38 replies to this topic

[careysub]

Some thoughts on high etendue telescopes, HET as Mel Bartel calls them.

 

As a fan of rich field/wide angle astronomy viewing, and visually observing low contrast DSOs, and a tinkerer, I am interested in exploring the different possibilities with these. Additionally, havin taken up night vision image intensifier viewing, it brings in considerations that imagers  have about fast optics to best illuminate their sensors.

 

There is no standard unit for measuring etendue, various systems of units are used depending on the application. Mel Bartel has a computation which I will call a "Bartel etendue figure of merit" which can be used to roughly compare different telescope confiugrations.

 

BEFOM = aperture^2 * TFOV^2

 

A "proportionality" symbol instead of an equality symbol might be better here since this is not geometrically accurate, omitting the Pi/4 factor for round optics. It also ignores the aperture loss from the central obstruction in Newtonians (which are the telescopes Mel is applying them to), though that would not be greatly different between the scopes likely to be compared for this.

 

Formulated this way, for a visual observer, the eyepiece is an integral part of the calculation since it determines the true field of view with the telescope focal length modified by the coma corrector.

 

Taking that into account the BEFOM of a system is determined by:

 

BEFOM = (Field Stop)^2/(Focal Ratio)^2

 

The coma correction factor can be applied to either the field stop (as Mel recommends) to decrease it or to the focal ratio, to increase it, as people more commonly do, with the same effect.

 

Now the aperture does not appear at all in the equation. It shows that HET potential is unrelated to aperture (relative CO being held constant).

 

What larger aperture does is provide the same etendue at higher magnification for the same field stop, which should always be as large as possible.

 

The additional constraint on this is that the exit pupil should be very close to the maximum pupil size of the observer, to get the benefit of the HET system. Much smaller, or larger, then the maximum potential of light capture in viewing is not obtained.

 

BTW - using the hex key method of measurement my exit pupil (age 62) is greater than 5.5 mm but less than or equal to 6 mm. Maybe it is 6 mm, but I suspect that might be a trifle optimistic and at any rate it is not going improve, so in planning a system I will want to use for the next 20 years I use 6 mm as a hard upper limit, and aim to be just below that. But effective BEFOM-wise though, going slightly larger has the same effect of going slightly smaller, if you take the "wasted light" into consideration.

 

Exit Pupil = (EP Focal Length)/(Focal Ratio)

 

Again with the coma correction factor applied to either value as you please.

 

There are only a few eyepieces on the market (or even optically feasible) that are candidates for maximizing BEFOM for any particular primary diameter and focal length and coma corrector combination, and there are only three suitable coma correctors on the market, each with a different magnification factor (1.15, 1.10, 1.06).

 

In optimizing telescope configurations then, this is a type of integer optimization problem (for those of you in to operations research, or algorithms) where there are only a few discrete choices to be made among available eye pieces that are close to what is optimal for a particular diameter and focal length. Or to look it a little differently, there are only a few EP/CC options on the market that can be matched to a feasible primary to optimize BEFOM.

 

On Mel's site he discusses the use only of Televue optics, the 21 mm Ethos and the Paracorr family,

 

Since Mel started writing on this subject Explore Scientific has come out with an eyepiece and a coma corrector that achieves equivalent BEFOM with an F/4 mirror, which are much more readily available.

 

I have built one such myself, using a 6" F/4 (well, 152mm F/3.92) mirror from GSO.

 

Looking toward a larger project, an 8" or a 10", I am wondering if I am missing something in using this combination instead of the F/3 Paracorr 20/21 mm 100 degree EP solution.

 

One argument might be that the overall optical quality of the ES 100 25mm, and the HRCC, is not quite as good as the Televue. And I will agree that Televue sets the standard. But if I find them acceptable (and I have thus far not found cause for complaint), is there anything else?

 

One advantage of using F/4 mirrors is that they are available at very reasonable prices. From GSO an 8" is $230 and and 10" is $480. I use one for a build AND drop in a replacement later, from Zambuto say, at a cost of $1475 and $2150, respectively. F/3 mirrors would run higher than that.

 

An advantage of the F/3 mirror is simply that it makes a more compact telescope. But in the 8-10" range this is not that big a factor, since even a 10" F/4 is all that long.

 

I am planning on building a 10" HET, and seeing if anyone can make the case for going with an F/3 mirror. I would sort of like to be convinced, but haven't been able to convince myself.

 

Attached are the optimum HET configurations of the telescopes under discussion. The lower field stop value is the calculated value for the ES 100 20 mm (which I have) and the higher is for the ES 100 25 mm (ditto).

 

P.S. Just after posting this it occurred to me another possible advantage of the F/4 mirror -- using if without the CC for an extended object like Andromeda or Barnard's Loop. You could do this with an F/3 also, but the comatic effect would likely be much less noticeable with the F/4. It would be interesting to compare the views though for a definitive judgement.

Attached Thumbnails

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Edited by careysub, 10 April 2020 - 12:15 AM.

[mark cowan]

Hmm... I would take the approach that since Mel sort of discovered this thing SFAIK after he built and deployed a small f/2.6 (or f/2.8) mirror that the focal ratio does indeed play an important role in the actual high entendue system and that you can't actually duplicate it with the slower mirrors, for some unobvious reason.  But that's just intuition and it of course can be misleading.

[careysub]

Hmm... I would take the approach that since Mel sort of discovered this thing SFAIK after he built and deployed a small f/2.6 (or f/2.8) mirror that the focal ratio does indeed play an important role in the actual high entendue system and that you can't actually duplicate it with the slower mirrors, for some unobvious reason.  But that's just intuition and it of course can be misleading.

Possibly - but can someone explain how this is so? I am willing to believe it, but have to understand it, The BEFOM figure does not show it. What would?

 

Maybe its use with NV II tubes is the "killer app" for an F/3 mirror. 

 

The II tube has a fixed sensor size, 18 mm, smaller than the field stop of a super wide angle lens, so it has an narrow field of view and thus a lower etendue. But it can be raised by making the focal ration as low as you like. There are no restrictions placed by available eyepiece focal lengths, or human pupils sizes. Lower is always better.

Edited by careysub, 10 April 2020 - 12:02 AM.

[Jon Isaacs]

I have thought about this more than a little bit.  Some of the issues I see:

 

- Mirror reflectivity, central obstruction. If one is comparing reflectors to refractors then this must be included.

 

- At the very, very least, the entire field of view must be visible simultaneously.  If the entire field is not visible, only the field that is being seen can be included in the calculation.  If you are not seeing it, it doesn't contribute. 

 

With some eyepieces, for example, the 20mm ES-100, even with the eyecup rolled down, I could not take in the entire field simultaneously, I had tilt my head and peek to see the edge of the field and I couldn't see the entire field doing that.  And it wasn't all that pretty even with a Paracorr.  The Ethos is better for me in this regard, the field is sharper and I can see the entire field in a "single glance", that means not moving my eye, not moving my head.

 

- Probably more relevant, how often are you actually seeing the entire field?  In my experience with 100 degree eyepieces, not very often.  With an 82 degree eyepiece with sufficient eye relief like the 31mm Nagler, I am seeing the entire field at least peripherally most of the time.  

 

- The calculation gives equal weight to the all parts of the field of view.  This ignores the workings of the eye. The eye is not a camera.  In terms of the sensitivity of the eye, the outer regions are less sensitive, have poorer resolution, they contribute less to the information to the brain than the regions closer in.  This is not a linear relationship, equal areas do not contribute equally to the image we see in our brains.  

 

- In my mind, the region outside of about 82 degree AFoV, is very much in the periphery and on a unit area basis contributes very little to the conscious part of the image.  I think it's also pretty clear that the region between 68 degree AFoV and 82 degree AFoV contributes less information on a unit area basis that the area inside the 68 degree AFoV. 

 

- In my mind, the question is how much real difference in what one actually perceives between a 82 degree AFoV and a 100 degree AFoV at the same magnification and exit pupil.  My own assessment is that it's not that much of a difference, most of the time, one is not seeing the 100 degree AFoV and even if the eye is positioned properly, that outer region is still very much in the periphery.

 

---------

 

I've thought about this whole thing quite a bit.  I have had my 12.5 inch F/4.06 for about 20 years, the 31mm Nagler more than 10 years.  When Mel built his 13 inch F/3 around the 21mm Ethos, I pondered how much difference there would really be and how much value that outer field actually has. Now that I have owned the 21mm Ethos for a while, it's a great eyepiece and I use it more than the 31mm Nagler but I think I am correct when I question the value of that outer field in terms of it's contribution to the visual image.  I rarely see the entire view when sweeping the sky or when observing an object.

 

---------

 

Measuring your dark adapted pupil.  There's a thread on it General Astronomy and Observing.  Tom Polakis used a similar method to mine.  A flash photograph taken in a closet and then measured on a computer screen. 

 

https://www.cloudyni.../#entry10102421

 

If done properly, it can be quite accurate.  When I measured mine, it closer to 8mm than to 7mm.  The 31mm Nagler without a Paracorr in my 12.5 inch F/4.06 provides a 7.6mm exit pupil which is a good match for my pupil.  

 

--------

 

Generally, one sees more with a smaller exit pupil and greater magnification.. A large dark adapted pupil is only useful in a very few situations. 

 

Jon

[careysub]

 

- Probably more relevant, how often are you actually seeing the entire field?  In my experience with 100 degree eyepieces, not very often.  With an 82 degree eyepiece with sufficient eye relief like the 31mm Nagler, I am seeing the entire field at least peripherally most of the time.  

 

- The calculation gives equal weight to the all parts of the field of view.  This ignores the workings of the eye. The eye is not a camera.  In terms of the sensitivity of the eye, the outer regions are less sensitive, have poorer resolution, they contribute less to the information to the brain than the regions closer in.  This is not a linear relationship, equal areas do not contribute equally to the image we see in our brains.  

 

- In my mind, the region outside of about 82 degree AFoV, is very much in the periphery and on a unit area basis contributes very little to the conscious part of the image.  I think it's also pretty clear that the region between 68 degree AFoV and 82 degree AFoV contributes less information on a unit area basis that the area inside the 68 degree AFoV. 

Thanks. I have thought that the 68 (or maybe 72) degree EP hits a "sweet spot" - one where the image completely fills your "normal" FOV and it stops seeming like you are viewing through a limiting hole.

 

I'll have to try recalculations with Panoptic or Delos style EP to see the effect.

[Jon Isaacs]

Thanks. I have thought that the 68 (or maybe 72) degree EP hits a "sweet spot" - one where the image completely fills your "normal" FOV and it stops seeming like you are viewing through a limiting hole.

 

I'll have to try recalculations with Panoptic or Delos style EP to see the effect.

 

I think the cut off is somewhere between 80 degrees and 100 degrees, at 80 degrees, the outer field is definitely contributing. 

 

Jon

[MitchAlsup]

On Mel's site he discusses the use only of Televue optics, the 21 mm Ethos and the Paracorr family,

 

Since Mel started writing on this subject Explore Scientific has come out with an eyepiece and a coma corrector that achieves equivalent BEFOM with an F/4 mirror, which are much more readily available.

 

I have built one such myself, using a 6" F/4 (well, 152mm F/3.92) mirror from GSO.

 

Looking toward a larger project, an 8" or a 10", I am wondering if I am missing something in using this combination instead of the F/3 Paracorr 20/21 mm 100 degree EP solution.

 

One argument might be that the overall optical quality of the ES 100 25mm, and the HRCC, is not quite as good as the Televue. And I will agree that Televue sets the standard. But if I find them acceptable (and I have thus far not found cause for complaint), is there anything else?

It is a confined space of choices.

 

I happen to have gone in this direction and have my 13" F/3 up and running. I designed this scope to have 1/2 of the focal length of my 20" F/4. 

 

A few years ago I started to notice that the 31NT5 was not performing as it should--in effect my eye's pupil had constricted to be smaller than the exit pupil created by the 20" F/4 and the 31NT5. I got a 25ES100 for "essentially" the same FoV and a smaller exit pupil. At F/4 young people can use 31NT5 {EP = 6.73mm with P2} but as one ages and your eye's pupil gets smaller there is a time when it right to switch to a higher power but wider FoV EP to be able to "see the same FoV" while getting it delivered through an exit pupil that is properly matched to your eye.

 

At F/3 my recourse is the 21E with a 6.08 exit pupil almost matching my eyes pupil. The 25ES100 {EP = 7.24} has too big of an exit pupil at F/3.

 

What you are noticing is that the "usable" entendue is controlled by exit pupil and your eye provides the size of the exit pupil you can use.

[Jon Isaacs]

At F/4 young people can use 31NT5 {EP = 6.73mm with P2} but as one ages and your eye's pupil gets smaller there is a time when it right to switch to a higher power but wider FoV EP to be able to "see the same FoV" while getting it delivered through an exit pupil that is properly matched to your eye.

 

 

An interesting study:

 

Dark-adapted pupil diameter as a function of age measured with the NeurOptics pupillometer

 

60 to 69 years (n=30), 5.58 mm (range: 3.5 to 7.5 mm)

 

In this study of dark adapted pupil as a function of age, they tested 30 people, the average was 5.58 but the range is 3.5mm to 7.5mm.  Some older folks have large dark adapted pupils. As a matter of fact, that would be me.. 

 

But in general, I am skeptical of the value of exit pupils matching the dark adapted pupil.  It is true, it will be the widest, brightest view possible but in my experience, in nearly every situation, increasing the magnification, decreasing the exit pupil and field of view will show me more.  

 

I suggest that despite a ~7.7mm dark adapted pupil, I would see more at the 6mm exit pupil that Mitch is designing for than at the 7.7mm exit pupil that would fill my pupil.  

 

Jon

[MitchAlsup]

But in general, I am skeptical of the value of exit pupils matching the dark adapted pupil.  It is true, it will be the widest, brightest view possible but in my experience, in nearly every situation, increasing the magnification, decreasing the exit pupil and field of view will show me more.  

 

I suggest that despite a ~7.7mm dark adapted pupil, I would see more at the 6mm exit pupil that Mitch is designing for than at the 7.7mm exit pupil that would fill my pupil.  

High etendue shows greater brightness only on extended objects and comes with whatever background "blackness" that is available at your site.

 

Increasing magnification increases the background blackness at the cost of FoV. Extended objects become bigger and dimmer, while stars stay at the same brightness levels.

 

It is a delicate balance.

 

One mental note I took while using the 13" F/3 is: "wow, that is a lot of FoV". It made it noticeably easier to find M33 for example in the 13" F/3 than the 20" F/4. With the 21E in the focuser, the 13" F/3 is a super-duper-finder, indeed.

[Jon Isaacs]

High etendue shows greater brightness only on extended objects and comes with whatever background "blackness" that is available at your site.

 

Increasing magnification increases the background blackness at the cost of FoV. Extended objects become bigger and dimmer, while stars stay at the same brightness levels.

 

It is a delicate balance.

 

One mental note I took while using the 13" F/3 is: "wow, that is a lot of FoV". It made it noticeably easier to find M33 for example in the 13" F/3 than the 20" F/4. With the 21E in the focuser, the 13" F/3 is a super-duper-finder, indeed.

Some thoughts:

 

- The contrast of a star increases with increased magnification.  

 

- The vast majority of DSOs are small..

 

- I have had my 12.5 inch F/4.06 for 20 years.  With the 31mm Nagler and the Paracorr, it's not that different from your 13 inch F/3 with the 21mm Ethos and the Paracorr:  

 

Your scope provides a 1.82 degree TFoV at 54x with a 6.1 mm exit pupil,  my scope provides a 1.62 degree TFoV at 48x with a 6.6 mm exit pupil.  

 

Now your scope provides 1.82 degree field where as mine "only" offers a 1.62 degree field.  But this is where my first post applies:

 

I can easily see the entire field of the 82 degree AFoV, 1.62 degree TFoV whereas seeing the entire 100 degree AFoV in a single glance, while possible, is not the normal mode.  If one is only seeing 90 degrees of the 100 degree AFoV, the TFoVs are indentical. 

 

This concept of high etendue Newtonians is really just an extension of the RFT concept.  For most observing, higher magnifications will show me more than the widest, brightest field will.  

 

Jon

[PEterW]

With NV using the afocal method you can get to the effectively very fast speeds for bright results, but without needing to have the fastest mirrors or use focal reducers. (Of course faster mirrors means you can get to even faster final speeds...

Peter

[CrazyPanda]

But in general, I am skeptical of the value of exit pupils matching the dark adapted pupil.  It is true, it will be the widest, brightest view possible but in my experience, in nearly every situation, increasing the magnification, decreasing the exit pupil and field of view will show me more.

 

Agreed with this. In terms of the tug of war between view brightness and magnification, magnification and properly framing the objects (or diving into it to see more details) wins. A low power, bright exit pupil view of M27? Meh. A moderate power, moderate exit pupil view of M27 in that same scope? Wow.

 

Regarding overall etendue, I feel like the equation has to account for sky brightness. A max exit pupil telescope is arguably worse under moderately bright skies than a smaller exit pupil (even at the same magnification).

 

A while back I did a comparison of M27 in my 8" F/10 and 12" F/5 at about 127x magnification. This produced a 1.6mm exit pupil in the 8", and a 2.4mm exit pupil in the 12". I live under class 4 skies. To my great surprise, I somewhat preferred the view of M27 in the 8" over the 12". The darker overall view tricked my brain into thinking the view had better contrast than it really did. I knew objectively that everything in the view was dimmer - nebula and all. But subjectively, I felt the view through the 12" looked washed out.

 

So depending on how bright your skies are, you could make the argument that for a given aperture, a smaller exit pupil will produce a subjectively better etendue. Or even a smaller aperture at the same magnification (thereby a lower exit pupil), will also produce a subjectively better etendue.

Edited by CrazyPanda, 10 April 2020 - 10:38 PM.

[Mike Lockwood]

I invented my own telescope metric some time ago, it was called the Coefficient Of Various Instrument Dimensions, but that acronym was an unfortunate choice for the times. 

[macdonjh]

I invented my own telescope metric some time ago, it was called the Coefficient Of Various Instrument Dimensions, but that acronym was an unfortunate choice for the times. 

But if you started using that measurement "some time ago", it would have been, for example, the Coefficient Of Various Instrument Dimensions-17.  Or something similar.  As we all know, that was a non-event.

[Mike Lockwood]

I think that sometimes we get too hung up on formulae.

 

The main issue that I see with the etendue formula is that there is no penalty term for poor image quality across the field.  This is a very practical consideration.

 

My philosophy has been to use the fastest optic that will produce good views, which is mostly determined by corrector and eyepiece performance.  I like f/3.0, but f/2.8 can work too.  So, one gets the widest high-quality field of view, but high magnifications are still achievable and will show more detail on small objects as John noted above.

 

Then, simply choose the aperture for the size of field of view that you want.  Mitch has taken this approach, too, and has his excellent 13" f/3.0 finished up and larger f/3.0s left to build.

 

So, I've got a 10" f/3.0 (dedicated for nightvision) that I need to build, I have a 20" f/3.0, and I hope to have a larger f/3.0 in the future.

 

Nightvision, used afocally, has really removed the limitation on exit pupil size, and fortunately f/3.0 works quite well for that, too.

[MitchAlsup]

Some thoughts on high etendue telescopes, HET as Mel Bartel calls them.

 

<snip>

There is no standard unit for measuring etendue, various systems of units are used depending on the application. Mel Bartel has a computation which I will call a "Bartel etendue figure of merit" which can be used to roughly compare different telescope confiugrations.

 

BEFOM = aperture^2 * TFOV^2

BEFoM = (Aperture^2-Obstruction^2)*TFoV^2

Taking CO into account.

 

Taking that into account the BEFOM of a system is determined by:

 

BEFOM = (Field Stop)^2/(Focal Ratio)^2

BEFoM = ( FieldStop^2 / FocalRatio^2 ) * (1 - Obstruction^2/Aperture^2)

 

Taking CO into account.

 

Now if we had a relationship between Central Obstruction and F/ratio we would have enough to figure out where the optimum etendue point would be.

[Jon Isaacs]

I don't really see a need to get too deeply into this. 

 

It only applies to one exit pupil, one situation.  It maximizes the field of view when the exit pupil matches the observer's dark adapted pupil. 

 

How often do you make critical observations under those circumstances? I certainly don't do it very often. 

 

I mostly observe at smaller exit pupils and higher magnifications.  Maximizing etendue is not important enough to me to design a scope around it.

 

If you wanted to write an equation it would need to include the secondary obstruction, the reflectivity of the mirrors and some sort of model of the relationship between the eye and the AFoV would be needed.  In the current equation, say you developed an eyepiece with a 160 degree AFOV.

 

On paper, it would seem like it had a huge advantage over an eyepiece with a 100 degree AFoV but if you see can't the entire 100 degree AFoV, a 160 degree AFOV would have no advantage. 

 

Jon

[careysub]

BEFoM = (Aperture^2-Obstruction^2)*TFoV^2

Taking CO into account.

 

BEFoM = ( FieldStop^2 / FocalRatio^2 ) * (1 - Obstruction^2/Aperture^2)

 

Taking CO into account.

 

Now if we had a relationship between Central Obstruction and F/ratio we would have enough to figure out where the optimum etendue point would be.

My spreadsheet excerpt does take the CO into account.

 

CO has a weak relationship to FR since longer focal ratios can use a smaller CO, which is why my 10" F/4 has a slightly higher value than the other 8" and 10" configurations.

 

But given that secondaries also come in fixed size increments, it just slightly enlarges the integer programming problem - it did not create significantly more freedom in my spreadsheet of reasonable configurations.

 

One of my takeaways presented here is that for a range of HET designs, the actual etendue value is essentially the same limiting value. And since you are shrinking FR to boost BEFoM, tending to increase the CO, they tend to offset each other.

Edited by careysub, 11 April 2020 - 10:48 PM.

[havasman]

This concept of high etendue Newtonians is really just an extension of the RFT concept.  For most observing, higher magnifications will show me more than the widest, brightest field will.  

 

Jon

I see these as another RFT-style instrument too. And I see these scopes as specialist instruments best applied to specialist observations: widefield observing of faint, large extended objects done from darkest available sites. The light gathering enabled by larger apertures than a typical RFT like an NP101 carries value for that limited class of observations. 

 

My thoughts on these scopes do not place much value on them as primary instruments for general observing or observing from poorer sites. But as additions to a capable kit they may, for some, be useful.

[careysub]

I see these as another RFT-style instrument too. 

 

I regarded the whole HET concept as a formalization of what "rich field" means, which was otherwise imprecise.

 

The rich field concept was not framed around large extended objects but rather wide, rich star fields.

 

Another thing to recall (which allude upthread, about looking at performance with shorter FL EPs) is that they are in no way limited to the limiting design case of maximum exit pupil/longest FL EP. They work perfectly well at higher magnification.

Edited by careysub, 12 April 2020 - 12:08 PM.

[MitchAlsup]

It only applies to one exit pupil, one situation.  It maximizes the field of view when the exit pupil matches the observer's dark adapted pupil. 

Let us compare 2 telescopes of the same aperture.

 

One has a 20% CO, the other has a 50% CO.

Both are the same focal length.

We use the same EP in both.
So the exit pupil is the same in both.

 

One has significantly better etendue than the other! 

[Jon Isaacs]

Let us compare 2 telescopes of the same aperture.

 

One has a 20% CO, the other has a 50% CO.

Both are the same focal length.

We use the same EP in both.
So the exit pupil is the same in both.

 

One has significantly better etendue than the other! 

 

Very true.

 

And surprisingly, if you do the real calculation and not the simplified calculation being discussed here, the scope with a larger CO will very likely have the greater integrated etendue.

 

One of the underlying assumptions in this simplified equation is that the field is fully illuminated.  But that is very unlikely with a fast scope with a 20% central obstruction whereas a scope with a 30% CO can be fully illuminated. 

 

The real calculation needs to integrate something like:

 

2 Pi K FI[r] r dr where FI is the Field illumination and K is a transmission constant which includes transmission losses to to reflectivity and the size of the secondary shadow.  

 

This would calculate total throughput of the scope rather than assuming the transmission was uniform across the field.  But it highlights another issue with the simplified equation, the outer field illumination which has a large contribution to the integrated etendue because of the large area, has little importance visually and contributes very little to the eye's perception.

 

Jon

[MitchAlsup]

The real calculation needs to integrate something like:

 

2 Pi K FI[r] r dr where FI is the Field illumination and K is a transmission constant which includes transmission losses to to reflectivity and the size of the secondary shadow.  

It is likely that K is a function of off axis distance, also:: So::

 

2 Pi K[r] FI[r] r dr

[careysub]

The field illumination is something I did want to add, but had not gotten around to looking into the correct calculation. Thanks to both of you for proposing a formula.

[Jon Isaacs]

It is likely that K is a function of off axis distance, also:: So::

 

2 Pi K[r] FI[r] r dr

 

K is a constant which includes reflectivity and the secondary shadow.

 

FI®  is the Field Illumination function of r. 

 

Jon