Focussing Lens without Spherical Aberration
Narasimham
monochromatic rays aplanatically to a focus without spherical
aberration? Please give net link. Thanks in advance.
Amadeus Train-Owwell Zirconium
doesn't the abberation have to do
with different colors?
.
--The Pinochet Plan for Social Security!
http://larouchepub.com
Christopher Green
wrote:
>What shape does a plano-convex or bi-convex lens have to bring parallel
>monochromatic rays aplanatically to a focus without spherical
>aberration? Please give net link. Thanks in advance.
The least spherical aberration that can be had would be had in an
aspheric lens. These can be made in many designs, including
plano-convex, biconvex, and meniscus. Aspheric surfaces defy simple
description (and, for that matter, defy conventional grinding, so
aspheric lenses are often molded). They are generated by CAD programs
and may be curve-fits involving polynomials of degree as high as 10.
See http://www.janostech.com/files/Optical%20Design%20Data.pdf
For a spherical lens, spherical aberration is minimized by choosing
the optimum Coddington shape factor for the index of refraction and
the desired object and image distances. See many sources, including
http://journals.tubitak.gov.tr/physics/issues/fiz-01-25-3/fiz-25-3-12-9911-1.pdf
--
Chris Green
Narasimham
> doesn't the abberation have to do with different colors?
To exclude it I asked for monochromatic, or same color. Now looking
only at geometric optics.
> Narasimham wrote:
> > What shape does a plano-convex or bi-convex lens have
> > to bring parallel monochromatic rays aplanatically to a
> > focus without spherical aberration? Please give net
> > link. Thanks in advance.
>
redbe...@yahoo.com
The standard shape used in the industry is a conic surface,
with additional small corrections in the form of a polynomial
(even-order terms only).
The equation is given at
http://www.thorlabs.com/Equation.cfm?Section=2&Ref=12
(They had the wrong equation for a while in their catalog,
but have got it right again.)
An example of a lens, with values for the equation parameters
for both surfaces:
http://www.thorlabs.com/ProductDetail.cfm?&DID=6&ObjectGroup_ID=800&Product_ID=152
FYI, the meaning of the terms in the equation (given in 1st URL):
curv = curvature at center of lens, = 1/(radius of curvature)
K = conic constant:
hyperboloid: K < -1
paraboloid: K = -1
ellipsoid, major axis along optical axis: -1 < K < 0
circle: K = 0
ellipsoid, minor axis along optical axis: K > 0
A, B, C, D: polynomial coefficients
These lenses typically are designed to focus rays parallel to the
optical axis (or conversely, to collimate a source that is on-axis).
They're great for collimating and focussing diode lasers. For off-axis
sources, the aberrations can be significant.
-- Mark
redbe...@yahoo.com
The standard shape used in the industry is a conic surface,
rrl...@yahoo.com
choosing the correct conic constant for the convex surface. The conic
constant is a function of the lens refractive index.
redbe...@yahoo.com
A thousand apologies for the double-post . . .
Narasimham
> On 28 Dec 2004 16:48:45 -0800, "Narasimham" <math...@hotmail.com>
> wrote:
>
> >What shape does a plano-convex or bi-convex lens have to bring
parallel
> >monochromatic rays aplanatically to a focus without spherical
> >aberration? Please give net link. Thanks in advance.
>
> The least spherical aberration that can be had would be had in an
> aspheric lens. These can be made in many designs, including
> plano-convex, biconvex, and meniscus. Aspheric surfaces defy simple
> description (and, for that matter, defy conventional grinding, so
> aspheric lenses are often molded). They are generated by CAD programs
> and may be curve-fits involving polynomials of degree as high as 10.
Is z= curv R^2/(1+sqrt(1-(k+1)R^2)+ A R^6 +....+ D R^10 supposed to be
an equation to a conic of revolution?
> See http://www.janostech.com/files/Optical%20Design%20Data.pdf
Thanks. But I do not get this at all.
For hyperbola it is given in above design data 1/s1 + 1/s2 = constant (
1/eff. focal length), but we know s1 - s2 is a constant for a
hyperbolic surface !
Like wise, for an ellipse it is again given 1/s1 + 1/s2 = constant (
1/eff. focal length), but we know s1 + s2 is a constant for an
elliptic surface. Can you please explain?
Narasimham
Thanks.
But,
Is z= curv R^2/(1+sqrt(1-(k+1)R^2)+ A R^6 +....+ D R^10 supposed to be
an equation to a conic (ellipse,hyperbola) of revolution when A,B,C,D
vanish? Cannot a conic be more simply desribed by (z/a)^2 +/- (R/b)^2
=1?
rrl...@yahoo.com
conic, with k being the "conic constant"
Christopher Green
solution (I've heard there is one for certain aspheric Fresnel lenses,
but I don't know the details). I believe the CAD programs that are used
to design these lenses start from a conic, then ray-trace an optimized
lens, and produce an 8th or 10th-degree polynomial curve fit to the
optimized result.
--
Chris Green
Christopher Green
aberrations, obtain the desired correction in a thinner lens, and
create such nifty eyewear as progressive lenses and bifocal contacts.
It's all in what you optimize the lens for.
--
Chris Green
redbe...@yahoo.com
>Is z= curv R^2/(1+sqrt(1-(k+1)R^2)+ A R^6 +....+ D R^10 supposed to be
> an equation to a conic
> (ellipse,hyperbola) of revolution when A,B,C,D
> vanish? Cannot a conic be more simply desribed by (z/a)^2 +/- (R/b)^2
> =1?
Correct on both counts. However, the equation with "k" and "curv"
seems to be standard in the optics industry. I'm not entirely sure
why. However, it does have the following desireable features:
1. The surface curvature appears explicitly in the equation.
2. z=0 at R=0, so that for other values of R, calculating z will
directly give the surface's displacement relative to the center of the
lens.
3. In lens design and ray tracing software, simply change the value of
k to change the type of surface. No need to edit the equation and
change "+" to "-" or vice-versa.
Mark
rrl...@yahoo.com
avoid division by zero during computation.
Narasimham
Thanks again, Mark, Green and rrl. Following questions still seem to
linger:
For the above equation of lens profile defintion:
1) Do all rays collimate/focus to a single focal point at 1/(2*curv)
distance below z=0 reference plane for all conic k values? If not,what
is the focal point?
2) Is k the same as eccentricity of conic? What actually determines
choice of k?
3) What role do coefficients A to D play? Is impression that < pure
conics are aplanatic, making rays at points far removed from optic
axis also come a focal point in same phase > correct, say in case of
plano-convex?
4) The conic equation should be y = curv x^2/(1+sqrt(Q)),
Q = 1 - (1+k)(x/a)^2 where a is the maximum radius in case of
ellipsoidal lenses needed in design. Else even the physical dimension
does not tally. Right?
Wishing you all a happy New 2005.
Narasimham
Narasimham
Narasimham wrote:
> redbe...@yahoo.com wrote:
> > Narasimham wrote:
> >
> > >Is z= curv R^2/(1+sqrt(1-(k+1)R^2)+ A R^6 +....+ D R^10 supposed
to
> be
> >
> > > an equation to a conic
> >
> > > (ellipse,hyperbola) of revolution when A,B,C,D
> > > vanish? Cannot a conic be more simply desribed by (z/a)^2 +/-
> (R/b)^2
> > > =1?
> >
> > Correct on both counts. However, the equation with "k" and "curv"
> > seems to be standard in the optics industry. I'm not entirely sure
> >
> > 1. The surface curvature appears explicitly in the equation.
> > 2. z=0 at R=0, so that for other values of R, calculating z will
> > directly give the surface's displacement relative to the center of
> the
> > lens.
> > 3. In lens design and ray tracing software, simply change the
value
> of
> > k to change the type of surface. No need to edit the equation and
> > change "+" to "-" or vice-versa.
> >
> > Mark
>
> 2) Is k the same as eccentricity of conic?
saw pamphlet, it gives k=- eccentricity^2.
What actually determines
> choice of k?
>
> 3) What role do coefficients A to D play? Is impression that < pure
> conics are aplanatic, making rays at points far removed from optic
> axis also come a focal point in same phase > correct, say in case of
> plano-convex?
>
> 4) The conic equation should be y = curv x^2/(1+sqrt(Q)),
> Q = 1 - (1+k)(x/a)^2 where a is the maximum radius in case of
> ellipsoidal lenses needed in design. Else even the physical dimension
> does not tally. Right?
Sorry, just saw pamphlet, Q = 1 - (1+k)(x * curv)^2 but then, rmax =
Sqrt(1/(k + 1))/curv is where one stops for both elliptic and
hyperbolic spherical aberration free lenses?
A surface profile for this purpose from geometric optics is easily
obtained, and so am curious.
redbe...@yahoo.com
> 1) Do all rays collimate/focus to a single focal point at 1/(2*curv)
> distance below z=0 reference plane for all conic k values? If
not,what
> is the focal point?
If the surface is a mirror, this is true for parabolic (k=-1) surfaces,
and only for rays parallel to the axis.
For other k values, it's approximately true for rays close to the
center of the mirror and parallel to the axis.
For lenses, if I remember correctly, the surface should be a hyperbola
to focus rays to a common point. Not sure where the focal point is
measured from in that case, however and the focal length will be
different than 1/(2*curv) -- it must depend on the index of refraction
somehow.
> 3) What role do coefficients A to D play? Is impression that < pure
> conics are aplanatic, making rays at points far removed from optic
> axis also come a focal point in same phase > correct, say in case of
> plano-convex?
A conic surface can form a perfect image for objects lying on the
optical axis, but still have unacceptable aberations for off-axis
objects. These extra terms can modify the surface to reduce the
off-axis aberations, at the expense of introducing small-but-acceptable
aberations for on-axis objects.
> 4) The conic equation should be y = curv x^2/(1+sqrt(Q)),
> Q = 1 - (1+k)(x/a)^2 where a is the maximum radius in case of
> ellipsoidal lenses needed in design. Else even the physical dimension
> does not tally. Right?
The dimensions do work out in the original equation as given on the
Thorlabs site. Since curv has dimensions (1/length),
the term (curv^2)*(R^2) is dimensionless, consistent with the other
dimensionless terms in the denominator.
> Wishing you all a happy New 2005.
> Narasimham
And to you as well.
Mark
John Savard
<Qnc...@netscape.net> wrote, in part:
>doesn't the abberation have to do
>with different colors?
That's chromatic aberration. Spherical aberration and coma are two other
aberrations.
>Narasimham wrote:
>> What shape does a plano-convex or bi-convex lens have to bring
>parallel
>> monochromatic rays aplanatically to a focus without spherical
>> aberration? Please give net link. Thanks in advance.
A meniscus lens will take light rays already coming to a focus, and
bring them to a nearer focus, aplanatically - that is, without spherical
aberration or coma, if it has just the right shape.
http://members.shaw.ca/quadibloc/science/opt0503.htm
To bring parallel rays to a focus without spherical aberration, you will
need an aspheric surface.
John Savard
http://home.ecn.ab.ca/~jsavard/index.html
Narasimham
2) .. What actually determines choice of k?
I repeat my earlier query that has not been addressed so far : about
how geometry of a,b axes of hyperbola or its eccentricity is fixed for
an apalnatic (no spherical aberrration,coma for given material
refractive index and wavelength of light (color)).
Since sin(i)/sin(r)=refractive index mu = sin(i)/cos(ph)
(where i is angle in less dense air medium)is fixed , i-> 90 degree, ph
tends to complement of critical angle ,and I tend to think mu is
related to eccentricity or k. That is why I ask what actually
determines choice of k .. mu should be k or some function of k, or so
it appears to me.
CLT
> aberrations.
Spherochromatism is SA that varies by wavelength.
Clear Skies
Chuck Taylor
Do you observe the moon?
Try http://groups.yahoo.com/group/lunar-observing/
Are you interested in understanding optics?
Try http://groups.yahoo.com/group/ATM_Optics_Software/
************************************
Amadeus Train-Owwell Zirconium
doesn't spherical abberation have to do
with different colors?
redbe...@yahoo.com
> doesn't spherical abberation have to do
> with different colors?
Nope. It is chromatic aberration that has to do with different colors:
rays are not all focussed to the same point because the refractive
index of glass changes with wavelength.
It's probably best to explain spherical aberration with a picture; see
Figure 1 at
http://www.vanwalree.com/optics/spherical.html
In this case, rays are not focused to the same point because the lens
surface is a sphere. It has nothing to do with wavelength.
Narasimham
> I repeat my earlier query that has not been addressed so far : about
> how geometry of a,b axes of hyperbola or its eccentricity is fixed
for
> an apalnatic (no spherical aberrration,coma for given material
> refractive index and wavelength of light (color)).
> Since sin(i)/sin(r)=refractive index mu = sin(i)/cos(ph)
> (where i is angle in less dense air medium)is fixed , i-> 90 degree,
ph
> tends to complement of critical angle ,and I tend to think mu is
> related to eccentricity or k. That is why I ask what actually
> determines choice of k .. mu should be k or some function of k, or
so it appears to me.
My calclations on the basis of intrinsic conic equations indicate (for
given material,color of light,plano-convex lens, SA free collimation)
that lens should be geometrically profiled as per:
|----------------------------------------------------------|
| eccentricity of hyperboloid lens = refractive index |
|----------------------------------------------------------|
May I again invite comments from all, in particular Mark, Chris Green,
John Savard and CLT who have given valuable comments regarding present
design before we may consider the thread closed. Regards
Narasimham
redbe...@yahoo.com
> eccentricity of hyperboloid lens = refractive index
You have finally succeeded in getting me to go look at my old copy of
Optics by Hecht & Zajac, and sure enough they have two pages devoted to
refraction by aspheric surfaces. There they give the same result for
hyperbolic surfaces, eccentricity = refractive index.
The section of interest is in the chapter on Geometrical Optics --
Paraxial Theory.
-- Mark
Spagyrique
"Narasimham" <math...@hotmail.com> wrote
> May I again invite comments from all, in particular Mark, Chris Green,
> John Savard and CLT who have given valuable comments regarding present
> design before we may consider the thread closed.
Your original query follows:
"What shape does a plano-convex or bi-convex lens have to bring parallel
monochromatic rays aplanatically to a focus without spherical aberration?"
You were given the answer that a plano-hyperboloidal lens was indeed going
to bring parallel monochromatic rays to a focus without w/o spherical
aberration, the convex side facing the focus.
(the incident rays being parallel to the focal axis of the above
hyperboloid).
A classical result first enunciated by Descartes 375 years ago (*).
However you include the word "aplanatically" which leads one to believe that
you are also concerned with coma in addition to spherical aberration.
Therefore you are also considering incident rays emanating from infinity and
which are not parallel to the optical axis of the above lens.
Now if you want a lens (singlet)
***simultaneously free of spherical aberration and free of coma for
infinity***,
I regret to inform you that there is none (&).
In other words while the above plano-hyperboloidal lens does look fine on
paper its practical use may entail certain problems.
Now if you meant "stigmatically" when you wrote "aplanatically" that's
another story.
I just wanted to call your attention on this point.
I assume that you have a practical application in mind, in which case a
little ray-tracing might clarify the situation.
(*) "The geometry of Rene Descartes" Dover Publications, New York, NY.
(&) Handbuch der Physik, Band XXIV, Grundlagen der Optik, Springer Verlag,
Berlin. page 62.
probably more accessible would be:
A.E. Conrady, Applied Optics and Optical Design, Part One, Dover Pubs etc.
p.331 etc.
BC
***simultaneously free of spherical aberration and free of coma for
infinity***,
I regret to inform you that there is none (&)."
A single element lens can be completely corrected for third order coma
and all orders of spherical aberration with a single aspheric surface.
In a fast lens there will be higher order coma, which can be
effectively corrected with a bi-aspheric design. This is the basis of
the objective lenses used in all CD/DVD players.
Brian
www.caldwellphotographic.com
Narasimham
be included in more common elementary geometry references for curves
and surfaces. E.g. when we want to include off-axis parallel rays and
want to know which aplanatic curve containing P became spherical in the
particular 'shallow' case.(A is any point in the mirror aperture plane)
AP+PF=constant is known for a curved mirror that calculates
geometrically to a parabola, but not so for a lens ( plano convex plane
contains point A) AP+mu*PF = constant, which calculates to a hyperbola
with eccentricity equalling refractive index... surprising that it was
common knowledge from times of Descartes. Regards.
Spagyrique
"BC" <brian...@aol.com> wrote in message
> A single element lens can be completely corrected for third order coma
> and all orders of spherical aberration with a single aspheric surface.
The other being...spherical, planar?
By corrected for all orders of SA do you actually mean stigmatic on axis for
a source at infinity?
> In a fast lens there will be higher order coma, which can be
> effectively corrected with a bi-aspheric design. This is the basis of
> the objective lenses used in all CD/DVD players.
Concerning the CD/DVD lenses how do they take care of the large SA caused by
the plastic layer, especially at high NA? By introducing a certain amount of
SA of the opposite sign?
BC
"BC" <brianc1...@aol.com> wrote in message
> A single element lens can be completely corrected for third order
coma
> and all orders of spherical aberration with a single aspheric
surface.
The other being...spherical, planar?
In general spherical, although if you are free to vary the refractive
index you could make the rear surface plano.
By corrected for all orders of SA do you actually mean stigmatic on
axis for
a source at infinity?
Yes, stigmatic on axis for a source at infinity.
> In a fast lens there will be higher order coma, which can be
> effectively corrected with a bi-aspheric design. This is the basis of
> the objective lenses used in all CD/DVD players.
Concerning the CD/DVD lenses how do they take care of the large SA
caused by
the plastic layer, especially at high NA? By introducing a certain
amount of
SA of the opposite sign?
The cover layer just becomes a part of the design, and poses little
difficulty. The only time it really becomes an issue is if you want to
reduce the lens size as much as possible, and you have to start
worrying about the BFL.