Читать книгу Optical Engineering Science - Stephen Rolt - Страница 73

The ‘front end’ of many high power microscope objectives exploits the principle of single surface aplanatic points through the use of a hyperhemisphere co-located with the object. The hyperhemisphere consists of a sphere that has been truncated at one of the aplanatic points which also coincides with the object location, as illustrated in Figure 4.3.

Using the hyperhemisphere, we wish to create a ×20 microscope objective for a standard optical tube length of 200 mm. In this example, it is assumed that two thirds of the optical power resides in the hyperhemisphere itself; other components collimate the beam. In other words:

Figure 4.3 Hyperhemisphere objective.

The refractive index of the hyperhemisphere is 1.6. What is the radius, R, of the hyperhemisphere and what is its thickness?

For a tube length of 200 mm, a ×20 magnification corresponds to an objective focal length of 10 mm. As two thirds of the power resides in the hyperhemisphere, then the focal length of the hyperhemisphere must be 15 mm. Inspecting Figure 4.2, it is clear that the thickness of the hyperhemisphere is −R × (n + 1)/n, or −1.625 × R. To calculate the value of R, we set up a matrix for the system. The first matrix corresponds to refraction at the planar air/glass boundary, the second to translation to the spherical surface and the final matrix to the refraction at that surface. On this occasion, translation to the original reference is not included.

From the above matrix, the focal length is −R/0.6 and hence R = −9.0 mm. The thickness, t, we know is −1.625 × R and is 14.625. In this sign convention, R is negative, as the sense of its sag is opposite to the direction of travel from object to image space.

The (virtual) image is at (n + 1) × R from the sphere vertex or 2.6 × 9 = 23.4 mm.

In summary: