Hello to everyone,

I am confronted with someone stating that the cos^4 law is "a
meaningless rule of thumb" and lens design (for photographic lenses) can
be done such that at the boundary of the image circle one can have more
light throughput than in the center (no filters of course...).
Before I get drunk to understand this, I better ask - any idea what that
guy could possible have ment?
Thanks!
George

From:	BC
Subject:	Re: cos^4 law
Date:	11 Jan 2005 10:51:25 -0800

"That is a bold statement. Perhaps your "someone" can present
a few examples?"

I'm the "someone", and I'd be happy to. I'm surprised that nobody
seems to have experience with lenses of this sort since telecentric
projection lenses are a pretty hot item lately.

U.S. Patent 5,969,876 has several examples with this behavior. US
Precision Lens built several lenses similar if not identical to some of
these examples back in the mid-1990s for wide-angle DMD projection
lenses. All of these lenses have very low distortion.

The folowing two-element example has a pronounced increase of
illumination toward the corners. This design has less than 0.006%
rectilinear distortion.

Radius Thickness Index Conic Constant
85.4982 3.0000 1.5000 0.6064936
21.6038 137.0893 -0.5535472
Stop 24.2982
25.7926 24.0000 1.5000 -1.4579459
-31.9469 28.4818 -3.5718662

Focal Length: 10.0mm
Aperture: f/4
File of View: 90 degrees (full field)
Waveband: Monochromatic

This is a very simplified monochromatic version of a lens type that
I've encountered numerous times when designing wide angle telecentric
lenses.

Brian
www.caldwellphotographic.com

From:	Paul van Walree
Subject:	Re: cos^4 law
Date:	Tue, 11 Jan 2005 20:31:02 +0100

The "BC" wrote:

>I'm the "someone", and I'd be happy to. I'm surprised that nobody
>seems to have experience with lenses of this sort since telecentric
>projection lenses are a pretty hot item lately.

I am sure there will be people who have experience with lenses of
that sort, but I am not one of them. I am mostly interested in
picture taking lenses, and have not (yet) seen examples in that
lens category.

>The folowing two-element example has a pronounced increase of
>illumination toward the corners. This design has less than 0.006%
>rectilinear distortion.
>
>Radius Thickness Index Conic Constant
>85.4982 3.0000 1.5000 0.6064936
>21.6038 137.0893 -0.5535472
>Stop 24.2982
>25.7926 24.0000 1.5000 -1.4579459
>-31.9469 28.4818 -3.5718662
>
>Focal Length: 10.0mm
>Aperture: f/4
>File of View: 90 degrees (full field)
>Waveband: Monochromatic

OK then. Thank you for the example.

>"I am curious to see an example of a lens that is telecentric, has the
>right type of pupil aberration, and that manages to overcome the light
>falloff due to optical (physical) vignetting all at the same time, to
>yield a stronger corner than center illumination."
>
>See my earlier post to you, where I've provided a very simple example
>along with a patent reference to several more.
>
>Hope I've managed to overcome your considerable skepticism.

Yes. I find it difficult to visualize illumination behavior
based on a few radii and other figures, but I have no reason to
doubt your statements.

--
Hell is oneself, hell is alone, the other figures in it, merely
projections. T.S. Eliot

From:	mg
Subject:	Re: cos^4 law
Date:	Tue, 11 Jan 2005 01:47:18 GMT

I suppose if you design a lens to have significant distortion of a certain
type, you mihgt be able to increase illumination at the image edges.

"Dr. Georg N.Nyman" wrote in message
news:3eadnQFbZbONlH7cRVn-rQ@comcast.com...
> Hello to everyone,
>
> I am confronted with someone stating that the cos^4 law is "a meaningless
> rule of thumb" and lens design (for photographic lenses) can be done such
> that at the boundary of the image circle one can have more light
> throughput than in the center (no filters of course...).
> Before I get drunk to understand this, I better ask - any idea what that
> guy could possible have ment?
> Thanks!
> George

From:	BC
Subject:	Re: cos^4 law
Date:	14 Jan 2005 05:45:02 -0800

The cos^4 law is only a theoretical constuct which explains the
illumination pattern of a certain type of idealized optical system.
This idealized optical system need not have anything to do with
specific real optical systems. Empirical evidence reveals that a large
number of lenses don't even come close to obeying this "law". Cos^4 is
certainly a good rule of thumb to bear in mind when designing lenses,
but it does not fundamentally restrict the designer. A competent
designer will know the theoretical basis of the cos^4 rule, and will
also know how to violate the rule when necessary to meet design
requirements.

Can you explain to me how the cos^4 law has any validity at all in a
lens exhibiting a cos^0 illumination pattern?
Brian
www.caldwellphotographic.com

From:	Paul van Walree
Subject:	Re: cos^4 law
Date:	Fri, 14 Jan 2005 17:43:37 +0100

The "BC" wrote:

>Can you explain to me how the cos^4 law has any validity at all in a
>lens exhibiting a cos^0 illumination pattern?

If the uniform illumination is due to a telecentric design, the
cos^4 law adequately accounts for the illumination pattern. Of
course, cos^4 is violated when pupil aberrations are at play.

--
I've been rich and I've been poor; rich is better. Sophie Tucker

From:	Mark W. Lund, PhD
Subject:	Re: cos^4 law
Date:	Mon, 10 Jan 2005 16:49:18 -0700

I have to admit I'm flummoxed on this one. So he proposes to have more
projected collection area at the edge of the image than the center if
you do things right? I would bet my money on the cosine^4 law.

I will agree that it is a rule of thumb, but in the sense that you
are not likely to get as much illumination at the edge of the field,
not more!!

Ah, the object is brighter at the edge of the field!

Best regards
mark

Dr. Georg N.Nyman wrote:
> Hello to everyone,
>
> I am confronted with someone stating that the cos^4 law is "a
> meaningless rule of thumb" and lens design (for photographic lenses) can
> be done such that at the boundary of the image circle one can have more
> light throughput than in the center (no filters of course...).
> Before I get drunk to understand this, I better ask - any idea what that
> guy could possible have ment?
> Thanks!
> George

--
----------------------------------------------------------------------
Mark W. Lund, PhD ** Battery Chargers
CEO ** Bulk Cells and Custom Battery Packs
PowerStream Technology ** Custom Power Supplies
140 S. Mountainway Drive ** DC/DC Converters
Orem Utah 84058 ** Custom UPS
http://www.PowerStream.com ** Engineering, manufacturing, consulting

From:	BC
Subject:	Re: cos^4 law
Date:	14 Jan 2005 08:43:50 -0800

I think the problem lies with inappropriate use of the word "law" in a
technical field. It sounds too much like something which cannot be
violated, and as a result people get easily mislead about what can or
can't be accomplished.

Brian
www.caldwellphotographic.com

From:	Paul van Walree
Subject:	Re: cos^4 law
Date:	Tue, 11 Jan 2005 18:57:47 +0100

The "Dr. Georg N.Nyman" wrote:

>I am confronted with someone stating that the cos^4 law is "a
>meaningless rule of thumb"

It's a rule of thumb alright, but not meaningless.

>and lens design (for photographic lenses) can be done such that
>at the boundary of the image circle one can have more
>light throughput than in the center (no filters of course...).

That is a bold statement. Perhaps your "someone" can present
a few examples?

>Before I get drunk to understand this, I better ask - any idea what that
>guy could possible have ment?

Why don't you ask a leading (hands-on) expert and consultant for Optics?
Here's one: http://www.gnyman.com/

--
An economist is an expert who will know tomorrow why the things he
predicted yesterday didn't happen today. Laurence J. Peter

From:	BC
Subject:	Re: cos^4 law
Date:	11 Jan 2005 11:15:56 -0800

"I am curious to see an example of a lens that is telecentric, has the
right type of pupil aberration, and that manages to overcome the light
falloff due to optical (physical) vignetting all at the same time, to
yield a stronger corner than center illumination."

See my earlier post to you, where I've provided a very simple example
along with a patent reference to several more.

Hope I've managed to overcome your considerable skepticism.
Brian
www.caldwellphotographic.com

From:	Steve Eckhardt
Subject:	Re: cos^4 law
Date:	11 Jan 2005 18:07:07 GMT

In article <3eadnQFbZbONlH7cRVn-rQ@comcast.com>, gnnyman@swissonline.ch
says...
>
>Hello to everyone,
>
>I am confronted with someone stating that the cos^4 law is "a
>meaningless rule of thumb" and lens design (for photographic lenses) can
>be done such that at the boundary of the image circle one can have more
>light throughput than in the center (no filters of course...).
>Before I get drunk to understand this, I better ask - any idea what that
>guy could possible have ment?
>Thanks!
>George

Arrgh! I've read the four replies posted so far. Are there no other lens
designers in this group?

The key to resolving this conflict is pupil aberration. It is possible to
design a lens that has a smaller pupil on axis than off-axis. A simple way to
do this is to introduce pupil coma. Those who have taken Roland Shack's UA
course in advanced aberration theory know that the side effect of pupil coma
is image distortion.

The common example of using pupil coma is the fisheye camera lens. The image
is almost uniformly bright on the film, but the distortion is horrible. (If
you can't fix it, feature it!)

There are some newer wide angle 35mm lenses that don't have significant
distortion, so there is another plan of attack I'll have to figure out when I
need it.

I hope I haven't cost a bartender too big a tip!
--
Best regards,
Steve Eckhardt
skeckhardt at mmm dot com

From:	BC
Subject:	Re: cos^4 law
Date:	11 Jan 2005 10:01:55 -0800

In a telecentric lens without pupil aberrations you wind up with
perfectly uniform illumination. This can occur even when rectilinear
distortion is zero. The telecentricity ensures that there is no
obliquity in image space (hence no cosine factors), and if there are no
pupil aberrations then the exit pupil looks exactly the same at every
field point.

If you incorporate the right type of pupil aberration you can have
greater illumination in the corners than in the center. This sort of
thing happens in telecentric projection lenses all the time.
Brian
www.caldwellphotographic.com
..

From:	Paul van Walree
Subject:	Re: cos^4 law
Date:	Tue, 11 Jan 2005 20:02:37 +0100

The "BC" wrote:

>In a telecentric lens without pupil aberrations you wind up with
>perfectly uniform illumination. This can occur even when rectilinear
>distortion is zero. The telecentricity ensures that there is no
>obliquity in image space (hence no cosine factors), and if there are no
>pupil aberrations then the exit pupil looks exactly the same at every
>field point.
>
>If you incorporate the right type of pupil aberration you can have
>greater illumination in the corners than in the center. This sort of
>thing happens in telecentric projection lenses all the time.

I am curious to see an example of a lens that is telecentric, has the
right type of pupil aberration, and that manages to overcome the light
falloff due to optical (physical) vignetting all at the same time, to
yield a stronger corner than center illumination.

--
Since fear is mostly about ignorance, the best part is that it's as
temporary as you choose. Christine Comaford

From:	Phil Hobbs
Subject:	Re: cos^4 law
Date:	Tue, 11 Jan 2005 12:42:23 -0500

Dr. Georg N.Nyman wrote:

> Hello to everyone,
>
> I am confronted with someone stating that the cos^4 law is "a
> meaningless rule of thumb" and lens design (for photographic lenses)
> can be done such that at the boundary of the image circle one can have
> more light throughput than in the center (no filters of course...).
> Before I get drunk to understand this, I better ask - any idea what
> that guy could possible have ment?
> Thanks!
> George

If I recall correctly, the 4 powers of cosine are:

1 from foreshortening of the pupil (the projected pupil area goes like
cos theta)

1 from obliquity (Lambertian emitter)

2 from the change of image magnification (tall shadows at evening, 1/r**2)

Barrel distortion and telecentricity on the image side would help some,
it would seem.

Cheers,

Phil Hobbs

From:	jrayces at juno.com
Subject:	Re: cos^4 law
Date:	13 Jan 2005 20:52:32 -0800

I am surprised people still argue about the cos^4 law. In my humble
opinion whoever says it is an empirical law shows utter ignorance. The
whole subject was settled once for all, long ago, by Irvine C Gardner
in "Validity of the Cosine-Fourth-Power Law of Illumination" Research
Paper RP1824, Vol 39, September 1947, Journal of Research of the
National Bureau of Standards.

From:	jrayces at juno.com
Subject:	Re: cos^4 law
Date:	13 Jan 2005 20:47:14 -0800

From:	rrllff at yahoo.com
Subject:	Re: cos^4 law
Date:	10 Jan 2005 18:33:36 -0800

The effect of the cos^4 law can minimized, however in practical
designs, particularly photographic lenses, there will always be less
light at the field edge. Theoretically one can increase edge
illumination by introducing outward going coma for the aperture stop
imagery. However, this is not pracical.

From:	BC
Subject:	Re: cos^4 law
Date:	14 Jan 2005 13:54:01 -0800

I've always had a problem with using the image space chief ray angle as
the cosine argument (as opposed to the field angle) since it has to be
determined by ray tracing and is so easily complicated by exit pupil
spherical aberration. However, if you want to use that definition then
you're correct. Of course, by that definition even fisheye lenses will
obey the cos^4 rule so long as there is no exit pupil aberration, and a
number of textbooks will have to be revised!
Brian
www.caldwellphotographic.com

From:	Paul van Walree
Subject:	Re: cos^4 law
Date:	Sat, 15 Jan 2005 13:58:04 +0100

The "BC" wrote:

>>>Can you explain to me how the cos^4 law has any validity at all in a
>>>lens exhibiting a cos^0 illumination pattern?
>>
>>If the uniform illumination is due to a telecentric design, the
>>cos^4 law adequately accounts for the illumination pattern. Of
>>course, cos^4 is violated when pupil aberrations are at play.

>I've always had a problem with using the image space chief ray angle as
>the cosine argument (as opposed to the field angle) since it has to be
>determined by ray tracing and is so easily complicated by exit pupil
>spherical aberration. However, if you want to use that definition then
>you're correct. Of course, by that definition even fisheye lenses will
>obey the cos^4 rule so long as there is no exit pupil aberration, and a
>number of textbooks will have to be revised!

Fortunately, there are also text books that do not need revision. I
don't understand why people derive the cos^4 thing in image space, and
subsequently insert field angles to discover that the law doesn't work
well.

--
He who accounts all things easy will have many difficulties. Lao-Tzu

From:	Steve Eckhardt
Subject:	Re: cos^4 law
Date:	14 Jan 2005 15:59:07 GMT

In article <1105710302.350207.149120@c13g2000cwb.googlegroups.com>,
brianc1959@aol.com says...
>
>The cos^4 law is only a theoretical constuct which explains the
>illumination pattern of a certain type of idealized optical system.
>This idealized optical system need not have anything to do with
>specific real optical systems. Empirical evidence reveals that a large
>number of lenses don't even come close to obeying this "law". Cos^4 is
>certainly a good rule of thumb to bear in mind when designing lenses,
>but it does not fundamentally restrict the designer. A competent
>designer will know the theoretical basis of the cos^4 rule, and will
>also know how to violate the rule when necessary to meet design
>requirements.
>
>Can you explain to me how the cos^4 law has any validity at all in a
>lens exhibiting a cos^0 illumination pattern?
>Brian
>www.caldwellphotographic.com
>
Brian,
The law is certainly valid for a simple lens with the stop at the
lens. There are, of course, ways to circumvent the law, which is what you're
discussing. A similar case is Brewster's law. My company, 3M, makes
materials specifically designed to circumvent Brewster's law, which they do
quite nicely ("Giant Birefringent Optics in Mulitlayer Polymer
Mirrors", Science Magazine, vol. 287 # 5462, pp. 2365-68). In other words, a
"law" frequently depends on certain common conditions that can be changed.
--
Best regards,
Steve Eckhardt
skeckhardt at mmm dot com

From:	BC
Subject:	Re: cos^4 law
Date:	15 Jan 2005 10:20:51 -0800

"Fortunately, there are also text books that do not need revision. I
don't understand why people derive the cos^4 thing in image space, and
subsequently insert field angles to discover that the law doesn't work
well. "

I think its easy to understand why. Using the image space definition
robs the cos^4 concept of practial usefulness in the sense that its
more difficult to use it to compare illumination qualities in different
lenses. It simply becomes a different way of expressing the effect of
pupil aberration. Thus, two lenses covering a 150 degree field of view
may both have zero pupil aberration and both obey the cos^4 rule in the
image space sense, but they could easily differ in relative corner
illumination by at least two orders of magnitude.

Well known texts by Smith and Kingslake are confused on the issue, so
its probably one of those things in optics that still isn't widely
agreed on.

Brian
www.caldwellphotographic.com