Optical Engineering Science. Stephen Rolt

error contributions at the second surface are given by:

      (4.23)

      The total wavefront error is then simply given by the sum of the two contributions. This is expressed in standard format, as below:

(4.24)

The important conclusion here is that a flat plate will add to system aberration, unless the optical beam is collimated (object at infinite conjugate). This is of great practical significance in microscopy, as a thin flat plate, or ‘cover slip’ is often used to contain a specimen. A standard cover slip has a thickness, typically, of 0.17 mm. Examination of Eq. (4.24) suggests that this cover slip will add significantly to system aberration. In practice, it is the spherical aberration that is of the greatest concern, as θ₀ is generally much smaller than NA₀ in most practical applications. As a consequence, some microscope objectives are specifically designed for use with cover slips and have built in aberration that compensates for that of the cover slip. Naturally, a microscope objective designed for use with a cover slip will not produce satisfactory imaging when used without a cover slip.

      Worked Example 4.2 Microscope Cover Slip

A microscope cover slip 0.17 mm thick is to be used with a microscope objective with a numerical aperture of 0.8. The refractive index of the cover slip is 1.5. What is the root mean square (rms) spherical aberration produced by the cover slip? The aberration is illustrated in Figure 4.7.

      From Eq. (4.24):

Figure 4.7 Spherical aberration in cover slip.

Substituting the above values we get: K_sa = 0.003 22 mm or 3.2 μm.

      The wavefront error (in microns) is thus given by:

      where p is the normalised pupil function.

      For reasons that will become apparent later, in practice, wavefront errors are usually expressed as a fraction of some standard wavelength, for example 589 nm. The above wavefront error represents about 0.4 × λ when expressed in this way. An rms wavefront error of about λ/14 is considered consistent with good image quality. This level of aberration is, therefore, significant and measures must be taken (within the objective) to correct for it.

      4.4.2 Aberrations of a Thin Lens

We extend the treatment already outlined to analyse a thin lens. A thin lens can be considered as combination of two refractive surfaces, where the distance between the two surfaces is ignored. In practice, this is a reasonable assumption, provided the thickness is much less than the radii of the surfaces in question. Of course, the wavefront error produced by the two surfaces is simply the sum of the aberrations of the individual surfaces. A schematic for the analysis is shown in Figure 4.8.

The wavefront error contribution for the first surface is very easy to compute; it is simply that set out in Eqs. (4.5a)–(4.5d). To compute the contribution for the second surface, one can analyse this using the same methodology as in Section 4.2, but exploiting natural symmetry. That is to say, one can analyse the second surface by rotating the whole surface about the y axis, such that z → −z and x → −x. In this event, for the second surface, R → −R₂, u → v, θ → −θ. It is then simply a case of substituting these values into the formulae in Eqs. (4.5a)–(4.5d) and adding the wavefront error contribution of the first surface. The total wavefront error for the thin lens is then:

Figure 4.8 Aberration analysis for thin lens.

(4.25a)

      (4.25b)

      (4.25c)

(4.25d)

      4.4.2.1 Conjugate Parameter and Lens Shape Parameter

In terms of gaining some insight into the behaviour of a thin lens, the formulae in Eqs. (4.25a)–(4.25d) are a little opaque. It would be somehow useful to express the aberrations of a thin lens directly in terms of its focusing power and some other parameters. The first of these other parameters is the so called conjugate parameter, t. The conjugate parameter is defined as below:

(4.26)

      As we are dealing with a thin lens, we can use the thin lens formula to calculate the focal length, f, of the lens:

      This, in turn, leads to expressions for u and v:

(4.27)

Figure 4.9 illustrates the conjugate parameter schematically. The infinite conjugate is represented by a conjugate parameter of ±1. If the conjugate parameter is +1, then