#DevOptical Part 29: Apertures as Optical Filters

When it comes to the wonderful world of optical engineering, it is rare to get some consideration for apertures elements. Most people will get ecstatic at the idea of large freeform mirrors for telescope designs, or with diffractive achromatic germanium infrared lenses, but very few will recognize the extreme beauty of apertures and how they make possible complete fields of modern optical instruments! It is not to say the least that I have spent a large amount of my time these last 10 years on apertures – more specifically on crafting and analyzing light using apertures!

The premises of this post can be found in an [»] earlier article that I posted in 2021. Back then, I showed that optical lenses systems with the object and image planes at the front and back focal length of the system had a very peculiar ABCD matrix. In terms of matrix, this can be written without any loss of generality as

where EFFL is the effective focal length of the system, BFL the back focal length of that system and FFL the front focal length of that same system.

Such a system is shown in Figure 1 where I annotated the back and front distances. The double arrow represents here “any imaging-forming device”. This can be a single lens or an assembly of multiple lenses.

As mentioned, this works for any imaging-forming device but is more easily introduced with a thinlens which has the property EFFL=BFL=FFL=f₁. I’m denoting this lens by the subscript 1 because more lenses will be added to the system as we progress into the theory.

Rewriting the previous equations we get

If we now trace a ray (y, u) through this system we get the peculiar inverting property of this system:

The system of Figure 1 transforms an input ray of height y into an output ray of slope u’=-y/f₁. Alternatively, an input ray of slope u will be transformed into an output ray of height y’=f₁*u. I will focus on the former transform first (height to slope), and we will add the second transform into the mix later.

Now, imagine you have a device that allows you to create a uniform field of light but where you can modulate the amplitude of the light at any position in the XY plane. I will note this modulation function A(x,y). Since the paraxial system above is only 1D, we will consider radial coordinates only for now on. The function is now A(r) with r=√(x²+y²). This device, when used in conjunction with the system of Figure 1, allows you to craft lights rays where the output rays have slopes u’=r/f₁ and are modulated according to A(r).

While there are many devices that can produce this light modulation (a computer display being one of them), backlights with apertures are the easiest ones to handle and manufacture in an optical workshop. Using an aperture, you either block the light or let it go through. There are an infinite number of variations of apertures that you can make but I will focus on three of them first:

1/ The bright spot aperture, or regular aperture, which is a hole of radius R in a thin sheet of metal. This aperture has the modulation function

The bright spot aperture produces a cone of light with slopes magnitude comprised in the range [0, R/f] as illustrated in Figure 2.

2/ The dark spot aperture, which is a thin disk of metal of radius R onto a transparent glass substrate (it is difficult to have free floating elements in the real world so it is common to use a transparent substrate). This aperture has the modulation function

The dark spot aperture is therefore the exact opposite of the bright spot aperture. It produces an inverted cone of light with slopes magnitude comprised in the ranges [R/f, +∞]. An illustration is given in Figure 3 although the cone of light should theoretically extend to infinity.

3/ The annular aperture which is a mix of a dark spot aperture of radius R₁ and a bright spot aperture of radius R₂. Its modulation function is

The annular aperture therefore produces a cone of light in the range [R₁/f, R₂/f]. This is illustrated in Figure 4.

In any practical case, the annular aperture is often preferred to the pure dark spot one because it constrains the eventual aberrations of the optical system and avoids stray light which would not have the clean inverting property we are looking for. Please note also that all the raytraces shown above truly are cones of light with 360° revolution around the optical axis.

Concerning manufacturing aspects, there are multiple ways to produce apertures but the two main ones I used were (1) machining apertures in sheet metal, and, (2) using chrome on glass targets.

Chrome on glass targets are excellent when you need precise control of the pattern you want to convert to slopes and also allows you to get uniform patterns in the slope space (more on this below) but it can be difficult to achieve high extinction coefficients for the light modulation function and the optical density of the chrome is often limited to OD3 or OD4. This is fine for most applications but needs to be taken into account at system level. Also, producing chrome on glass targets can be relatively expensive for small batches and take several weeks if not months. As an order of magnitude, producing a single 1” target costs somewhere between 1k€ and 3k€ depending on the technology used but the prices can drop rapidly if you group multiple apertures in a single batch.

Machining offers more material choice (steel, gold etc.) and doesn’t suffer from the optical density issue of chrome on glass targets but there are more restrictions on the patterns you can obtain this time. For instance, it is not possible to have free floating features so patterns like the dark spot aperture or the annular aperture will require branches to maintain the central part connected to the rest of the aperture. These branches introduce both weaknesses in the aperture and break the uniformity of the produced slopes. A tradeoff must be found between uniformity of the slopes and strength of the aperture, in particular its response to vibration and shocks.

Nonetheless, a big advantage of machined apertures is that you can easily prototype them using a CNC router provided the features are not too small or too intricate. I even made a [∞] web application to generate ISO 6983-compliants NC programs for bright spot apertures and annular apertures. Example apertures made using this program are given in Figure 5.

Before we move on to the second part of this post, I need to make a brief comment on the output height of the rays in the system of Figure 1.

Output height is directly related to the input slope at height coordinate r, so, r’=f₁*u with intensity A(r). The illumination extent at the image plane of the system of Figure 1 is therefore limited by how much input slopes we have. Using a diffuser or an integrating sphere, we can generate a relatively homogeneous slope content at each ray position but will be bound anyway to some value. Diffusers are more common but will have Gaussian-like or Lorentzian-like falloffs depending on the grit you choose. Figure 6 lists some typical ground glass performances of Thorlabs diffusers to give an order of magnitude of what can easily be achieved in an optical lab using stock components. About 95% of the emitted energy of the diffusers in Figure 6 occurs within a 40° cone.

Using the exact same system as the one of Figure 1, we can convert input slopes to output heights. The idea is now to put two systems back-to-back such that we generate slopes using the first system and convert these slopes into height positions using the second system. By introducing another aperture at the end of the second system, we produce a filter. This is illustrated in Figure 7 with an annular aperture as source and a regular aperture as filter.

The second system equations can be written as

Looking at the equations above and the schematic in Figure 7, the overall concept might look like we are not doing anything (thinking so would be largely underestimating your guest :)) : starting rays are emitted with modulation A₁(r) before being converted into slopes u’=-r/f₁ by the first lens, and these slopes are converted back to heights r⁽³⁾=f₂*u’ where they are filtered by the second aperture with modulation function A₂(r⁽³⁾). Ray starting from position r therefore ends-up at position r⁽³⁾=-(f₂/f₁)*r with total modulation A₁(r)*A₂(r⁽³⁾). For equal focal lengths (f₁=f₂), the total modulation is then A₁(r)*A₂(r), the product of the two apertures. For binary modulations (apertures can be only “0” or “1”), this corresponds to the AND combination of the two apertures: the light passes only when both apertures are open. In the system of Figure 7, this never occurs because we chose an annular aperture and a bright spot aperture that mutually excludes each other.

You will insist to know what’s the point in the system of Figure 7. This becomes clearer as we insert in the mid-plane separating the two lenses f₁ and f₂ a non-refracting object with a small defect as some position h that will refract light according to a function

where the terms u’+α on the right can be thought of as a Taylor expansion of the refraction law and the Dirac function δ(r-h) reminds us that the defect only occurs at height h.

If we insert this equation between the two lenses we get:

and therefore

Since we theoretically generate any input slopes u, there will be a specific f₁*u value that will end up at height h of the defect. Position r⁽³⁾ is therefore offset by value f₂*α. The consequence is that some light might pass depending on the aperture’s dimensions and the value of α.

If the highly mathematical aspects of these equations trouble you, consider that the two apertures are in conjugated planes. In the absence of perturbating elements, the first aperture is imaged onto the second one. If they are mutually exclusive (e.g. a dark spot larger than its corresponding regular aperture), no light passes through. However, as you introduce a thin prism in the mid-plane between the two lenses, rays are deviated and the image of the first aperture on the second one becomes shifted. If the shift is strong enough, there will be an overlap between the two apertures that will let some of the light pass. Any elements that depart from a perfect wedge will have the same effect but with a distorted aperture image. This is illustrated in Figure 8.

Figure 8 shows the importance of keeping a tolerance margin in the system such that it stays sensitive to defects in the object but not as sensitive as it would let misalignments of elements or lenses aberrations let light pass through.

Last but not least, if you look at the system of Figure 7, you realize that it is making an image of the source aperture which is not very helpful on its own. By adding a third lens f₃ to the system, we can produce an image of the object at the (final) image plane with magnification ratio f₃/f₂. This system is built as a 4f system such that we benefit from its metrological properties (bi-telecentricity as discussed [»] here). This might look like a lot of effort to make a 4f system but recall that lens f₁ acts as an angular source and that the sub-system composed of f₁ and f₂ acts as an angular filter. The overall system therefore has more properties than a traditional 4f system.

This last system is shown in Figure 9 and is known as a dark-field imaging setup. It is made such that no light passes through in the absence of scattering objects but is kept sensitive enough such that any defects (scratches, bacteria, lensplates etc.) will let some light pass. The resulting images therefore has an overall dark background, and all the defects pops up brightly on the image – hence the name “dark-field” in opposition to traditional “bright-field” microscopy where the background is illuminated.

Up to now, I introduced the concept using macroscopic defects that refract rays in a specific direction. But as the defects become smaller, [∞] scattering enters the place. Scattering typically happens when the defect (or object) is on the order of, or smaller than, the wavelength of the light used. With scattering, incoming rays do not refract along a specific direction but many rays are emitted in a lot of directions. For large defects, the rays goes mostly forward but as the object gets smaller and smaller, wider angles are produced. Our system will collect this scattered light and make it visible, even if the object is smaller than the spatial resolution of the imaging instrument! Dark-field imaging is therefore typically useful at revealing extremely small defects that would not appear in traditional (bright-field) imaging.

Dark field imaging is only one of the many arrangements of the generic principle illustrated in Figure 9. Up to now, I limited myself to apertures that could be described by radial modulation functions A(r). The principles developed here can however be generalized to any aperture’s shapes.

Let’s illustrate this with the apertures set of Figure 10. Here, light is emitted by an aperture made of two decentered slits and collected by a round aperture. The decentered slit will produce light rays with slopes along one axis only at many different slopes’ magnitudes (positive and negative), controlled by the length of the slit. The condition to capture light is that the object under analysis should refract light rays along the slit direction only. Light refracted to a different direction will not pass through the filtering aperture and won’t yield an image. Depending on the aperture orientation relative to the object, different features will become visible. By mounting this target on a rotation mount, or by placing the object on a rotation table, we can analyze the object features in terms of refracting directions.

We could continue investigating the different possible apertures combinations, but the list would be infinite. Some combinations are more useful than others, but I keep them for another time. I hope you now share a bit of the passion I have for apertures and that this post might help you invent the next big concept in optical instruments :)

If you would like to discuss this further, don’t hesitate to join our [∞] community board!

I would like to give a big thanks to Young, Sebastian, Alex, Stephen, Lilith, James, Jesse, Jon, Cory, Karel, Sivaraman, Samy, Kausban, Michael, David, Shaun, Themulticaster, Tayyab, Onur, Marcel, Zach, Benjamin, Sunanda, Dennis, M, Natan and Max 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!