◄ 11.10. Houghton telescopes   ▐    11.13. Plano-symmetrical HCT ►
 

11.12. Houghton corrector: secondary spectrum reduction

As the ray plots clearly show, chromatism of the aplantic single-glass Houghton corrector becomes unacceptably large as the mirror relative aperture reaches 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,

                                                           1/ƒ=(1/ƒ1)+(1/ƒ2)-p/ƒ1ƒ2                               (150)

with the focal lengths of the front and rear lens given by:

                                           ƒ1=(n-1)[(1/R1)-(1/R2)+(n-1)t1/nR1R2)]                   (150.1)   and

                                           ƒ2=(n-1)[(1/R3)-(1/R4)+(n-1)t2/nR3R4)]                   (150.2)

respectively, and the quantity p, the separation between the second principal plane of the front lens and the first principal plane of the rear lens (FIG. 111) being:

                                                           p=(n-1)[(ƒ1t1/R1)-f2t2/R4]+s                        (150.3)

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, t1 and t2 the respective lens thicknesses, s the lens separation and R1, R2, R3 and R4 the respective lens surface radii. With 1/f1 and 1/f2 being very close numerically, it is the value of p that mainly determines corrector's focal power (Eq. 150), and hence its secondary spectrum.

As can be seen from Eq. 150.3, size of p it is not affected by the change in index, or radii, but changes 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 R4, but it would require the rear element to be a negative out-curving meniscus, which would make corrector with acceptable coma impossible.

In the equal-curves 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/p (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

                                               df~{[(n-1)n'/(n'-1)n]-1}ƒc                             (151)

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 ~fc/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 Fc being the corrector's focal ratio number, approximated by Fc~ƒc/D~ƒ1ƒ2/pD. 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 aplantic 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 a near-contact or contact doublet, requiring ƒ1/ƒ2=V2/V1. Since the aplantic 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 V2 for minimizing the secondary spectrum should be different in approximately the same proportion.

Assuming a 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 V2 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 the needed properties of the achromatizing glass obtained from:

                                                              V2/(n2-1)=V0ƒ1/(n0-1)ƒ2e                              (152)

with n0, V0 and n2, V2 being the refractive index (optimized wavelength) and Abbe number for the starting single-glass aplantic corrector, and the achromatized negative element, respectively.

A helpful indicator of the cumulative effect of the new glass type on secondary spectrum, that can be called its relative chromatic power, is given by 1/ç=(n2-1)V0/(n0-1)V2. 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 the index change in opposite directions with more significant index changes, it becomes difficult to impossible to find a glass with the relative chromatism ç close to 1). Once the 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 the lens thickness and separation also being possible factors (FIG. 112). 

The achromatizing is still worthwhile at ~ƒ/3 mirror focal ratios. By replacing the rear lens glass (BK7) by PK3, and increasing lens spacing from 1.4mm to 3mm in the ƒ/3/10 symmetrical aplanatic Houghton (Fig.110c), the 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 allowing some system coma (0.28 and 0.075 wave RMS for the h- and r-lines, respectively). Note that the corrector type with all radii different allows for still better color correction in a single-glass arrangement.

As mentioned, secondary spectrum is the 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 the typical symmetrical aplantic 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 the corrector's chromatism.

The achromatizing glass often will, to some extent, also result in a change of spherochromatism, compared to than 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 secondary spectrum level.
 

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