9.1. DESIGNING DOUBLET ACHROMAT

Designing doublet objective is quite complex procedure. The reason is obvious: there is four optical surfaces, whose synergistic effect needs to correct both monochromatic and chromatic aberrations. Follows one possible solution. The goal here is a coma-free air-spaced doublet achromat (contact type), also corrected for spherical aberration. This doublet type is known as Fraunhofer. Correction of astigmatism will not be required, since it is typically low, unless a doublet is of large relative aperture, in which case it is the secondary spectrum that is more detrimental optically.

Starting with the requirement for minimized chromatism, the individual focal lengths of the two lens elements need to relate as the reciprocal of their Abbe numbers:

with ƒ1, ƒ2 being the focal length of the front and rear element, respectively. The individual focal lengths relate to the doublet focal length ƒ as ƒ=ƒ1ƒ2/(ƒ1+ƒ2). Note that ƒ2 is numerically negative. In terms of the system focal length, the individual lens focal lengths are ƒ1 = (V1-V2)ƒ/V1 and ƒ2 = (V2-V1)ƒ/V2. While formally any combination of Abbe numbers could be used, the practical requirement is that the two Abbe numbers are significantly different in their nominal values, roughly in V1:V2~2:1 ratio (the inverse ratio - Steinheil type doublet - requires significantly more strongly curved lenses). The lower ratio, the more strongly curved lenses have to be.

 As Eq. 46 indicates, the amount of secondary spectrum varies - possibly significantly - with the relative partial dispersion of the two glass types. The common choice are n~1.52 common crown and n~1.62 flint (for instance, BK7 and F2), resulting in near-minimum secondary spectrum possible with common glasses. Graph below shows main groups of the Schott glass production, as a function of index of refraction (n) and dispersion (as Abbe number, V). To the left, relative partial dispersion PF,e. The smaller PF,e differential, and the greater V differential between the two doublet glasses, the lower its secondary spectrum. A doublet made from BK7 crown and F3 flint will have secondary spectrum close to the minimum possible with common glasses (~1/2000ƒ); if F3 is replaced with, say, KZFS1 (several times more expensive), secondary spectrum will be further reduced by nearly 40%. In effect, the amount of secondary spectrum produced by a doublet is proportional to the ratio of height vs. base of the right-angle triangle whose hypotenuse connects the two glasses (colored pink for BK7 and F3).

Once the glass combination is determined, both Abbe numbers and refractive indici - V1 and V2, n1 and n2, for the front and rear lens, respectively - are known.

The next step is choosing correction type, that is, which two separate wavelengths will be brought to a common focus. This in turn determines the lens bending parameter k in terms of refractive indici for the two distant wavelengths to be brought to a common focus. If we choose to bring together F and C lines, the bending parameter ratio is given by:

with k=(1/R1)-(1/R2), R1,2 being the front and rear lens radius of curvature, respectively, and nF1, nC1 and nF2, nC2 refractive indici of the selected distant wavelengths for the front and rear lens, respectively (F and C spectral lines are the usual choice; better overall alternative is to bring together d and F lines, which would result in slightly different lens bending ratio).

With the front and rear surface radius of each lens element being related in terms of the lens shape factor q=(R2+R1)/(R2-R1) as R2=(q+1)R1/(q-1), substituting  R2=(q1+1)R1/(q1-1) and R4=(q2+1)R3/(q2-1) in k2/k1 gives:

with q1 and q2 being the lens shape factor for the front and rear lens, respectively. This determines the relation between the first and third radius needed to bring selected wavelengths to a common focus.

The lens shape factor q is determined according to the requirement for corrected monochromatic aberrations. For corrected coma, the sum of aberration coefficients c1 and c2 for the front and rear lens, respectively, must be zero. Coma aberration coefficient for thin lens is:

with n being the refractive index, p the position factor, and ƒ the lens focal length.

Position factor p=1-(2ƒ/i), with ƒ being the lens focal length, and i the lens-to-image separation, is given by p1=1-(2ƒ1/i1)=-1 for the front lens (since image distance i1 equals the focal length ƒ1 for distant objects), and with ƒ2=(V2-V1)ƒ/V2 from Eq. 43, position factor for the rear lens p2=1-(2ƒ2/i2)=(2V1-V2)/V2 (since i2=ƒ).

Setting c1+c2=0, substituting for the refractive indici and position factors, allows to extract the ratio between the shape factors for the front and rear lens needed to cancel coma as q2=xq1.

The next step is setting the aberration coefficients for spherical aberration to zero and, after substituting xq1 for q2 in the rear lens aberration coefficient expression, consolidate the expression into quadratic equation and solve it for q1. Aberration coefficients  s1 and s2 for primary spherical aberration for each lens element are obtained from the general coefficient relation:

after substituting the appropriate values for n, p and ƒ. Obviously, the calculation is extensive, but fairly straightforward. Once the sum of two coefficients is consolidated to s1+s2=aq12+bq1+c=0 quadratic form, the solution for q1 is given by q1,2=[-b±(b2-4ac)1/2]/2a. Of the two solutions for the shape factor q1, the one to use is numerically negative, usually roughly midway between -0.5 and zero (for biconvex lens with the rear surface more strongly curved). For the rear lens, the shape factor q2=xq1 is numerically positive, around ~1.6, being negative meniscus with the first surface strongly curved, and weak rear surface curvature.

After the shape factor q is determined for both lens elements, the appropriate surface radii are obtained from:

with R1, R2 being the front and rear surface radius of the lens element, respectively. This defines relation of the front and rear radius for the two lens elements; it now needs to be combined with the relation between the front lens radii (q1+1)R1/(q2+1)R3=-(nF1-nC1)/(nF2-nC2) needed to bring the two selected wavelengths to a common focus.

All these radii relations are relative to each other. Based on the R1 to R3 ratio, the R2 and R4 are determined relative to R1 and R3, respectively. What is left to do is determine relative relation of one of those radii with respect to doublet's focal length, so that specific value for that radius, and for the other three can be obtained. With the focal length of a thin lens ƒ=R1R2/(n-1)(R2-R1), substituting for ƒ1 and ƒ2 in the relation for doublet's focal length ƒ=ƒ1ƒ2/(ƒ1+ƒ2) gives doublet focal length defined in terms of lens indici and the four radii as:

Next step is substituting R2=(q1+1)R1/(q1-1) and R4=(q2+1)R3/(q2-1), which defines (1/ƒ) in terms of indici and R1 and R3; after substituting R3=-(nF2-nC2)(q1+1)R1/(nF1-nC1)(q2+1), from the relation needed to bring the two selected wavelengths to a common focus, the doublet focal length is defined in terms of indici, the front lens shape factor and first radius as (1/ƒ)=2[(n1-1)-(n2-1)(nF1-nC1)/(nF2-nC2)]/(q1+1)R1. This  defines R1 in terms of the doublet focal length ƒ as:

SUMMARY

To summarize the steps, the design procedure unfolds in the following order:

1 - decide the focal length ƒ of the doublet, and calculate needed f.l. for each lens for chosen glasses

2 - decide type of correction, i.e. which two colors will be brought to a common focus; this will determine R1/R3 ratio

3 - set the sum of coma coefficients for the two elements to zero and extract the ratio of shape factors q2=xq1 needed to correct coma

4 - substitute xq1 for q2 in the rear lens aberration coefficient for spherical aberration and set the sum of the coefficients for the two lenses to zero to get quadratic equation in a form aq12+bq1+c=0 with two solutions for the front lens shape factor, q1,2=[-b±(b2-4ac)1/2]/2a; use numerically negative solution q1=y, which determines the rear lens shape factor as q2=xy

5 - calculate the radii from:

Note that R1 and R3 are determined for C and F lines focusing together.

Finding the radii concludes preliminary design process. The doublet should be at the optimization level with respect to its optical quality. For F>10 systems such preliminary design could be quite usable without optimization, but it is always good to confirm that with ray trace. In F<10 systems, sensitivity to deviations increases, and higher-order spherical aberration can become significant in the optimization procedure: ray trace is a must.

Actual lens making is covered in detail in the series of articles on Bob May's site.