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10.2.4.3. Houghton telescopes
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10.2.4.6. Plano-symmetrical HCT
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#
**10.2.4.5. Houghton corrector: secondary spectrum reduction**

As the ray plots
clearly show, chromatism of the aplanatic (symmetrical) single-glass Houghton corrector becomes unacceptably
large as the mirror relative aperture approaches and exceeds ~ƒ/3. While both,
secondary spectrum and spherochromatism contribute to it, it is the
former that dominates. The cause of the
Houghton secondary spectrum is that its lens elements are not ideal thin
lenses with near-zero thickness and separation. Consequently, the thin
lens focal length formula given by (1/ƒ)=(1/ƒ1)+(1/ƒ2)
doesn't apply. Instead, the effective focal length is given by the
thick
lens focal length formula for a pair of lenses,

with the individual focal lengths of the front and rear lens, as the
actual thick-lens focal lengths, given by:

respectively, with the quantity **Π**
being the separation between the
second principal plane of the front lens and the first principal plane
of the rear lens (**FIG. 202**) given by:

with
ƒ1=(n-1)[(1/R1)-1/R2]
and
ƒ2=(n-1)[(1/R3)-1/R4]
being the front and rear lens focal length, respectively, **t****1**
and **t****2**
the respective lens center thicknesses, **s** the lens separation and R1,
R2,
R3
and R4
the respective lens surface radii. With 1/ƒ1
and 1/ƒ2
being very close numerically, it is the value of **
Π**
that mainly determines corrector's power (**Eq. 150**), and hence its
secondary spectrum.

**FIGURE 202**: Houghton corrector
principal plane separation **Π**, between the second principal
plane of the front lens (**P**1**'**)
and the first principal plane of the second lens (**P**2),
is the main determinant of the effective non-zero focal length of
the doublet. According to **Eq. 151**, the principal plane separation increases with
lens thickness and lens spacing (**s**), reducing further corrector (positive) focal length,
nearly in proportion to
~ƒ1ƒ2/Π.
This in turn results in the increase of corrector's secondary
spectrum. The principal plane separation - and secondary spectrum - for given
aperture also increase with more strongly curved lens surfaces, needed to
correct spherical aberration and coma of faster mirrors. When the
two lenses are in contact, the separation **s** is likely to be,
roughly, somewhat less than a half of the value of **
Π**.
This means that any relative increase in lens separation will result
in less than half as much of a relative increase of secondary
spectrum. With the typical values of
ƒ1/R1~1.3
and
ƒ2/R4~0.5,
change in the thickness of the front lens results in secondary
spectrum larger by a factor of ~2.5 than with identical thickness
increase of the rear lens. Also, any relative change in the front
lens thickness will result in ~1.3 times greater secondary spectrum
than identical increase in lens separation, while thickness increase
of the rear lens will result in the increase of secondary spectrum
smaller by a factor of 0.5 than identical increase in lens
separation.

As can be seen from **
Eq. 151**, the size of **Π** it is not affected by
a change in
index, or radii, but it does change in proportion to lens thickness. Since the
lenses can only be so thin, there is a practical minimum level of
secondary spectrum that can not be further lessened with a single-glass
Houghton corrector. It could only be cancelled by changing the sign of
**R**4,
but it would require the rear element to be a negative
out-curving meniscus, which would make corrector with acceptable coma
impossible.

In the
symmetrical corrector
configuration, the positive element has
slightly less power than the negative element as a single lens, but
when combined the two have a weak positive power. Neglecting the smaller
(negative) value of (1/ƒ1)-1/ƒ2
in **Eq. 150**, the corrector focal length is approximated by
ƒc~ƒ1ƒ2/Π
(at the end of this section is explained in more details why taking
somewhat smaller value is likely to give better end
result), with the variation in the focal length for non-optimized
wavelengths approximated by

where **n** and **n'** are the refractive index of the optimized
and non-optimized wavelength, respectively. It places the secondary
spectrum of the Houghton corrector - as the axial separation of its
paraxial foci - roughly at ~ƒc/200,
or some 10 times greater than in a doublet achromat. It is only due to
the weak corrector's power that it induces relatively small amount of
chromatism. Nominally, the secondary spectrum is negative for shorter
(than the optimized) wavelengths, and positive for longer wavelengths -
a consequence of the positive effective power of the corrector. It
simply means that the former focus shorter, and the latter longer than
the optimized wavelength.

Should the secondary
spectrum be the only chromatic aberration present, the
**transverse chromatic blur
diameter** in units of the green e-line Airy disc diameter would be approximated by
B~745(Δƒ)/Fc2,
with **F****c**
being the corrector's focal ratio, approximated by Fc~ƒc/D~ƒ1ƒ2/ΠD. In
reality, secondary spectrum is always combined with a certain amount of
sphero-chromatism, in which case the actual secondary spectrum is
measured by the separation between best aberrated foci for different
wavelengths, and the blur size results from the combined size of defocus
(secondary spectrum) and spherical aberration (sphero-chromatism) for
the wavelength.

Houghton corrector
secondary spectrum can be minimized by either slightly weakening
chromatic power of the front lens, or by slightly strengthening the rear
lens. The two options for reducing the secondary spectrum of the
Houghton corrector are: (1) abandoning symmetrical radii, and (2)
using two different glass types for the lenses. Another option is to use
plano-symmetrical corrector
type, which has generally less of the residual power, hence smaller
secondary spectrum. Abandoning equal radii design is a practical
disadvantage fabrication-wise (also requiring more complex calculations), while plano-lens design puts constraints
on coma-correction. Thus, the most effective way of reducing the
Houghton secondary spectrum is by using two different glass types.

Starting out with a typical single-glass
aplanatic Houghton corrector, **minimizing the secondary spectrum** requires
small change in either power, or
dispersion (or both) of one of the two lens elements. Lens power changes with (n-1), and its dispersion with
1/V, **n** being the glass refractive index and **V** its nominal dispersion.
The measure of needed chromatic power change can be obtained from the
general rule of achromatism for near-contact or contact doublet, requiring
ƒ1/ƒ2=V2/V1.
Since the aplanatic Houghton doublet acts as a thin lens pair in which
the rear lens focal length is somewhat longer than that of the front
lens, resulting in a weak positive power of the doublet, the appropriate
dispersion **V****2**
for minimizing the secondary spectrum should be different in
approximately the same proportion.

Assuming thin-lens
doublet, the rear lens effective focal length **
ƒ2e**
is obtained from 1/ƒc=(1/ƒ1)+1/ƒ2e,
as 1/ƒ2e=1/ƒc-(1/ƒ1).
The appropriate rear lens dispersion **V****2**
can be obtained from
ƒ1/ƒ2e=V2/V0,
with V0
being the dispersion of the basic single-glass corrector. However, glass
of different dispersion will almost invariably also have different
refractive index. Since it is also a factor affecting chromatic power
(secondary spectrum), it needs to be taken into account. With it, the
achromatic relation takes the form
ƒ1/ƒ2e=(n0-1)V2/(n2-1)V0,
with **needed properties of the achromatizing glass** obtained from:

with **n****0**,
**V****0**
and **n****2**, **V****2**
being the refractive index (optimized wavelength) and Abbe number for
the starting single-glass aplanatic corrector, and the achromatized
negative element, respectively.

Helpful
indicator of the effect of new glass type on secondary spectrum is given by 1/ç=(n2-1)V0/(n0-1)V2.
This parameter can be called corrector's
**relative
chromatic power**. If it is the front lens to be achromatized, **ç** will be slightly
smaller, and for the rear (negative) lens slightly greater than 1 (rough
average with n~1.5 and mirrors in the
ƒ/2.5-ƒ/3 range is
±1%).
It reflects needed change in the relative chromatic power of one of the
lens elements for near-optimal balancing of the combined secondary
spectrum of the corrector.

In general, the second glass alternative for minimizing the secondary
spectrum will be of very similar index and dispersion
values to those of the single-glass corrector to be achromatized (as the nominal dispersion and index change in opposite
directions with significant index changes, it becomes difficult to
impossible to find a glass with relative chromatism **ç** close
to 1). Once suitable glass is found, it will make possible
significant reduction
of the secondary spectrum. An integral part of the optimization is
tailoring out best combination of the secondary spectrum and spherical
aberration, with lens thickness and separation also
being possible
factors (**FIG. 203**).

**FIGURE
203: **
Effect of achromatizing on the chromatism of a HCT with symmetrical
aplanatic Houghton corrector (compare with **Fig**.
132a** **and** **
133a). By replacing the BK7 negative
lens element glass with BK8, the wavefront error is reduced from 1.3 and
0.38 to 0.45 and 0.048 wave RMS for the **h** and **r**
spectral lines, respectively (system **a** in the Appendix). By
increasing lens separation, the
chromatism is further reduced and optimized (balanced) to 0.14 and
0.12 wave RMS for the **h** and **r** lines, respectively, as
shown on the plot (system **b** in the Appendix).
The h-line correction is now nearly twice better than in
non-achromatized plano-lens version, or at the level of a 4"
ƒ/200 achromat.
While still
approximately double the chromatism of a comparable SCT, it is vastly
improved over the non-achromatized version
(note
that further reduction could be possible). An
ƒ/2.5 relative aperture for this aperture
size is probably near-limit for all-spherical Houghton system. At
this primary focal ratio,
higher-order spherical, responsible for
most of the scattered rays, cannot be corrected significantly better
than 1/30 wave RMS without adding aspheric surface term. SPEC'S

Achromatizing is
still worthwhile at ~ƒ/3
mirror focal ratios. By switching a rear lens
from BK7 to PK3, and increasing lens spacing from 1.4mm to 3mm (to
compensate for the induced spherical aberration) in the
ƒ/3/10 symmetrical aplanatic Houghton (Fig.
133c),
corrector's chromatism is reduced from 0.37 and 0.1 to 0.085 and 0.043
wave RMS wavefront error for the violet h-line and red r-line,
respectively. That compares very favorably to the reduction of
chromatism by compromising it with some coma (0.28
and 0.075 wave RMS for the **h-** and **r-**lines, respectively). Note that the corrector type
with all four radii different allows for still better color correction in a
single-glass arrangement.

An alternative to increasing lens separation of the achromatized
corrector is slightly (typically less than 1%) relaxing second radius.
It is more likely to bring best foci of different wavelengths closer
together, but the choice is best made using ray-trace.

As mentioned, secondary spectrum is main, but not a lone contributor to the chromatism of
the Houghton corrector. The other is **sphero-chromatism**.
As a consequence, the secondary spectrum - and chromatism in general -
is minimized by bringing together best foci for different wavelengths,
not the paraxial foci. This would certainly require more involved
optimization than the one outlined above. In general, however,
considering typical symmetrical aplanatic Houghton LA properties, even this crude
form of optimization alone - in particular with the change in relative
chromatic power purposely reduced to ~2/3 of the power differential indicated
by the **ç ** value - should result in a significant reduction
of corrector's chromatism.

The achromatizing glass
often will, to some extent, also change the spherochromatism of the basic (single-glass) corrector. It can be for
better, or for worse, but the effect is, in general, secondary to that
of the change in size of secondary spectrum.

◄
10.2.4.3. Houghton telescopes
▐
10.2.4.6. Plano-symmetrical HCT
►

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