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Darren Drake
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what is 3rd, 5th, and 6th order sperical aberratio
      #5775458 - 04/03/13 12:34 PM

I've been seeing these terms in Suiters book and now Richard Berry's new book and I still don't get what these terms really mean. I suspect 3rd order is general SA and 5th represent zones but I'm not sure.

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EJN
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Darren Drake]
      #5775506 - 04/03/13 12:54 PM

See:

http://www.telescope-optics.net/higher_order_spherical_aberration.htm

and

http://en.wikipedia.org/wiki/Optical_aberration


They represent terms in a (mathematical) expansion series describing a conic surface.


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kfrederick
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: EJN]
      #5775511 - 04/03/13 12:57 PM

http://www.handprint.com/ASTRO/ae4.html#astigmatism

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Darren Drake
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: kfrederick]
      #5775728 - 04/03/13 02:41 PM

These are pretty technical explanations. Without reading a PHD thesis can this question be answered in a way that is more easily understood?

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Ajohn
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Darren Drake]
      #5775798 - 04/03/13 03:19 PM

You might say that lenses and mirrors etc don't much care about them as they just produce spherical aberration. Optical people do as the more orders they include the nearer the model they use will be to reality.

An easier one to consider that has similar terms is petzval curvature. A simple approximation is part of the surface of a sphere. In reality the real surface is a parabolic type shape so adding orders makes any modelling more accurate.

Third order theory is similar. Sines are approximated rather than using actual values. 1st order is more approximate than 3rd order as that uses more terms and gives a closer approximation. In practice 5 terms might be used if that was viable. I always wonder why they didn't use a PC.

John
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Mike I. Jones
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Ajohn]
      #5775868 - 04/03/13 03:55 PM

Whether it's refraction or reflection, a ray bends at a surface according to good old Snell's law:

(Index 1) x sin(A1) = (Index 2) x sin(A2)

where angles A1 and A2 are measured away from the 3D perpendicular vector at the surface where the ray intercepts it.

The sine of an angle "x" in radians is computed with this series:

sin(x) = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + dot dot dot

For small angles, sin(x) = x, and Snell's law becomes

(Index 1)(A1) = (Index 2)(A2)

Raytracing becomes linear and simple, called "first order" or "paraxial" raytracing. This gives a good approximation where rays would go with no aberrations (other than due to changes in wavelength and index of refraction).

Adding in the 3rd order term to sin(x) makes Snell's law

(Index 1) [A1 - (A1^3/3!)] = (Index 2) [A2 - (A2^3/3!)]

This is called 3rd order optics, and is where spherical aberration, coma, astigmatism and distortion, and variations of these with wavelength, begin to show up. Third order precision is OK for systems with very shallow ray angles and high focal ratios, which is why it was adequate for refractor designing before the days of computers.

So you get the idea: 5th order optics includes the x^5/5! term, and so on. The more terms that are included, the more accurately the ray trajectory is calculated. But it reaches a point of ridiculous beyond 5th and 7th order.

Then fortunately, along comes 3D skew raytracing and computer implementation in the late 40's, using only square roots that converge rapidly to unbelievable precision, and polynomial analysis becomes obsolete, or at best, of historical curiosity. Aspherics are also easily accomodated by iterative skew tracing. All modern optical programs have 3-D skew raytracing at their heart, but still include 3rd and 5th order aberration outputs if you really have some reason to use them.

Simple enough explanation?
Mike


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Mark Harry
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Mike I. Jones]
      #5775948 - 04/03/13 04:36 PM

-THAT'S- what we have you around for, MIKEY! You can explain this to us duma$$es!

Ha-ha!
M.


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Mike I. Jones
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Mark Harry]
      #5775959 - 04/03/13 04:43 PM

Yeah, plus the pay's so good!

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Chriske
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Mike I. Jones]
      #5776973 - 04/04/13 02:54 AM

Quote:

Whether it's refraction or reflection, a ray bends at a surface according to good old Snell's law:

(Index 1) x sin(A1) = (Index 2) x sin(A2)

where angles A1 and A2 are measured away from the 3D perpendicular vector at the surface where the ray intercepts it.

The sine of an angle "x" in radians is computed with this series:

sin(x) = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + dot dot dot

For small angles, sin(x) = x, and Snell's law becomes

(Index 1)(A1) = (Index 2)(A2)

Raytracing becomes linear and simple, called "first order" or "paraxial" raytracing. This gives a good approximation where rays would go with no aberrations (other than due to changes in wavelength and index of refraction).

Adding in the 3rd order term to sin(x) makes Snell's law

(Index 1) [A1 - (A1^3/3!)] = (Index 2) [A2 - (A2^3/3!)]

This is called 3rd order optics, and is where spherical aberration, coma, astigmatism and distortion, and variations of these with wavelength, begin to show up. Third order precision is OK for systems with very shallow ray angles and high focal ratios, which is why it was adequate for refractor designing before the days of computers.

So you get the idea: 5th order optics includes the x^5/5! term, and so on. The more terms that are included, the more accurately the ray trajectory is calculated. But it reaches a point of ridiculous beyond 5th and 7th order.

Then fortunately, along comes 3D skew raytracing and computer implementation in the late 40's, using only square roots that converge rapidly to unbelievable precision, and polynomial analysis becomes obsolete, or at best, of historical curiosity. Aspherics are also easily accomodated by iterative skew tracing. All modern optical programs have 3-D skew raytracing at their heart, but still include 3rd and 5th order aberration outputs if you really have some reason to use them.

Simple enough explanation?
Mike




Mike,

Could you now explain again for left-handed people please...?



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Mark Harry
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Chriske]
      #5777097 - 04/04/13 07:15 AM

Chris, most of that completely went over my little head. Really want to know?
M.


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Napersky
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Mark Harry]
      #5777274 - 04/04/13 10:09 AM

I think what Mike is saying that as you go up in powers over the cubic you are looking at smaller and smaller miniscule abberations that are very hard to manage and control. How opticians manage figuring lenses above the 3rd order I have no clue.

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Ajohn
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Mark Harry]
      #5777346 - 04/04/13 10:46 AM

Thought my answer was a lot simpler Mike . Unlike some on else I know you wont find that offensive.

I edited it to add a bit more but it seems to have gone awol. The terms are often used to describe the shape of ray intercept curves as well. Hopefully where that is done there are also examples without them or with others. I think there are series of various mixes shown in oslo documentation. One shows the effect of zonal spherical aberration. The ideal intercept curve is a flat line across the entire field of view. Zonal spherical aberration puts an up wards bulge in it some where across the field. It has no order as such but more orders can make the shape of the bulge more complex.

Lens design "orders" are based round the fact that small angles expressed in radians are very close to the sine of the angle. The error gets larger as the angle gets bigger, 1 degree in radians is 0.0174533, the sine of 1 degree is 0.0174524, 5 degrees is 0.0872665 radians and the sine is 0.0871557, still accurate to better than 1 part in 1000. 10 degrees 0.174533 radians, 0.173548 sine. Still good enough for "roughing out" maybe on few optical surfaces. More orders in this respect just means more terms in a sine approximation series that goes
sine(x)=x-x^3/3!+x^5/5!-x^7/7!................
An example of Taylor's approximation. That number of terms is accurate to better than 1 part in 100,0000. PC's etc use similar polynomials to calculate all sorts of things but to a lot more terms. Seidel came along and used this to calculate basic aberrations. People who use that sort of thing have used even more terms. The advantage of this approach is that an optical system can be expressed as an equation and solved for certain characteristics. The approach is still used followed by computer optimisation. It helps ensure that the computer finishes up with a solution in the intended area. The manual approximations are better at ensuring that end result really is the best solution for a given set of optical components. There are also approximations for specific things like conic mirrors that can be as accurate as needed - just to make life difficult for anyone who wants to get interested in this sort of thing.

John
-

Edited by Ajohn (04/04/13 10:54 AM)


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Mike I. Jones
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Ajohn]
      #5777973 - 04/04/13 03:42 PM

Nah. To offend me you have to be technically wrong and be so dense you don't realize you're wrong, but continue to puff out your chest in delusional expertise.

Any raytracer algorithm consists of two sections:
1. Ray intersection solver, and
2. Ray bender

Ray Intersection: calculating the intersection of a ray with a surface is essentially root solving in 3D: finding a point in (x,y,z) common to both the ray and the surface. Some surfaces, such as planes, spheres and conics, have closed-form solutions that are as accurate as the source code compiler and processor architecture are capable of. That's usually 15-16 digits. Higher-order aspheres have no closed-form solutions, and iteration is required to find the common (x,y,z) point of intersection to a specified precision (usually 12 digits or so). The surface normal, or perpendicular, at the 3D ray/surface intersection point is calculated during the root solving section.

Ray Bending: This is where 3rd, 5th, 7th, etc. order optics enters; to what precision are the sines in Snell's law calculated? The ray is bent using Snell's law to some precision, giving a new ray trajectory, and then you're done with that surface.

As John shows above, that precision drops with increasing angle, and rather rapidly. Aberration polynomial analysis is useful when someone wants to break down wavefront errors into big named chunks, like "spherical aberration", "astigmatism", etc., and understand which surface is the weakest and strongest at producing these aberrations. But even in Conrady's era in the late 19th/early 20th century, optical designers were moving away from polynomial approximations and starting to use six and seven-digit lookup tables published in big thick books. Want to see some of that agony? Look at Wyld's article in ATM-3, or find a copy of Conrady to read. Pure misery, and that's how I originally taught myself in 1970.

Today's modern codes calculate ray/surface intersections to roughly the diameter of a proton in accuracy, and bend the ray at full 16-digit precision, using much faster square-root coding rather than much slower sines and cosines. Light goes pretty much exactly where the raytracer says it will.

Does not using polynomial analysis cause any loss in understanding and intimacy with a given lens as it is designed? Yes, to some extent, but truthfully, except in the hobbyist world, who needs it? Spherical aberration at each surface might be academically interesting for a simple 2 or 3 element lens, but that kind of understanding gets lost in the mush when you're designing a 20-element 10X zoom lens at five different temperatures and multiple object distances! All that counts is to get the thing designed and toleranced while staying on requirements, budget and schedule for a paying customer. Modern codes can even scan down a 30-element system and determine which surface would best benefit the system by being an aspheric, again using 3D full-precision raytracing. Lens design has changed incredibly from Conrady's days, and from when James Wyld wrote his excellent little article in ATM-3.

My 20 millidollars anyway,
Mike


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wh48gs
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Darren Drake]
      #5778156 - 04/04/13 05:15 PM

Quote:

I've been seeing these terms in Suiters book and now Richard Berry's new book and I still don't get what these terms really mean. I suspect 3rd order is general SA and 5th represent zones but I'm not sure.




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 lower-order) 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 higher-order 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 sagitta-related 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


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Ajohn
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: wh48gs]
      #5778406 - 04/04/13 07:26 PM

One thing I don't understand about that method Vla is that it doesn't account for slope so how does that fit in with refractive index etc? ie Z is the sagitta from either a plain or the vertex of an optic.

I have seen it used very recently to cope with ray angles off conic mirrors but am having difficulty seeing how it derives an angle. - As soon as refractive optics were encountered more or less seidel techniques were used so the conic mirror ray angles were converted to the usual series of aberrations. Then summed as usual to remove as many as is possible over a given field angle.

John
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wh48gs
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Ajohn]
      #5779125 - 04/05/13 08:41 AM

Conic sagitta is not just the depth. It is a function that describes conic surface. Thus it determines the conic slope at given zonal height, i.e. the direction of local (zonal) surface radius (all conics but sphere have zonal radii each intersecting the axis at a different point), which in turn determines the angle of incidence and reflection for given object distance. So the sagitta function directly affects the angle. The actual calculation is extensive, although straightforward (Schroeder shows some of it in "Astronomical Optics" 1st ed p62). When you see the extent, you realize the practical benefit of using simpler sagitta approximations.

Vla


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Ajohn
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: wh48gs]
      #5779419 - 04/05/13 11:22 AM

Thanks. At Ł4-72 I will try the book Vla. I'll probably wonder why he doesn't show all of the calculation.

John
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Loren Chang
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Ajohn]
      #5795685 - 04/13/13 12:09 AM

Quote:

One thing I don't understand about that method Vla is that it doesn't account for slope so how does that fit in with refractive index etc?




Hello John,

As Vla said, it is a wavefront aberration. You have to caculate Optical Path Difference between reference and general ray. Optical path equals refractive index times distance ray travels in space. That's why refractive index shows up.
For your reference, In google book search for "Fundamental optical design" by Michael J Kidger. Go to Chapter6 finishing reading p101-107. Will give you a very basic concept how it comes.




Hello Mike,

Ouch! What you said really hurts me, a lens hobbyist. On the other hand, I had been told same words by other designers almost 15 years ago. Want to say something for this out of date analysis. Deduce 3rd order aberration is tedious and obscure so you don't need to understand fully. However, final results is very important. That's why they always appears in optical design textbooks. For example, many of us familiar with field curvature is related to refractive index and power of lens. Actually it comes from Seidell aberration coefficient S4! So to learn a little about 3rd order aberration is more or less helpful.




Hello Vla,

Thanks for your explanation which clarified my basic concept. My question is can we say it's 3rd order because ray aberration(lower by one to wavefront) is 3rd order?


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jpcannavo
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Loren Chang]
      #5795959 - 04/13/13 05:53 AM

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 nth-ary vs. (n+1)th-ary aberrations.

Joe

(Excuse the peculiar notation, falling back on what little analysis I remember to try and frame some coherent questions!)


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wh48gs
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: Loren Chang]
      #5797165 - 04/13/13 05:07 PM

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.

Vla


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wh48gs
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Re: what is 3rd, 5th, and 6th order sperical aberratio new [Re: jpcannavo]
      #5797219 - 04/13/13 05:39 PM

Joe,

Quote:

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 nth-ary vs. (n+1)th-ary 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 higher-order 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 higher-order spherical cannot be corrected, only minimized by balancing it with lower-order. In such case, the P-V wavefront error of balanced aberration is 4.3 times smaller than that of pure higher-order 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


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

Quote:

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 lower-order) 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 higher-order 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 sagitta-related 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 “third-order” 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)


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

Quote:

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.


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

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

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

Quote:

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?


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

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

Quote:

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.)

"Higher-order spherical aberrations of reflecting (shown) or refracting conic surface result from corrections to the lower-order surface approximation."

http://www.telescope-optics.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.


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

Quote:

I disagree; there is no name difference between transverse and wavefront aberrations.




You just proved there is

Quote:

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 “third-order” 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 third-order 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


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

Quote:

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 Schmidt-type aspheric on the surface, or minimized by balancing it with lower-order 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 off-axis aberrations are in the similar manner minimized by balancing them with their respective lower-order forms.

Vla


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




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 third-order 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 “third-order” 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 “fourth-order 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 “Third-order” 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 4th-order 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)


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

John,

Quote:

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 “third-order” 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 “fourth-order 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 “Third-order” 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 third-order if they are (as they are now) calculated using the modern approach. Again, the "third-order" 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 third-order 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


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

Quote:

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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