Focus Shift Induced from an Optical Window

In the limit of a relatively narrow converging beam (i.e., a large f-ratio, meaning about f/3.5 or higher in this instance) the focus shift, FS, is:

FS = d · [1 - 1 / n]

where d is the thickness of the window and n is the index of refraction of the glass out of which the window is made. n ~ 1.53 for regular crown glass, implying

FS = 0.346 · d

The focus shift is different for different parts of the converging beam, which means the window also introduces a small amount of spherical aberration, more for very fast f-ratio's.

Here's the Derivation

Refer to the diagram at left. Look first at where the converging beam first makes contact with the optical window. Snell's Law tells us that:

Sin(Theta₁) = n · Sin(Theta₂)

which we can rearrange to give

Theta₂ = ArcSin[ Sin(Theta₁) / n ]

From the geometry of the refracted beam while it's still inside the window,

s = d · Tan(Theta₂)

From the geometry of the beam in the absence of the window:

x = d · Tan(Theta₁)

Thus, the horizontal offset in the beam due to the window is:

x - s = d · [ Tan(Theta₁) - Tan(Theta₂) ]

Now, looking at the focal plane region, we can see that also

x - s = FS · Tan(Theta₁ ),

where FS is the focus shift. Eliminating x - s by setting the last two equations equal to each other, rearranging, and then substituting for Theta₂ gives the focus shift as:

FS = d · [ 1 - Tan( ArcSin( Sin(Theta₁) / n ) ) ÷ Tan(Theta₁) ]

Finally, we make use of the basic property of lenses

Theta₁ = 1 / ( 2 · f-ratio )

and we can solve for FS for a given d, f-ratio., and index of refraction.

Note that the 3-D bundle of rays coming from the lens is comprised of rays with angles from 0 up to Theta₁, so rays coming from different parts of the lens have different focus shifts. This is the definiton of spherical aberration, so the optical window introduces some small amount of it. In principle we need to integrate over all angles of the beam, weighting by area (of the lens), to get the mean focus shift for the lens as a whole and, thus, the position of best focus. But that's beyond the scope of this simple analysis. It does help to know that the limit of FS as the f-ratio goes to infinity (Theta₁ going to zero) is just

FS = d · [1 - 1 / n]

which provides an asymptotic limit to FS. The following table illustrates this for n = 1.53:

f-ratio	FS / d		f-ratio	FS / d
1.00	0.396		2.8	0.352
1.12	0.385		3.2	0.351
1.26	0.377		3.6	0.350
1.41	0.371		4	0.349
1.59	0.366		4.5	0.349
1.78	0.361		5	0.348
2.0	0.358		5.6	0.348
2.2	0.356		11	0.347
2.5	0.354		inf.	0.346

Thus, for an f/3.6 lens the range of shifts for different rays in the converging beam is very small -- less that half a thousandth of an inch for d=3mm -- and we can safely take FS/d for the lens as a whole as being the mid-point of the range from the len's f-ratio to infinity, which is 0.348 in this case.

For much faster lenses this mid-range estimate may only be indicative since more of the light at focus comes from rays with large off-axis angles.

This derivation was reformatted from a web page in the internet archive. I believe the original author was Chris Wetherill.