Lenses For Flow Cytometry

About

Flow cytometers use lasers to illuminate particles confined to the centre of a stream. The beam is focused down to a small spot using lenses, with the effect of increasing the overall intensity of the light for any particle traversing it. The beam width is the diameter at which the laser beam's intensity is 1/e² of the energy found at the centre of the beam when the laser is in TEM₀₀ mode. To find the beam diameter look in the manual, for example a Coherent I90 Argon Ion laser beam diameter tuned to 514.5nm is 1.5 mm.

Figure 1 shows the energy characteristics of a TEM₀₀ laser beam spot.

The coherent light from the laser is considered invariant for a certain distance before diverging. This invariant length is known as the "Rayleigh Range" of the laser beam. After this distance the beam will diverge and it's diameter will become larger, see figure 1.

Figure 2 shows the characteristics of the laser beam.

The focal spot size of the laser beam used in flow cytometry dictates the irradiance of an area that a particle will be exposed to when it traverses the laser interrogation point. If you have 100mW beam with a diameter of 1mm and shrink this beam down by a factor of 10 to 100µm, the equivalent energy density increases 10 fold.

In Flow Cytometry the spot size of the laser beam at the focus depends the diameter of the laser at the back of the focusing lens, the focal length of the lens and the wavelength of the laser light. The bigger the laser diameter at the back of the focusing lens the smaller the spot size. Similarly the shorter the focal length of a lens then the smaller the spot size.

When we analyse particles it's useful to know just how big the beam diameter is in comparison to the particle. For example, if we have a spot size of 50µm and our cells are 100µm, then we will not be able to accurately analyse these cells and so we must switch our focusing lens to give a greater spot size of say 400µm. Similarly it makes sense to improve the illumination of small particles such as chromosomes with smaller beam diameters of 20µm or so.

Use the lens calculator to calculate the spot size of our laser at the laser interrogation point.

Figure 3 shows the lens calculator in action.

The "Depth of Focus" field reported is the "Rayleigh Range" for the focus, in the above example it is 7.953mm (expressed in µm in the app).

Ray tracing

Raytracing is a simple way to understand the light path through optics. A cartoon drawing that obeys reflection and refraction by recursively following the path that light takes as it bounces through an environment in our case through lenses. Because we are speaking of laser systems, then this simple idea of ray tracing becomes realistic. We can steer a laser beam by deflecting it off of mirrors, we can alter the beams diameter by passing the light through a lens or series of lenses. It is useful at this point to introduce a couple of simple lenses to illustrate raytracing and ultimately controlling the size of spot we want at the laser intersection point.

Lens

Figure 4 below illustrates what happens to a ray as it passes through a convex lens. Each ray is a line that takes a specific pathway which is traced through the lens. The ray intacts with the lens following Snells Law which deviates the path of the ray. Pathways through the centre of the lens are unaffected, line 1 in figure 4 illustrates this. All non-centre pathways that are parallel to the optical axis will be deflected to the focal point of the lens, line 2 and 3 in the figure 4. Ray 3 passes through the focal point of the lens before going into the lens and passes out horizontally, parallel to the lens's optical axis.

FIgure 4 shows the path that three differrent rays from an object take through a thin convex lens depending on their souce position.

If we consider laser beams we assume that the light is travelling in parallel rays to the optical axis of the focusing lens. Analogously to an "object" that is positioned so far away (infinity) that the observed light from it is parallel.

Thin Positive Lenses

Convex lenses focus the light down to a focal spot f (figure 5).

Figure 5 the light from parallel rays are deflected inwards to a focal point after passing through the lens.

Thin Negative Lenses

Concave lenses do the opposite from the convex lenses and spread the light out rather than focusing it down to a spot (figure 6). The focus spot f, can be traced to in left hand side of the lens

Figure 6 the light from the focal point is deflected outwards spreading the rays as they pass through the lens.

Beam Expanders

If we need to reduce the spot size at the interrogation point, we can either use a lens with a shorter focal length or use a beam expander to enlarge the beam diameter at the back of the focusing lens.

Beam expanders are used to alter the diameter of the laser spot size by expanding the beam. A beam expander would be placed between the output of the laser at the back of the focusing lens in a flow cytometer. Some cytometers are more ameanable to inserting optics as the have a standard optical bench.

There are two basic types of beam expander Galilean and Keplerian. In the Galilean type the expander is configured with a negative and positive lens. In the Keplerian type, both lenses are positive. The second lens is the collimating lens in both types of beam expander.

Galilean beam expander:

Kaplerian beam expander:

As you can guess, there is a couple of obvious mathematical relationships that are apparent in these devices. One is that the focal lengths of the lenses dictate the size of the optical device so:

where L is the length between the lenses, f1 and f2 are the focal lengths of the lenses. The lens with the shortest focal length is first in the beam expander.

And the other relationship is that of the magnification (M) of the system:

We can use the Simple Lens Calculator to calculate the Beam Diameter needed for a given spot size. Looking at the instrument manuals will give us the spot size of the laser and the focal length of the focusing lens.

Figure 7 shows the Simple Lens Calculator screen ready to calculate the Beam diameter needed for a 30µm focus spot.

Fill in the spot size required, the focal length of the focusing lens leaving the Beam Diameter in mm empty and calculate...

Figure 8 shows the calculation complete. Our beam laser diameter should be approximately 2.5mm go give the 30µm focal spot.

If we want a spot size of 30µm we need to expand our laser beam's diameter. Say our laser beam diameter is 1.25mm - we assume that the beam will be in the Rayliegh range when it hits the focusing lens. We dividing this value into the required beam diameter (2.49mm) we get a magnification of approximately 2. Using equation to calculate M, the magnification, we need to choose one lens at 2 times the focal length of the other. For example, lens 1 has a focal length of at 50 mm and lens 2's focal length is 100mm. This will also give us the dimensions of the beam expander we need, in this case a Kaplerian beam expander, the lenses will be 15 cm apart.

Galilean beam expanders are very compact and are good for compact spaced as they use a negative lens as the first lens reducing the overall length of the device. The benefit of using the Kaplerian beam expander is that we can introduce a diaphragm at the focal point of first lens. The diaphragm is used to block unwanted laser light from the laser resulting in a better spot profile, this is called "Spatial filtering".