Читать книгу Optical Engineering Science - Stephen Rolt - Страница 94

At this point we illustrate the design of an air spaced achromat by looking more closely at the previous example where we analysed a 200 mm achromat design. We are to design an achromat with a focal length of 200 mm working at the infinite conjugate, using SCHOTT N-BK7 and SCHOTT SF2 as the two glasses, with the less dispersive N-BK7 used as the positive ‘crown’ element. Again, the Abbe numbers for these glasses are 64.17 and 33.85 respectively and the n_d values (refractive index at 589.6 nm) 1.5168 and 1.647 69. From the previous example, we know that focal lengths of the two lenses are:

The two conjugate parameters are straightforward to determine. The first conjugate parameter, t₁, is naturally −1. Eq. (4.53) can be used to determine the second conjugate parameter, t₂. This gives:

We now substitute the conjugate parameter values together with the refractive index values (ND) into Eq. (4.30a). We sum the contributions of the two lenses giving the total spherical aberration which we set to zero. Calculating all coefficients we get a quadratic equation in terms of the two shape factors, s₁ and s₂.

(4.54)

We now repeat the same process for Eq. (4.30b), setting the total system coma to zero. This time we get a linear equation involving s₁ and s₂.

(4.55)

Substituting Eq. (4.55) into Eq. (4.54) gives the desired quadratic equation:

(4.56)

There are, of course, two sets of solutions to Eq. (4.56), with the following values:

 Solution 1: s1 = −0.194; s2 = 1.823

 Solution 2: s1 = 3.198; s2 = 2.929

There now remains the question as to which of these two solutions to select. Using Eq. (4.29) to calculate the individual radii of curvature from the lens shapes and focal length we get:

 Solution 1: R1 = 121.25 mm; R2 = −81.78 mm; R3−81.29 mm; R4 = −281.88 mm

 Solution 2: R1 = 23.26 mm; R2 = 44.43 mm; R3−58.91 mm; R4 = −119.68 mm

The radii R₁ and R₂ refer to the first and second surfaces of lens 1 and R₃ and R₄ to the first and second surfaces of lens 2. It is clear that the first solution contains less steeply curved surfaces and is likely to be the better solution, particularly for relatively large apertures. In the case of the second solution, whilst the solution to the third order equations eliminates third order spherical aberration and coma, higher order aberrations are likely to be enhanced.

The first solution to this problem comes under the generic label of the Fraunhofer doublet, whereas the second is referred to as a Gauss doublet. It should be noted that for the Fraunhofer solution, R₂ and R₃ are almost identical. This means that should we constrain the two surfaces to have the same curvature (in the case of a cemented doublet) and just optimise for spherical aberration, then the solution will be close to that of the ideal aplanatic lens. To do this, we would simply use Eq. 4.29, forcing R₂ and R₃ to be equal and to replace Eq. 4.55 constraining the total coma, providing an alternative relation between s₁ and s₂. However, the fact that the cemented doublet is close to fulfilling the zero spherical aberration and coma condition further illustrates the usefulness of this simple component.

The analysis presented applies only strictly in the thin lens approximation. In practice, optimisation of a doublet such as presented in the previous example would be accomplished with the aid of ray tracing software. However, the insights gained by this exercise are particularly important. For instance, in carrying out a computer-based optimisation, it is critically important to understand that two solutions exist. Furthermore, in setting up a computer-based optimisation, an exercise, such as this, provides a useful ‘starting point’.