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▪ ** **CONTENTS
◄
4.8.3. Ronchi test
▐
4.8.5. Hindle sphere test
► #
**
4.8.4. Waineo null test**
Probably the simplest way to null paraboloid is to use another
reflecting surface. By placing point source of light at the infinity
focus of a paraboloid, the collimated light output can be reflected back
to it, to produce focus that can be nulled. The disadvantage of this
setup is that it requires a flat as large as the paraboloid.
A setup with the flat replaced by a smaller, easy to make reflecting
surface, is more practical. And this is what Waineo null test
offers: spherical aberration of a concave paraboloid - or a surface with
a negative conic in general - is compensated for by a
smaller concave sphere (**FIG. 55**).
**FIGURE 55**: Waineo null test setup: the sphere
is facing test surface, both centered around common axis, with the
light source placed inside the focus of the sphere. Diverging light
reflected from the sphere is focused by the test surface; if the
aberration contribution of a sphere is at the level of mirror tested,
and of opposite sign, the final focus will be aberration-free, and the
surface tested - if sufficiently accurate - will null when the
converging light is intercepted at the focus.
While the setup is seemingly simple, it is fairly complex optically,
with a number of interrelated factors affecting the aberration level:
source to sphere separation is the object distance (**L**S)
for the sphere, with the image formed by it being object for the test
surface (with its corresponding object distance** L**T=M+IS,
**I**S
being the sphere-to-image separation and **M** the mirror separation), which in turn determines distance from test
surface to its focus (**I**T),
when the null focus (**F**) is formed. The two surfaces have to
produce the same amount of aberration of the opposite sign and, at the
same time, to be properly aligned (centered, and at the proper
separation ensuing that the beam width nearly coincides with surface) in order for the diverging cone from
the sphere to match nearly exactly the test surface.
According to Eq. 9,
the combined primary spherical aberration contributions of the two
surfaces as the P-V error at paraxial focus is given by:
**W**(S+T)**= **
[(ΩS-1)2(kd)**4**/4RS**3**] + [K+(ΩT-1)**2**d**4**/4RT**3**]
**= **[(ΩS-1)**2**(k**4**/ρ**3**)+K+(ΩT-1)**2**](d**4**/4RT**3**)
where **Ω** is the inverse
object (point source for the sphere, and image formed by the sphere for
the test surface) distance (**L**S
and **L**T)
in units of the radius of curvature **R **(subscripts S and T refer
to the sphere and test surface, respectively), **k** is the ratio of
sphere vs. test surface effective diameter, **d** is the effective
pupil radius of the test surface, **K** is the test surface conic and
**ρ** is the ratio of curvature radii **R**S/**R**T
(since the reflective index **n** and radius **R** both change
sign going from one surface to another, the index from **Eq. 9** can
be omitted and **R** can be assumed numerically positive for both
surfaces).
The left side of the sum represents the wavefront error of the sphere,
and the right side that of the test surface. Since all parameters except
the test surface conic **K** are numerically positive, the condition
for zero sum is [K+(ΩT-1)**2**]<0,
i.e. K<0 and |K|>(ΩT-1)**2**.
Since the object for the test surface is the imaginary focus of the
sphere **F**S,
and the requirement for a real final focus means that distance from this
imaginary focus to the test surface has to be larger than its focal
length, a two alternatives for the sphere's imaginary focus location are
either somewhat inside test surface's center of curvature, or somewhat
outside of it. Since the latter allows for much smaller mirror
separation (in part due to the need to avoid too large obstruction of
the converging beam by the sphere when it is inside the beam), it is the
more practical one. The final focus forms at relatively short distance
behind the sphere, which requires either a central hole on it, or
diverging mirror in front of it.
Consequently, object separation **L**T
for the test surface is larger than its radius of curvature **R**T,
and ** Ω**T<1.
For paraboloid, **L**T
is closer to 2RT,
closer to **R**T
(larger) for prolate ellipsoids and around 2RT
for mild to moderate hyperboloids for hyperboloids. Toward
a mild prolate ellipsoid test surface, sphere's imaginary focus and the
final focus are shifting closer to the test surface's center of
curvature, as the result of the sphere becoming smaller in order to
match lower spherical aberration of such surface (and opposite for more
strongly aspherized surfaces). With **L**T
in the 1.1-2**R**T
range from weak ellipsoid to weak hyperboloid test surface, **
Ω**T
ranges from 0.9 to 0.5, gravitating toward 0.7.
Consequently, (ΩT-1)2
ranges from 1/100 to 1/4 from weak prolate ellipsoid to a mild
hyperboloid as the test surface.
Similarly, since the sphere forms a diverging beam, its object - the
point source - is inside its focus, hence **L**S
is more than twice smaller than **R**S,
and **Ω**S>2.
Roughly, **L**S
ranges between RS/5
for weak prolate ellipsoids to RS/3
for paraboloids (and larger for hyperboloids), with **Ω**S
in the 5 to 3 range, and, correspondingly, (ΩS-1)2
is in the 16 to 4 range, respetively. Since the sum [(ΩS-1)2(k4/ρ3)+K+(ΩT-1)2]
has to be zero for a null, the much larger value of (ΩS-1)2
has to be offset by the correspondingly smaller value of **k**4
(for given sphere radius, i.e. radii ratio
**ρ**), in order to match numerically opposite in sign value of
the test mirror contribution, proportional to
K+(ΩT-1)2.
In other words, as the test mirror conic increases toward zero, and its
aberration contribution diminishes, the effective aperture of the sphere
DS=kDT
also decreases, theoretically to zero for K=0.
Calculating needed setup parameters is a pretty involved procedure,
since the radii, separations and effective aperture values need to
result in near zero spherical aberration sum for the given test surface
conic.
Fortunately,
Waineo2006
freeware program (Richard, UK) makes determining Waineo null test setup parameters
a breeze ("knife position" in the input refers to the back focus
distance **B** on the above illustration). The program balances
lower- and higher-order aberration components, making the null focus
near perfect. Large, fast paraboloid, say, 400mm
ƒ/4, can be null
tested with less than half its diameter high-quality sphere, say
ƒ/4, to a 1/175
wave RMS setup accuracy; a 500mm
ƒ/3 paraboloid can be nulled with a
9-inch
ƒ/3.6 sphere
to better than 1/40 wave RMS accuracy, and so on.
Additional advantage of the test vs. regular tests for paraboloid at the center of
curvature is that the length of the setup is smaller than the radius of
curvature of test surface, by nearly 1/3 on the average.
Sensitivity of the test to setup errors is moderately high but,
similarly to Hindle sphere test, can be rather easily controlled by
keeping the final focus within several mm of its specified distance from
the test surface. This is due to magnification of the secondary mirror -
i.e. test surface - being very sensitive to the changes in its object
distance, which is the image of the light source projected by the sphere
(**F**S
on the above illustration). Specific change of the test surface
magnification **m**T
is given by:
** **mT
**= **-ƒT/(LT-ƒT)
**ƒ**T
being the test surface focal length. Change in the magnification is caused
by the change in test surface object distance **L**T=IS+M,
due to the change in separation of the image formed by the sphere, IS=RS/[2-(RS/LS)]=RS/(ΩS-2),
with **L**S
and **R**S
being object separation (i.e. source-to-sphere separation) and curvature
radius for the sphere, respectively. The effect of change in separation between the
two mirrors is comparatively negligible.
Note that mirror-to-image separation **I** in
terms of mirror radius of curvature **R** and object distance **o**
is given by I=R/[2-(R/o)] - or by I=ƒ/[1-(ƒ/o)]
in terms of the focal length
**ƒ** - where a negative **I**
value for **R** and **o** of the same sign indicates object
location inside the focal point and diverging rays (i.e. forming virtual
image). As the setup illustration indicates, this is the manner in which
the Waineo sphere forms its image of the source. In this configuration,
according to the sign convention, mirror-to-image separation is
numerically positive. Since n/R is positive for the sphere, making **R**S
effectively positive, **I**S
is also made positive by expressing it as I=RS/(ΩS-2).
For illustration, in the above Waineo setup for 400mm
ƒ/4 paraboloid,
1mm spacing error between the sphere and the source induces ~1/125 wave
RMS error, and nearly 3.5mm shift in the location of final focus. For
the 500mm
ƒ/3 mirror, the setup is, as expected, more sensitive: 1mm
spacing error between the source and sphere induces about ~1/80 wave RMS
error, with the final focus shift of nearly 4mm.
According to the wavefront error equation above, for given curvature
radii and test surface conic, the only spacing-related variables are the
effective mirror apertures and the relative inverse object distance **
Ω**. Changes in the
effective aperture at the two surfaces are in the same direction, thus
relatively insignificant. Changes in the **
Ω** factor, with
ΩS=RS/LS
and ΩT=RT/LT=RT/(M+IS)=RT/{M+[RS/(ΩS-2)]},
are directly dependant on the source-to-sphere separation **L**S
in the case of **Ω**S,
and less directly but significantly in the case of **
Ω**T.
Denoting the change in (ΩS-1)2
due to the change in the source-to-sphere separation **L**S
by **Δ**S,
and the corresponding change in K+(ΩT-1)**2**
as **Δ**T,
the resulting P-V wavefront error of primary spherical aberration at
the best focus is:
**W**_{Δ}=
(ΔSDS**4**/256RS**3**)+(ΔTDT**4**/256RT**3**)
**= **[(ΔSk**4**/ρ**3**)+ΔT]DT**4**/256RT**3**
with the parameters needed for calculation given by the specifics of the
particular setup. In the above setup for a 400mm
ƒ/4 paraboloid (DT=400,
RT=3200,
K=-1), with the test sphere parameters DS=85mm
(k=0.425), Rs=1600mm (ρ=0.5), LS=552mm,
M=2431mm, and
ΩS=RS/LS=2.89855,
-1mm change in **L**S
results in **
Ω**S
change of +0.0053 and (QS-1)2
change of +0.02, which is the value of **Δ**S.
At the test surface, the +0.0053 change in **
Ω**S
and -1mm change in mirror separation **M** result in **
Ω**T
change from 0.7598 to 0.7254, i.e. by -0.034. It causes (ΩT-1)2
change by -0.0177, which is the value of change at the test surface, **
Δ**T.
The corresponding P-V wavefront error of primary spherical aberration
(best focus) induced is W=(0.26**Δ**S+**Δ**T)DT4/256RT3=-0.000038mm,
or 1/14.4 wave in units of 550nm wavelength. Raytrace gives about twice
smaller error, the difference probably coming mainly from the best focus
being optimally balanced with the higher-order spherical aberration.
◄
4.8.3. Ronchi test
▐
4.8.5. Hindle sphere test
►
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