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How is focal length defined in an aberrating lens (system)

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#1 bitnick

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Posted 17 January 2020 - 03:38 PM

The Lensmaker's_equation mathematically defines the focal length of a spherical lens. Here's a ray trace of rays passing through a spherical lens with equal front and back curvatures, chosen so that its focal length is 0.2 according to the lensmaker's equation:

The lens is centered at z = -0.2 and the rays end at z = 0. Apparently the calculated focal length is valid for paraxial rays (rays close to the optical axis) where spherical aberration isn't noticeable?

What about aspherical lenses, and systems of lenses: how is focal length defined there? OSLO, for example, calculates an effective focal length for this doublet. How is it done and what does it mean (for which rays is it valid)?

The reason for my question is that I'm experimenting with writing software to do optical raytracing (the image above is generated by gnuplot with data from that program). Just for fun, and to learn. Is it possible to calculate effective focal length from the ray data (e.g. from where some ray intersects the optical axis)?

#2 Oregon-raybender

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Posted 17 January 2020 - 04:07 PM

I would check the optical design books to give you guidance. You are asking many questions that would take a book of writing to answer. Check William-Bell is a good start, or the old Edmund books.

Warren Smith wrote one of the best books for engineers and Newbies, Modern Optical Engineering.

Starry Nights

https://www.amazon.c...3XSP99KCPCTRBR1

https://www.ebay.com...AyABEgLeJvD_BwE

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#3 MitchAlsup

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Posted 17 January 2020 - 04:44 PM

The FL is defined by the point where the central (axial) ray reaches focus. That is the narrow aperture focal length.

#4 ButterFly

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Posted 17 January 2020 - 05:07 PM

At the distance of the "circle of least confusion" for a given wavelength.  It is very rare that any lens focuses on at one point for all rays of a given wavelength - there is always some abberation.

#5 TOMDEY

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Posted 17 January 2020 - 05:47 PM

Two definitions:

In classical geometrical optics (e.g. Seidel-Space), it is the distance separating the focal point (axial paraxial image of an infinitely distant [axial] object), from its corresponding principal point, with absolutely no nuances involving aberrations.

In Zernike wave optics, the focal point is (re)located to where the unique axial Zernike wavefront power term is identically zero. This is more meaningful, because that is where the focused image is most concentrated (closely related to best achievable focused Strehl, etc.) and indeed accounts for all other aberrations, compound fields, etc.

Optics guys generally stick with the approximate geometrical model interpretation, because it is easier to visualize and gives OK approximate answers, when doing first-order design. Then they optimize for actual build, using quasi-wave optics (aka OPDs) to get very good and predictive parametrics for the build. In extreme systems, the designer will/must grind through the entire three-dimensional wave function... not for the faint of heart. This generally is reserved only for such things as laser-energetics, optical wave-guides, holography, and the (most-generalized) discipline of ~non image forming coherent passive optical systems~ like when we want to direct energy to a remote "target" as hot as possible, on the receiving end. That later stuff was where I wound up, half way through my working decades.

The practical answer to your insightful question is that Seidel Spherical aberration shifts best focus (and therefore deltas effective focal length and magnification). Zernike best focus is not influenced by the others. Which is the beauty of Zernike-Space. All of the (infinitely many) aberrations are all independent/orthogonal.    Tom

#6 Benach

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Posted 17 January 2020 - 06:00 PM

At the distance of the "circle of least confusion" for a given wavelength.  It is very rare that any lens focuses on at one point for all rays of a given wavelength - there is always some abberation.

All rays, of the entire FOV and wavelength? Impossible even!

#7 bitnick

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Posted 17 January 2020 - 06:07 PM

Okay, I'm a bit confused now, but if I understand you all correctly:

* There's the "classical", paraxial focus (and the distance to the focus is the focal length). This ignores aspheric sag and aberrations.

* Then there's a best focus, which defines the effective focal length, and this is at the circle of least confusion?

* Then there's a Zernike focus which is something else again. (Seems interesting but way over my head at the moment.)

Did I get that right?

Edited by bitnick, 17 January 2020 - 06:09 PM.

#8 MKV

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Posted 17 January 2020 - 08:16 PM

The reason for my question is that I'm experimenting with writing software to do optical raytracing (the image above is generated by gnuplot with data from that program). Just for fun, and to learn. Is it possible to calculate effective focal length from the ray data (e.g. from where some ray intersects the optical axis)?

Well, you sure got a lot of answers, and may even be somewhat confused. :o) The paraxial focal length is the theoreticafocal length of a perfect lens (uneffected by aberrations). Its often referred to as the effective focal length or efl.

Once you introduce 3rd order aberrations, the best focal length will usually be half way between the paraxial focal length and the marginal focal length. This is where the "circle of least confusion" is located, and corresponds to the focal point of the 70.71% aperture zone. This is where you'd normally focus for the best image clarity.

When you introduce 5th order aberrations, this point slight shifts closer to the marginal focus.

For more details, and a better "feel" for this, consult the old books on optical engineering (such as Rudolf Kingslake Lens Design Fundamentals, or Warren Smith's Modern Optical Engineering). They may even be available online.

Good luck!

Af for writing your own software, there are some many asoftware exmaples available online, it may be an interesting project but it's been done over and above any need for a new one.

#9 bitnick

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Posted 18 January 2020 - 11:35 AM

Thank you all for your answers! I now understand that I have some reading to do. Smith's Modern Optical Engineering is incoming.

I also found this online MIT course which looks interesting. Now to find the time (and energy) to take it all in...

#10 Oregon-raybender

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Posted 20 January 2020 - 10:37 PM

I would also suggest looking at and down loading the OSLO Edu. I have and use it often for quick check. It is limited, but powerful enough to answer most, if not all questions. There are examples

and you can place the various lens design from Warren's book to play with. It runs on windows.

Good Luck and most of all, have fun. Designing optics can lead to years of "what if's"  Also check

Sam Brown's books on optics, some what dated, but light still works the same way as it did decades ago.

Starry Nights

https://sites.chem.c...optics 1970.pdf

https://www.lambdares.com/edu/

http://www.amoptics....ng_OSLO_EDU.pdf

https://oslo-edu.sof...former.com/6.6/

Edited by Oregon-raybender, 20 January 2020 - 10:41 PM.

#11 SandyHouTex

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Posted 22 January 2020 - 01:44 PM

The FL is defined by the point where the central (axial) ray reaches focus. That is the narrow aperture focal length.

The central axial ray is always at focus.

#12 bitnick

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Posted 22 January 2020 - 07:27 PM

Ok, I've had the opportunity to take a look in Smith's Modern Optical Engineering.

This is what it says [pp 23]:

The effective focal length (efl) of a system is the distance from the principal point to the focal point.

If the rays entering the system and those emerging from the system are extended until they intersect, the points of intersection will define a surface, usually referred to as the principal plane. /.../ The intersection of this surface with the axis is the principal point.

As for calculation of the (effective) focal length:

The book defines "effective" focal length (efl) as mathematically equal to "focal length" [pp 39]:

efl = f = -y1/u'2, where:

y1 is the height of an incident ray parallell to the optical axis,
u'2 is the angle (in radians) of the same ray after passing the optical system.

If n and n' are indices of refraction before and after an optical surface,
u and u' are beam angles relative the optical axis before and after the surface,
and R radius of curvature of the surface, then

n'u' = nu - y(n'-n)/R

These definitions assume paraxial rays, so the term "focal length" by definition only applies to paraxial rays, I guess?

When it comes to "effective focal length" vs "focal length", perhaps someone thought that "focal length" sounded too boring and felt the need to prepend "effective"?

The book also gives an excellent justification for using the paraxial approximation [pp 32] (my emphasis):

The paraxial region of an optical system is a thin threadlike region
about the optical axis which is so small that all the angles made by the
rays (i.e., the slope angles and the angles of incidence and refraction)
may be set equal to their sines and tangents. At first glance this con-
cept seems utterly useless, since the region is obviously infinitesimal
and seemingly of value only as a limiting case. However, calculations
of the performance of an optical system based on paraxial relation-
ships are of tremendous utility. Their simplicity makes calculation and
manipulation quick and easy. Since most optical systems of practical
value form good images, it is apparent that most of the light rays orig-
inating at an object point must pass at least reasonably close to the
paraxial image point.

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