lens equation of tilted mirror lens
j4murali
Hi All
I have a newbie question. I would like to find out what is the lens
equation of the following optical setup. Specifically I would like to
find out what is the effective focal length of such a setup.
In a very simple periscope I have 2 mirrors tilted at 45 degrees. Now
instead of planar mirrors, i have two curved mirrors (concave mirrors)
with each focal length of -F. What is the effective focal length of
the combined system?
Assuming that the two mirrors are placed at a distance of 2F(between
two mirror centers).
Since the optical axis is tilted and shifted, I would like to also
hear some pointers as to what kind of aberrations become dominant?
Any comments/feedback is appreciated.
best regards
-mJ
Michael Koch
> Since the optical axis is tilted and shifted, I would like to also
> hear some pointers as to what kind of aberrations become dominant?
Astigmatism, so severe that this design is not usable.
Michael
j4murali
> > Since the optical axis is tilted and shifted, I would like to also
> > hear some pointers as to what kind of aberrations become dominant?
>
> Astigmatism, so severe that this design is not usable.
Suppose if the curved mirrors are replaced my segmented mirrors, is it
possible to overcome the astigmatism?
best regards
-mJ
Michael Koch
> Suppose if the curved mirrors are replaced my segmented mirrors, is it
> possible to overcome the astigmatism?
Theoretically you need strongly astigmatic mirrors. However isn't it
much easier to leave the mirrors flat, and put (not tilted) lenses in
the beam?
Michael
Helmut Wabnig
Did you look into OSLO?
http://www.lambdares.com/education/oslo_edu/.
(dont know if it can do tilted lenses)
w.
AES
<3041132c-626e-4d32...@15g2000yqa.googlegroups.com>,
j4murali <j4mu...@gmail.com> wrote:
> Hi All
>
> I have a newbie question. I would like to find out what is the lens
> equation of the following optical setup. Specifically I would like to
> find out what is the effective focal length of such a setup.
>
> In a very simple periscope I have 2 mirrors tilted at 45 degrees. Now
> instead of planar mirrors, i have two curved mirrors (concave mirrors)
> with each focal length of -F. What is the effective focal length of
> the combined system?
> Assuming that the two mirrors are placed at a distance of 2F(between
> two mirror centers).
You can work it out fairly simply using the ABCD matrices ("ray
matrices") for tilted mirrors, which are given in my LASERS text and
elsewhere -- but as someone else noted, very small amounts of tilt can
lead to surprising amounts of astigmatism.
A student of mine and I once worked out the same things for tilted
spherical or ellipsoidal dielectric surfaces also:
Gail A. Massey and A. E. Siegman, "Reflection and refraction of gaussian
light beams at tilted ellipsoidal surfaces," Appl. Opt., vol. 8, pp.
975--978 (May 1969). Derivation of ABCD matrices for gaussian beams
passing through tilted ellipsoidal dielectric surfaces (caution: does
not use reduced coordinates).
Gail set up a He-Ne ring-laser cavity in which two parallel sides of the
ring cavity passed through a baseball sized glass sphere sitting in an
air-bearing cup, with the two transits through the sphere being
perpendicular to and located symmetrically on opposite sides of the
rotation axis of the sphere. He compensated the astigmatism of the four
intersections with the sphere surface by specifying appropriate
astigmatism in the external mirrors and added a small low-gain He-Ne
tube to make the ring oscillation. The optical components he'd ordered
arrived in the morning FedEx delivery; the ring was lasing shortly after
lunch; then we spun up the air bearing . . .
j4murali
> Did you look into OSLO?
>
> http://www.lambdares.com/education/oslo_edu/.
> (dont know if it can do tilted lenses)
>
> w.
Many thanks for the pointer. Eventually I found someone with access to
Zemax and I think it can give the effective focal length.
However, I was wondering if there was any simple equation that could
be used analytically for two tilted curved mirrors.
best regards & thanks
-mJ
anorton
"j4murali" <j4mu...@gmail.com> wrote in message
news:cd2fe59e-d20a-4508...@g26g2000yqn.googlegroups.com...
As others have pointed out, it is usually useless to analyze tilted mirrors
with paraxial equations due to the astigmatism and coma.
However, if the aperture is small, there is an approximation for the focal
length of a single tilted mirror in the two planes. If A is the tilt angle
between incident ray and mirror normal, and R is the radius of curvature.
Fp is the focal length in the plane of the tilt, Fs is the focal length in
the perpendicular plane.
Fp = R/(2*Cos(A)) and Fs = Cos(A)*R/2
--
Adam Norton
Norton Engineered Optics
www.nortonoptics.com
(Remove antispam feature before replying)
Michael Koch
> be used analytically for two tilted curved mirrors.
If the object and image points are at infinite distance, the two mirrors
are off-axis segments of very fast paraboloids.
Michael
anorton
"anorton" <ano...@removethis.ix.netcom.com> wrote in message
news:2cOdnXC4y_7Nlx7W...@earthlink.com...
I want to add this: I re-read your original post and realized the system you
describe without any mirror tilts would be afocal (i.e. infinite focal
length, collimated light in, collimated light out). If the spherical
mirrors are tilted and separated by their paraxial focal length, (R/2), then
in one plane you will have a very large negative focal length and in the
other a very large positive focal length. To calculate what those are in
each plane, use the above approximations for the focal lengths in each plane
and find the effective focal length of the combined system in each plane.
Here is the cheat sheet with the equations:
http://cvimellesgriot.com/products/Documents/TechnicalGuide/Lens_Combination_Formulas.pdf
krokodyle
"AES" <sie...@stanford.edu> wrote
> You can work it out fairly simply using the ABCD matrices ("ray
> matrices") for tilted mirrors, which are given in my LASERS text and
> elsewhere -- but as someone else noted, very small amounts of tilt can
> lead to surprising amounts of astigmatism.
>
> A student of mine and I once worked out the same things for tilted
> spherical or ellipsoidal dielectric surfaces also:
>
> Gail A. Massey and A. E. Siegman, "Reflection and refraction of gaussian
> light beams at tilted ellipsoidal surfaces," Appl. Opt., vol. 8, pp.
> 975--978 (May 1969).
> Derivation of ABCD matrices for gaussian beams
> passing through tilted ellipsoidal dielectric surfaces (caution: does
> not use reduced coordinates).
>
> Gail set up a He-Ne ring-laser cavity in which two parallel sides of the
> ring cavity passed through a baseball sized glass sphere sitting in an
> air-bearing cup, with the two transits through the sphere being
> perpendicular to and located symmetrically on opposite sides of the
> rotation axis of the sphere. He compensated the astigmatism of the four
> intersections with the sphere surface by specifying appropriate
> astigmatism in the external mirrors and added a small low-gain He-Ne
> tube to make the ring oscillation. The optical components he'd ordered
> arrived in the morning FedEx delivery; the ring was lasing shortly after
> lunch; then we spun up the air bearing . . .
And what happened? still lasing after spinning? ;-)
I have made a little sketch as per your description but I am not sure about
the proportions, the artcle in question has a description I assume.
OK, I will have a look at the article in question at my favorite library.
Funny beacause at the time I was messing up with astigmatic laser beams
and 4 by 4 (not 2 by 2) ABCD matrices as per the approach described by
Arsenault etc (article in Optics Communications).
AES
"krokodyle" <fenouillard@ath�n�e.somnif�re.org> wrote:
> > tube to make the ring oscillation. The optical components he'd ordered
> > arrived in the morning FedEx delivery; the ring was lasing shortly after
> > lunch; then we spun up the air bearing . . .
>
> And what happened? still lasing after spinning? ;-)
>
Yes -- and we observed a beat signal between the laser light traveling
in the CW and CCW directions around the ring with a beat frequency
proportional to the rotation rate, caused by Fresnel drag of the light
passing through the sphere.
Massey and AES, "Fresnel drag technique for determining
the spin-axis orientation of a spherical rotor," IEEE JQE,
vol. QE-6, pp. 500--506 (August 1970).
Massey was a real optical genius; I've never gotten over his having the
whole complex ring resonator lasing away an hour after the last
specially designed lens arrived. Unfortunately, the technique had too
many error sources to be useful for the Gravity Probe B laser gyro
relativity experiment.
Helmut Wabnig
The other techiques (the one they actually used) also
>had too
>many error sources to be useful for the Gravity Probe B laser gyro
>relativity experiment.
w.
Ron
>>
>> As others have pointed out, it is usually useless to analyze tilted
>> mirrors with paraxial equations due to the astigmatism and coma.
>>
>> However, if the aperture is small, there is an approximation for the
>> focal length of a single tilted mirror in the two planes. If A is the
>> tilt angle between incident ray and mirror normal, and R is the radius of
>> curvature. Fp is the focal length in the plane of the tilt, Fs is the
>> focal length in the perpendicular plane.
>>
>> Fp = R/(2*Cos(A)) and Fs = Cos(A)*R/2
>> --
>> Adam Norton
>>
>> Norton Engineered Optics
>> www.nortonoptics.com
>>
This discussion brings back memories of the 1970's for me. As the only
Optical Engineer in a small company, I was faced with supporting the
manufacture of a range of optical systems that were based on tilted
spherical mirrors. The apertures and tilt angles were not small, and there
were typically four mirrors per system. Thanks to my first class education,
I knew that these cheap and cheerful systems could not possibly work because
of all the astigmatism, coma, distortion, etc, but there they were, in
production with unescapably good performance. I always struggled to
understand these systems, and struggled even more to modify the designs and
design new systems (fortunately this was only a small part of the job).
The application was optical accessories for spectrophotometers, generally
creating a new focus for the beam and then re-creating the original optical
path back into the spectrophotometer. Not exactly diffraction-limited stuff,
but the quality of the intermediate images and the throughput efficiencies
were pretty impressive.
My boss I think had just stolen the designs from a previous employment, and
the only hint I ever got from him was that he understood that they had been
designed using Coddington's equations, as given by Adam above. I never found
any published work on these designs. This of course was in the days before
PCs, and raytracing computers were way beyond the resources of this small
outfit. I traced rays graphically, and tried to write raytracing code for a
Commodore PET (anyone remember that?) but got nowhere near modelling these
systems before I moved on. I guess I could do it nowadays in Zemax.
Ron Gibbs
--
Gibbs Associates
Optical Design Consultant
www.gibbsassociates.co.uk
AES
Helmut Wabnig <hwabnig@ .- --- -. dotat> wrote:
"The other techiques (the one they actually used) also
had too many error sources to be useful for the Gravity
Probe B laser gyro relativity experiment."
Personally, I think you got that one exactly right -- but then, it may
be that by the time GPB actually flew, all the relativistic effects it
was intended to measure had already been measured by other
experiments to substantially better than GPB's design specs; so that
GPB's performance was irrelevant anyway.