# Why are shadowgraph LED lenses so hard to focus?

I’m typing this out of a frustration that I’m sure many of my colleagues have already felt when assembling a shadowgraph or a Schlieren flow visualization system: Adjusting the LED focusing lenses! Let’s look at the optics of this problem and try to see the rather unintuitive trade-offs we are making when adjusting these lenses in practice.

### The objective function

There are two objective functions that are usually desirable when focusing an LED system: Firstly, we want to collect as much light as possible. Second, we want to focus the LED to a point as small as possible, because we want to pass this light through a pinhole. In fact, the two objectives are the same: Focusing to a pinhole without cutting much light is the same as maximizing the amount of light (first objective). Thus, there’s really only one objective function. Let’s look at the standard collimator arrangement utilized in a Z-type shadowgraph:

### Light losses

There are three locations where light can be lost that we need to consider in this type of setup:

1. Light that is not collected by the lens
2. Light that is blocked by the pinhole
3. Light that is not collected by the parabolic mirror

If you are not thinking of these issues while setting up your shadowgraph, then you will lose a lot of light – perhaps more than you can afford to. The result, then, is an alignment struggle!

The reason this problem is rather unintuitive is because we usually don’t think about (3) because we’re focused into getting as much light into the pinhole. But the light lost in the parabolic mirror (3) really is the key to designing the optimal shadowgraph LED lens!

### the ideal shadowgraph collector – single lens design

Let’s look at this problem backwards: Start with a parallel beam coming into the parabolic mirror. This beam will focus at where the pinhole should be, which only depends on the parabolic mirror focal length. The convergence angle of the beams, however, depends on the f-number of the parabolic mirror. As we’re going to see, the f-number of the mirror is crucial here.

We seek to put a lens after the focal point such that it focuses at the pinhole location with a beam convergence exactly equal to the f-number of the mirror. If we do so, there will be no loss of light at the parabolic mirror, meaning we literally eliminated a whole source of loss of light!

Let’s do some basic optics math to understand what are the parameters of this problem. We start assuming the f-number $n_{mirror}$ of the mirror is fixed, since the mirror is usually the most expensive – and less interchangeable – part of the shadowgraph. Then, we just need to select a lens diameter $D_{LENS}$ and focal length $f$ to focus the LED positioned at $d_L$ to a point at $d_I$ from the lens. The LED will be considered a circular source of diameter $D_{LED}$:

Using the thin lens equation, we have the following relationship between the LED distance $d_L$ and the image distance $d_I$:

$\displaystyle \frac{1}{f}=\frac{1}{d_L}+ \frac{1}{d_I}$

I’ll isolate the LED distance for now, as it is an important variable when aligning the lens assembly:

$\displaystyle d_L=\frac{d_I f}{d_I-f}$

We know that the beam downstream of the lens has to have a defined f-number $n_{mirror}$:

$\displaystyle \frac{d_I}{D_{LENS}}= n_{mirror}$

$\displaystyle d_I= n_{mirror} D_{LENS}$

Thus, we get an equation for the LED distance $d_L$ as a function of known variables:

$\displaystyle \boxed{d_L=\frac{ n_{mirror} D_{LENS} f}{ n_{mirror} D_{LENS} -f}}$

Well, we technically only know $n_{mirror}$. In fact, $D_{LENS}$ and $f$ are the lens parameters we’re after. Now let’s look at the illumination by examining the light cone half-angle $\theta$ that the lens collects:

$\displaystyle \boxed{\tan{\theta}=\frac{D_{LENS}-D_{LED}}{2d_L}}$

The cone half-angle is related to the solid angle $\Omega$ by the following relation:

$\displaystyle \Omega=2\pi\bigg(1-\cos{\frac{\theta}{2}}\bigg)$

Well, most LED emitters (chip-on-board) are plane emitters. I am not considering, for simplicity purposes, the ones that have a built-in lens. Those usually have poor optical quality for shadowgraph applications and should be avoided, anyways.

Since the solid angle of a plane emitter is $2\pi$, then we get that the intensity collected by a LED lens is:

$\displaystyle \boxed{\eta=\frac{I_{LENS}}{I_{LED}}=1-\cos{ \frac{\theta}{2} }}$

This is what we want to maximize, which means we want to maximize $\theta$. One might think the obvious way to increase $\theta$ is to increase the lens diameter. But it’s not that simple! Let’s rearrange the first boxed equation to see why:

$\displaystyle d_L=\frac{ n_{mirror} D_{LENS} f}{ n_{mirror} D_{LENS} -f}$

Let’s multiply and divide the right-hand side by $D_{LENS}$:

$\displaystyle d_L=\frac{ n_{mirror} f}{ n_{mirror} -f/D_{LENS}}$

We’ll call $f/D_{LENS}=n_{lens}$, the f-number of the lens.

$\displaystyle d_L=\frac{ n_{mirror} f}{ n_{mirror} - n_{lens} }$

Divide both sides by $D_{LENS}$ again, now:

$\displaystyle \frac{d_L}{D_{LENS}}=\frac{ n_{mirror} f/D_{LENS}}{ n_{mirror} - n_{lens} }$

$\displaystyle \boxed{\frac{d_L}{D_{LENS}}=\frac{ n_{mirror} n_{lens}}{ n_{mirror} - n_{lens} }}$

Which basically means that for a given lens f-number, the LED distance from the lens scales linearly by the constant in the right-hand side. Thus, increasing the lens diameter without changing its f-number will simply move the lens farther from the LED, cancelling out the effect of light collection.

Thus, the lens f-number is really the only relevant variable in determining the light collection efficiency $\eta$ – assuming the lens is reasonably larger than the LED. Let’s plot the trend!

The plot above is for the following parameters: $D_{LENS}=25$ mm, $D_{LED}=5$ mm, $n_{mirror}=5$. Note the efficiency is inversely proportional to the f-number of the lens $n_{lens}$. Also, the spot size at the focal point $D_F$ is also smallest at the lowest f-number. This is great, because both objectives converge. A low f-number lens will accomplish the highest collection efficiency and the smallest spot size, which is precisely what we want for a shadowgraph.

Below, I’m plotting the same trend for a mirror f-number of 10. As we can see, the collection efficiency is weakly dependent on the mirror f-number; which is very likely to be between 5 and 10 for a typical shadowgraph. The spot size is even smaller for the higher f-number mirror, which is even more beneficial.

Unfortunately, the collection efficiency for a single-lens system is really limited to low single-digit percent because lenses with f-numbers of 1 or less are pretty hard to manufacture. Especially achromatic lenses with low spherical aberration, which are required for shadowgraphs with white LED light sources. But the lesson to be learned here is very simple: Just get your lowest f-number lens and make sure the beam f-number is close to the mirror f-number. Higher collection efficiencies might be possible with several lenses, but I haven’t really looked carefully into the math. Coming up, perhaps! In any case, hope this helps a fellow researcher!