# Impact of covering glass on lens performance

I've seen microscope lenses optimized for 0.17mm covering glass. I don't see what needs to be optimized here? As glass does not touch the lens (as in case of oil/water immersion) - it should just affect focal distance without introducing any aberrations.

Is that correct, or covering glass will cause aberrations imbalance, and will require recalculating the lens? (I can probably only think of very slight chromatic aberration, but it's not important in my case)

Same for water immersion: If we cover sample with 1mm of water, but water does not touch the lens, will it cause any additional aberrations and require recalculating the lens?

Same for diffraction-limited laser focusing optics: I've seen some of them are optimized for laser output windows - what is the nature of this optimization?

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## 1 Answer

Any time you know there will be something in your optical system, it is generally good practice to include it in your model during the design process. Of course, in many applications these things will not make a difference. The image quality of your point-and-shoot camera will not become unacceptable if somebody chooses to take a picture through their bedroom window, for example. In the cases you cited, however, the extra aberrations produced by the cover glass or laser window can indeed be significant. I'll explain why, starting with some background.

## Aberration Theory

Wavefront aberrations are usually expressed as a polynomial function 1:

$$W = \sum_{i,j,k} W_{i j k} h^i \rho^j \phi^k$$

where $h$ is the object height (or angle, in the case of an object at infinity), and $(\rho, \phi)$ are polar coordinates in the pupil, normalized such that $h=1$ is the maximum object height and $\rho = 1$ is the edge of the pupil.

For example, $W_{040}$ is the coefficient of spherical aberration. It is often called "third-order" spherical, because lens designers are concerned with the ray angle errors, which are given by the first derivative of the wavefront error; this also distinguishes it from higher order aberrations, like 5th order spherical, $W_{060}$.

There are a number of ways that aberration coefficients can be computed. Continuing with spherical aberration as our example, the contribution from a single surface can be expressed:

$$W_{040}= -\frac{1}{8} A^2 Y_a \Delta\left(\frac{u_a}{n}\right) \rho^4$$

where $A \equiv n i_a$, $n$ is the refractive index, $i_a$ is the angle of incidence of the marginal ray (the "a-ray") on the surface, $Y_a$ is the marginal ray height on the surface, $u_a$ is the (paraxial) marginal ray angle, and $\Delta(x) \equiv x' - x$, where primed variables represent quantities after refraction at the surface, and un-primed variables are quantities immediately before refraction.2

There are a number of techniques for computing aberration coefficients for complex optical systems, such as "Seidel Sums" or the more modern and efficient technique of "G-sums,"3 but the important point here with respect to your question is this: the aberrations due to a surface depend just as much on the characteristics of the rays at the surface as they do on the shape of the surface.

So, you can see that the aberrations introduced by a flat plate can be very significant indeed. In the formalism I have presented here, a high NA microscope system would have very large values for $u_a$, and the design must absolutely include these in balancing aberrations if very high performance is to be achieved.

This should also help make clear why liquid immersion is helpful in getting to very high NA without aberrations becoming a problem. However, in your question you also ask about the case of a water droplet over the object, without being in contact with the microscope objective as in a true liquid immersion objective. In this case you must keep in mind that the curved surface of the droplet would be a refractive surface and must be included in the optical calculations if the resulting lens is to function at all!

One last note, on the brief discussion in comments about the reason for cover glass. Cover glass serves a variety of purposes, but isn't really intended as a beneficial component of the microscope imaging system. It is primarily used to hold the sample still; protect it from air; and most importantly, to hold it flat, because a high NA microscope system will have a very shallow depth of field. It would be quite inconvenient for the microscope user to have micron-scale surface variations in his sample produce focus variations between adjacent features in the image. Cover glass can be accommodated by the optical design, but if it weren't for these practical considerations the lens designer would likely not call for it.

Footnotes:

1: These are Seidel aberrations; more modern approaches use an orthogonal set of polynomials as their basis functions, such as Zernike polynomials, which are more appropriate for numerical techniques. Seidel aberrations are left over from the time when aberrations had to be calculated by hand, and are still useful and well understood by modern lens designers.

2: The marginal ray is the ray from an object point on the optical axis, which goes through the edge of the pupil/aperture stop. Also important is the chief ray, which is a ray from the edge of the field of view through the center of the pupil/aperture stop. These rays are used heavily in the geometrical analysis of optical systems. For more on this, see "Modern Optical Engineering" by W.J. Smith.

3: Again, these are covered by Smith.

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