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# what is 3rd, 5th, and 6th order sperical aberratio

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### #1 Darren Drake

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Posted 03 April 2013 - 11:34 AM

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.

### #2 EJN

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Posted 03 April 2013 - 11:54 AM

See:

http://www.telescope..._aberration.htm

and

http://en.wikipedia....ical_aberration

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

### #3 kfrederick

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Posted 03 April 2013 - 11:57 AM

http://www.handprint...tml#astigmatism

### #4 Darren Drake

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Posted 03 April 2013 - 01:41 PM

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

### #5 Ajohn

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Posted 03 April 2013 - 02:19 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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### #6 Mike I. Jones

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Posted 03 April 2013 - 02:55 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

### #7 Mark Harry

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Posted 03 April 2013 - 03:36 PM

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

Ha-ha!
M.

### #8 Mike I. Jones

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Posted 03 April 2013 - 03:43 PM

Yeah, plus the pay's so good!

### #9 Chriske

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Posted 04 April 2013 - 01:54 AM

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...?

### #10 Mark Harry

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Posted 04 April 2013 - 06:15 AM

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

### #11 Napersky

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Posted 04 April 2013 - 09:09 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.

### #12 Ajohn

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Posted 04 April 2013 - 09:46 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
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### #13 Mike I. Jones

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Posted 04 April 2013 - 02:42 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

### #14 wh48gs

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Posted 04 April 2013 - 04:15 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.

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

### #15 Ajohn

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Posted 04 April 2013 - 06:26 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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### #16 wh48gs

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Posted 05 April 2013 - 07:41 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

### #17 Ajohn

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Posted 05 April 2013 - 10:22 AM

Thanks. At £4-72 I will try the book Vla. I'll probably wonder why he doesn't show all of the calculation.

John
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### #18 Loren Chang

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Posted 12 April 2013 - 11:09 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?

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?

### #19 jpcannavo

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Posted 13 April 2013 - 04:53 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!)

### #20 wh48gs

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Posted 13 April 2013 - 04:07 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

### #21 wh48gs

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Posted 13 April 2013 - 04:39 PM

Joe,

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

### #22 jhayes_tucson

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Posted 13 April 2013 - 07:59 PM

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!

### #23 Loren Chang

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Posted 13 April 2013 - 11:52 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.

Got it and Thank you, Vla.

### #24 Darren Drake

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Posted 14 April 2013 - 06:55 AM

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

### #25 jpcannavo

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Posted 14 April 2013 - 07:12 AM

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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