Notice the primary, secondary and tertiary wavefronts generated by the Airy disc. This is known as an Airy disc and is represented below. Instead, the image when viewed critically consists of a disc composed of concentric circles with diminishing intensity. Second, using even a "perfect" optical system, a point of light cannot be focused as a perfect dot. Notice that this causes the parallel wavefront to emerge from the aperture as a spherical wavefront.Īiry disc. Below is an example of how diffraction changes the wavefront in the presence of a small aperture. Diffraction results when a wavefront is impeded by any object, and of course the edge of the lens area constitutes an object, as does any superimposed aperture. First, it is impossible to achieve absolute focus using any optical system that uses particles with wave-like properties, because of diffraction and interference. It is desirable to understand several of the fundamental principles of light optics in order to understand the limitations of electron microscopy.ĭiffraction. Limits to Resolution in the Transmission Electron Microscope Two positive lenses can also be used in a Keplerian beam expander design, but this configuration is longer than the Galilean design.Limits to Resolution in the Electron Microscope This style of beam expander is called Galilean. To further reduce aberrations, only the central portion of the lens should be illuminated, so choosing oversized lenses is often a good idea. Θ 3 = θ 1|-f 1|/f 2 = (0.65 mrad)|-25 mm|/250 mm = 0.065 mradįor minimal aberrations, it is best to use a plano-concave lens for the negative lens and a plano-convex lens for the positive lens with the plano surfaces facing each other. Since real lenses differ in some degree from thin lenses, the spacing between the pair of lenses is actually the sum of the back focal lengths BFL 1 + BFL 2 = -26.64 mm + 247.61 mm = 220.97 mm. To expand this beam ten times while reducing the divergence by a factor of ten, we could select a plano-concave lens KPC043 with f 1 = -25 mm and a plano-convex lens KPX109 with f 2 = 250 mm. Note that these are beam diameter and full divergence, so in the notation of our figure, y 1 = 0.315 mm and θ 1 = 0.65 mrad. So, to expand a laser beam by a factor of five we would select two lenses whose focal lengths differ by a factor of five, and the divergence angle of the expanded beam would be 1/5th the original divergence angle.Īs an example, consider a Newport R-31005 HeNe Laser with beam diameter 0.63 mm and a divergence of 1.3 mrad. Is reduced from the original divergence by a factor that is equal to the ratio of the focal lengths |-f 1|/f 2. The divergence angle of the resulting expanded beam Or the ratio of the focal lengths of the lenses. In addition, diffraction may limit the spot to an even larger size (see Gaussian Beam Optics), but we are ignoring wave optics and only considering ray optics here. If this is not possible because of a limitation in the geometry of the optical system, then this spot size is the smallest that could be achieved. The only way to make the spot size smaller is to use a lens of shorter focal length or expand the beam. No improvement of the lens can yield any improvement in the spot size. We have already assumed a perfect, aberration-free lens. This is a fundamental limitation on the minimum size of the focused spot in this application. So, the diameter of the spot will be 33 µm. Thus, at the focused spot, we have a radius θ 1f = 16.5 µm. The KPX043 lens has a focal length of 25.4 mm. This Hene laser has a beam diameter of 0.63 mm and a divergence of 1.3 mrad. As a numerical example, let’s look at the case of the output from a Newport R-31005 HeNe laser focused to a spot using a KPX043 Plano-Convex Lens.
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