Resolution Assessment of a Microscope Based on a Raspberry Pi Camera

Theory of Optics

The resolution can be limited by two independent factors:

• pixel size;

• diffraction effects.

The larger of these determines the actual limitation of the system. In our case we know that the pixel size is 1.4 μm, so we now need to work out the diffraction limit, that is the smallest spot size which can be produced by the lens with the given specs. To calculate this, recall the Rayleigh criterion for a circular aperture: sin⁡θ=1.22 λ ⁄ d (image) Here λ~550nm is the wavelength of light, taking green for the middle of the visible spectrum, d=1.25mm is the diameter of the aperture and θ (small angle) is the angular radius of the spot, that is tan⁡θ=r ⁄ L Here r is the radius of the spot projected at a distance L from the aperture, which in our setup is actually the focal length f of the Raspberry Pi camera lens (and the spot is projected onto the CCD sensor). From first approximation for a small angle: sin⁡θ≈tan⁡θ 1.22 λ ⁄ d=r ⁄ f Rearranging this equation and plugging in the numbers gives the following diameter of the smallest resolvable spot: 2r≈3.8μm This is almost three times the size of the pixel on the CCD, which imposes the actual limit on the resolution. The pixels of the CCD outresolve the theoretical lens limits.

Compare this with a typical size of a chloroplast: 5-8μm diameter [1]. Our resolution will be just enough to image them, which is exactly what we have managed to do on this picture of Spirogyra cells. Note that these are larger than typical chloroplasts though. To obtain a better resolution, a lens with either larger aperture and/or shorter focal distance can be used, without the need of a better CCD. However, this is a tradeoff in terms of worse aberration and contrast. An improvement to the resolution will however be required in order to image bacteria, for example, which are of the order of 1μm in diameter [2].

[1] Wise, R. and Hoober, J. (2006). The structure and function of plastids. Dordrecht: Springer.
[2] Encyclopedia Britannica, (2015). bacteria :: Diversity of structure of bacteria. [online] [Accessed 30 Jul. 2015].

Inverting the Lens: Why and How:

The way a camera works is by focusing an image of a distant large object as a small set of points onto the CCD, which is positioned close to the lens (in its focal plane). Theoretically however, it might as well do the opposite (because light paths are reversible – a well known and intuitive physical principle): that is, inspect the CCD pixels and project their greatly enlarged image onto a distant screen.
The lens has a small aperture (1.25mm) at one end, and a larger one (4mm) offering a wider view angle at the other, which is required for viewing close up objects. This is normally oriented towards the CCD.

So, if we want to image a sample on a microscope slide, we need to:

• position the lens close to the sample – at roughly the focal distance, that is 3.6mm;

• orient the lens with its larger aperture towards the sample;

• position the sensor behind the lens, now at a much larger distance (roughly 2.8cm) – for this we have designed a special camera mount.

Now we have the Raspberry Pi camera working as a microscope!

The problem: how to unscrew the lens from the camera The Raspberry Pi camera is sold with the lens screwed (and lightly glued) into the sensor casing. To unscrew the lens, you will need the right type of pliers: ideally with ridged surface. Grip the top part of the plastic casing of the lens firmly, being careful not to crush it, and rotate counterclockwise. After the first small rotation, you should be able to unscrew it fully manually. ATTENTION: might not work from the first time! Do not try to cut out the lens or force it out in any other way.