#DevOptical Part 23: Plane Windows

Before we move on to our next case study, I need to cover one type of element that I neglected so far: plane windows.

By plane window, I mean any thickness of glass with zero power faces. Due to our formalism, we will also restrict ourselves to interfaces that are perpendicular to the optical axis. This englobes windows, filters and beamsplitter as well. It however does not include prism or tilted windows.

The transform matrix can be obtained from the one of a [»] thicklens with no curvature:

Plane windows therefore behaves like air gap of length T/n. Because their actual thickness is T however, they have the property to shift focus by the amount

If you place a window in the path of a focalized beam, the focus point will be made farther than without the window by an amount equal to ∆.

Plane windows will also affect aberrations when placed in a focalized beam. Just like with [»] thinlenses, the formulas of Warren J. Smith in Modern Optical Engineering do not follow the same convention as in Zemax OpticsStudio and will not yield the same results. Because the maths are really not complicated here, I propose to share the derivation with you.

We first need to evaluate the marginal (h,u) and chief (h, u) rays at each interface before we can compute the contributions of the aberrations using the [»] Seidel formula for third-order aberrations. Be sure to be familiar with these before proceeding.

Both the marginal and chief ray transform as follow at each of the interfaces (glass – thickness – glass):

We also have the quantities for each interface transitions

and

because the indices are n₀=n₃=1 (air), n₁=n₂=n (glass).

We also have

The total spherical aberration of the plane window is obtained as the sum of the three individual components

and so

Similarly, we obtain the coma

which gives

We obtain astigmatism in a similar way

Because the element has no power (c=0), the Petzval term is zero

And distortion is limited to the contribution of astigmatism

Looking at these equations we can derive a few conclusions:

- Plane windows do not introduce aberration for collimated beams (on-axis marginal rays, u₀=0);

- Plane windows introduce coma, astigmatism and distortion only for non-zero fields (u₀≠0);

- The amount of aberration does not depend on intercept height, only on angles;

- The amount of aberration is linearly proportional to the window thickness (T).

The consequence of the third point is that the exact position of the window in the beam is not important as the only important quantities are the angles u₀ and u₀ that strike the window.

Figure 1 illustrates the order of magnitude involved in the spherical aberration introduced by plane windows for a 1 cm thick window. The results do not depend widely on the refractive index as can be seen in the plot.

In the visible range, angles of typically 10° are required to produce 1λ of spherical aberration in a 1 cm thick window. Conversely, a 1 mm window will produce a 0.1λ spherical at the same wavelength and angles. Similarly, a 15° angle on a 1 cm thick window at that same wavelength will produce (15/10)⁴ = 5λ of spherical aberration or 0.5λ on our 1 mm thick window.

Finally, you should note that the stop-shift equations cannot be applied to the plane window because it is a thick element. If you check [»] how the stop-shift equations were derived, you will see that they apply only on thin elements.

That’s all for today! We now have all the elements necessary to tackle our next challenge: removing the spherical aberration from our acomatic anastigmat singlet lens!

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I would also like to give a big thanks to Naif, Young, Samuel, James, Andrew, Lilith, Sebastian, Hitesh, Jesse, Alex, Stephen,Sivaraman, Jon, Cory, Karel, Aviv, Themulticaster, Tayyab, Marcel, Kirk and Dennis who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!