to 7th order the dish has far steeper edges. It's worth bearing in mind that these are all to a certain degree of accuracy. Reality may be even more complex. It's a fact that in the past designers used far more than 3 sine approximation terms when they needed to. It's also a fact that a practical optical system will always have a strange mix of aberrations at some level of measurement and at that point there is no point in wondering what orders they are. Those are just design tools.
John 

Asbytec
Guy in a furry hat
Reged: 08/08/07
Loc: La Union, PI

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: Darren Drake]
#5798028  04/14/13 09:57 AM

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I still don't get it. Thanks all for trying though. And I'm usually pretty good at this stuff...
Darren, I had the same question. Simply and untechnically, I try to look at it this way. HSA is really classical SA and nothing weird, it's just a mathematical way to more accurately describe the actual performance of an objective that the lower order terms (an approximation of the actual surface) cannot accurately describe.
A lens or mirror will produce a focal point based on it's surface. That's lower order SA because mathematically it's term is found in the equation at the lower order. This term describes the wavefront resulting from this mirror or lens surface. The equation is an approximation because it is impossible to grind a perfect sphere. Te resulting difference from the mathematical expression and the actual surface is (or can be if signifigant error exists) handled in the equations higher order terms and is called higher order SA.
Now, there will always be some residual SA in the image because the actual surface deviates from the perfect sphere defined in the expansion series (equation) that describes this objective's performance. This residual SA is handled in some higher order terms and is called higher order spherical aberration.
For example, it is said a MCT corrector adds a lot of higher order spherical aberration. ANd it does. Why? Well the lower order SA defines the SA contribution of the spherical primary. It will have a given SA wavefront error and provide some description of the focus (caustic.)
Now, when you put a highly curved meniscus in front of that spherical primary you can imagine everything changes, the focal length will change, the caustic will become tighter, and the wavefront error will get better. So, the original lower order SA term no longer accurately describes the function of the primary (with a meniscus in place.) So, it's SA correction has changed. That changed SA contribution of the meniscus is handled in the higher order expressions (thus higher order SA) to bring the equation back into line with the performance of the objective...this time with the meniscus in place. Why? Because the meniscus changed the original mathematical approximation (and actual correction) of the primary as defined in the lower order term.
(Technically, what I said above is not really accurate, because once you put an meniscus  adding a lot of higher order SA  the original objective spherical surface will have to change, too...to correct for all that added SA coming from the meniscus. This is what they mean by adding lower order aberration of opposite sign. It brings the "system" SA back into a better mathematical approximation while leaving some residual HSA. I left out that concept trying to illustrate what LSA, classical spherical aberration, really is. Higher order is just the stuff that's left over after all the corrections to the lower order terms...approximations.)
"Higherorder spherical aberrations of reflecting (shown) or refracting conic surface result from corrections to the lowerorder surface approximation."
http://www.telescopeoptics.net/higher_order_spherical_aberration.htm
Third and fifth order (the odd terms) describe the ray, while fourth and sixth describe the wave front aspect of SA. Third and fourth order are lower order ray and wavefront descriptions (initial mathematical approximation), respectively, while fifth and sixth are the higher order ray and waveront, respectively, terms describing any correction to the objective's mathematical, approximate aberration.
This is how I view it...hope its clearer, Darren.

wh48gs
Carpal Tunnel
Reged: 03/02/07

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: jhayes_tucson]
#5798292  04/14/13 01:16 PM

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I disagree; there is no name difference between transverse and wavefront aberrations.
You just proved there is
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Piston (the constant term) is considered the zero order term and the next two terms (first and second order) of the expansion form the tilt (linear) and power (r^2) terms. So, the third order terms include spherical aberration, which varies as the fourth power of the radial coordinate in the pupil.
Piston is formally 4th order, but it does not constitute aberration with a single pupil, only with multiple pupils differing in phase (it is zero radial order in Zernike expansion schemes, but that is different concept).
You seem to be talking about Zernike expansion scheme, which is different animal. There, aberrations are classified according to the radial order, and which aberration belongs to which varies with the particular Zernike expansion scheme. For instance, primary spherical is 2nd order aberration in Wyant's scheme, and 4th order in Noll's and ANSI standard scheme (of course, it's all wavefront aberrations).
Quote:
So even though spherical aberration varies as r^4, it is called “thirdorder” SA. These third order properties were developed and named by Seidel and (to some extent) Petzval and are what allowed the early development of more sophisticated designs (like achromats.) This approach is still widely used to gain insight into aberration theory and optical properties; though, computer aided design has eliminated the need to consider low order aberrations during the design process.
Siedel calculation is called thirdorder because it uses sin/tan values calculated to the second term in the sin/tan expansion series (this 2nd term contains 3rd power of the angle, hence the name). But this trigonometric concept has nothing to do with the modern optical theory, established by Schwarzschild. Its modern aberration order notation is based on the modern aberration calculation. All wavefront aberrations have only even orders.
Vla

wh48gs
Carpal Tunnel
Reged: 03/02/07

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: Darren Drake]
#5798313  04/14/13 01:33 PM

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I still don't get it.
In the nutshell, higher order aberrations are simply corrections for the more exact approximations of surface shape (i.e. sagitta function). In most cases, conic surface approximated by the first two terms of the sagitta expansion series, yielding 4th order wavefront aberrations, is all that's needed. Inclusion of the third (6th order) term doesn't make appreciable difference. But as optical surface becomes more strongly curved, this third, and possibly even fourth term, become more significant. The 4th order aberration is still by far dominant, but after it's corrected, the residual 6th and possibly 8th order may be to large to neglect. Higher spherical can be either corrected by putting a Schmidttype aspheric on the surface, or minimized by balancing it with lowerorder spherical (which basically means correcting spherical surface so that it comes as close as possible to the actual surface needed for zero spherical). Higher order offaxis aberrations are in the similar manner minimized by balancing them with their respective lowerorder forms.
Vla

jhayes_tucson
sage
Reged: 08/26/12
Loc: Bend, OR

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: wh48gs]
#5799270  04/14/13 07:58 PM

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You seem to be talking about Zernike expansion scheme, which is different animal. There, aberrations are classified according to the radial order, and which aberration belongs to which varies with the particular Zernike expansion scheme. For instance, primary spherical is 2nd order aberration in Wyant's scheme, and 4th order in Noll's and ANSI standard scheme (of course, it's all wavefront aberrations).
…
Siedel calculation is called thirdorder because it uses sin/tan values calculated to the second term in the sin/tan expansion series (this 2nd term contains 3rd power of the angle, hence the name). But this trigonometric concept has nothing to do with the modern optical theory, established by Schwarzschild. Its modern aberration order notation is based on the modern aberration calculation. All wavefront aberrations have only even orders.
Vla,
1) I am not talking about Zernikes or Wyant’s organization table (which you may be misinterpreting.)
2) I agree 100% that all wavefront aberrations have only even mathematical orders and that OPD for the lowest order spherical aberration varies as the fourth power of the pupil radius.
3) I am trying to respectfully explain to you that the primary Seidel aberrations are universally called “thirdorder” for both transverse and wavefront (OPD) formulations. If you walk into a room full of optical engineers (including Wyant who was my dissertation advisor, coauthor, and business partner for 25 years,) they will not know what you are talking about if you refer to primary spherical aberration as “fourthorder spherical.” But you don't have to take my word for it. There are all sorts of respected references that make this point: Born and Wolf, "Principles of Optics", 7th edition p232 footnote states: “Since the ray aberrations associated with wave aberrations of this order are of the third degree in the coordinates, they are sometimes called the third order aberrations.” Warren Smith in “Modern Optical Engineering” derives the third order primary Seidels in Chapter 5 and discusses “Thirdorder” spherical aberration for OPD in Chapter 11. In “Reflecting Telescope Optics, Vol 1, chapter 3.2.2, R.N. Wilson reviews the Seidel Approximation with a derivation of the “third order coefficients” for wavefront aberration. I could give you a lot more references but I’ll stop there. I will add that in 25+ years as a professional optical engineer, I’ve never heard anyone refer to primary spherical aberration as 4thorder spherical aberration.
This is merely a discussion about the naming of a quantity so it doesn’t matter if you want to call it something different than everyone else, but I feel some obligation to speak up when I see others being told that this is the accepted, correct way to refer to the primary Seidels. It is not.
John
Edited by jhayes_tucson (04/14/13 08:01 PM)

wh48gs
Carpal Tunnel
Reged: 03/02/07

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: jhayes_tucson]
#5799669  04/14/13 11:51 PM

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John,
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1) I am not talking about Zernikes or Wyant’s organization table (which you may be misinterpreting.)
Possibly. But you are throwing in piston as zero order, and that can be seen only in Zernike expansion schemes.
Quote:
I am trying to respectfully explain to you that the primary Seidel aberrations are universally called “thirdorder” for both transverse and wavefront (OPD) formulations. If you walk into a room full of optical engineers (including Wyant who was my dissertation advisor, coauthor, and business partner for 25 years,) they will not know what you are talking about if you refer to primary spherical aberration as “fourthorder spherical.”
I find that hard to believe. The universal modern notation for the primary spherical aberration coefficient is 40 (e.g. w40), expressing its (4th power) dependance on pupil radius and no dependance on image radius. It is used in many online papers and articles, including Wyant's.
Quote:
But you don't have to take my word for it. There are all sorts of respected references that make this point: Born and Wolf, "Principles of Optics", 7th edition p232 footnote states: “Since the ray aberrations associated with wave aberrations of this order are of the third degree in the coordinates, they are sometimes called the third order aberrations.” Warren Smith in “Modern Optical Engineering” derives the third order primary Seidels in Chapter 5 and discusses “Thirdorder” spherical aberration for OPD in Chapter 11. In “Reflecting Telescope Optics, Vol 1, chapter 3.2.2, R.N. Wilson reviews the Seidel Approximation with a derivation of the “third order coefficients” for wavefront aberration. I could give you a lot more references but I’ll stop there.
Of course it is called  sometimes  "third order" and its origin is given in my previous post. It is third order in the context of manual trigonometric raytracing, which is now obsolete. The next higher order in this context, fifth, is different than the next higher order in the modern aberration calculation.
Quote:
This is merely a discussion about the naming of a quantity so it doesn’t matter if you want to call it something different than everyone else, but I feel some obligation to speak up when I see others being told that this is the accepted, correct way to refer to the primary Seidels. It is not.
There is no "secondary" Siedel aberrations, since he only made the third order calculations. Primary aberrations are Siedel aberrations only at the paraxial focus, and even then there's no reason to call them thirdorder if they are (as they are now) calculated using the modern approach. Again, the "thirdorder" term is only valid in the context of manual trigonometric raytracing. It is a relict from the past often used in a confusing manner in optics text. More proper way to put it would be "classical thirdorder aberrations" which at least hints at their origin. The modern calculation uses binomial expansion which, for these primary aberrations retains terms up to fourth order. Why should we call them something else?
Vla

Loren Chang
member
Reged: 04/28/09

Re: what is 3rd, 5th, and 6th order sperical aberratio
[Re: Darren Drake]
#5799898  04/15/13 05:33 AM

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I still don't get it. Thanks all for trying though. And I'm usually pretty good at this stuff...
Which part? 4th order and 3rd order? It represents highest order term uses when deduce the formula. For 4th order wavefront aberration you consider terms lower than y^4 in caculatiton. Same rules applies to 3rd order.


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