![]() In any experimental set up, R would typically be Prepared to tolerate in our phase threads is f =1/10 (a phase error of 36 degrees), then So for example, if we have an object which isġ00nm in size, the electron wavelength is 0.0025nm, and the phase error we are Recording plane) is very large, the diffraction pattern will not appear to move relative Unless we use a lens to form the diffraction pattern, then if R (the distance to the With the illuminating radiation) is moved laterally. Perspective: a Fraunhofer diffraction pattern does not move as the object (together We can formulate the definition of Fraunhofer diffraction from a completely different Other important differences between Fraunhofer and In electron microscopy, the scattering angles are generally small (one to ten degrees),Īnd so we can further approximate that L is about equal to D. The Fraunhofer condition is satisfied if. Sum of adding up all the various complex values of our phase threads will be radicallyĭifferent if the contributions from the extreme edges of the scattering object areĮrror to be much less than λ/2, say fλ, where f is a small fraction. Given by 2πδ/λ, where λ is the wavelength of our radiation. Clearly, this all depends upon how much of a phase error (causedīy the non-parallel threads) we are prepared to tolerate. We generally wish to know how large R has to be in order for the FraunhoferĬondition to apply. We see that (assuming 2R - δ is roughly 2R) Two threads inscribed in a circle of radius R, like this: Then the lengths PO times OQ equals SO times OT. ![]() Remember our elementary geometry, then for any two chords of a circle PQ and ST We see that the lower thread describes an arc of a circle, and the extra path lengthĪdded to it as a result of no longer being parallel with the upper thread is δ. Keeping the one upper stationary, we swing the lower thread upwards, like this: Then what is the effect of the path length change of the lower thread being at aĭifferent angle in the two diagrams? Well, imagine holding onto the two threads, If we suppose that the upper of the thread is at the same angle in the two diagrams, Scattering object, say at a distance R, like this: Now in the Fresnel condition, the threads meet up at a position relatively close to the In this diagram, L = D cosθ (we will use this quantity below). Threads subtending from opposite edges of the scattering object, of width D, at someĪngle, say θ, have a path length difference between them of p, as shown in the We can work out a rough expression for when the Fraunhofer condition applies byĬonsidering when the ‘parallel thread’ approximation breaks down. Validity limits of the Fraunhofer approximation: On the contrary, Fresnel diffraction is the term used whenever we cannot make this 'parallel thread' approximation, in other words when we want to calculate a wave near a source of scattering. Remember, we are imaging the back-focal plane, which by definition is where all parallel beams emerging from the specimen come to a focus: We have an easy way of making a Fraunhofer diffraction pattern in the electron microscope. How small and how large these dimensions are allowed to be depends on the wavelength, which determines the allowable error caused by the threads not being quite parallel. For all threads to be parallel, the object of interest (in the case above, the separation of the slits) must be small and the radius of the hemi-sphere must be large. ![]() The co-ordinates of Frauhofer diffraction are therefore angles (or, more precisely, direction cosines). The pattern we see would exist purely as a function of angle around the hemi-sphere. But if we were to make the hemi-sphere very, very large, then all the threads would be parallel. Well, the threads are not perfectly parallel here. We've already discussed one type of Fraunhofer pattern with our Young's slits experiment. Whenever all the phase threads are effectively parallel to one another, then we refer to the resulting diffraction pattern as a Fraunhofer, or Fourier domain, or far-field diffraction pattern. Our calculus of phase threads is a pretty general principle, but in practice, we often make certain approximations, which are referred to by different names. The Fraunhofer and Fresnel approximations Difference between Fraunhofer and Fresnel diffraction ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |