Notes on the point spread function and resolution for projection lens/corf. 22 April 2009 Dr. Raymond Browning

Transcription

1 Notes on the point spread function and resolution for projection lens/corf Abstract April 009 Dr. Raymond Browning R. Browning Consultants, 4 John Street, Shoreham, NY 786 Tel: (63) This is a calculation of the point spread function (PSF) of an immersion projection lens with a CORF. The PSF has very high spatial frequencies implying that the spatial resolution will always be determined by the signal to noise. Calculation of the Projection Lens/CORF optics response The NIST XPS microscope optics combines a magnetic projection lens with a CORF grid. Neither of these elements has a Gaussian like response. The PSF of the combined optics is very non-gaussian with a sharp fall off at small distances, and a long tail out to a defined maximum. Thus, when we define spatial resolution, we have different alternatives to those used with Gaussian optics. The PSF of the magnetic projection lens is limited by the cyclotron radius of the electrons emitted from the surface. The radius, R of the electron of total energy E emitted at an angle θ to the magnetic field direction is: ( me) / sin( θ ) R () eb Where B is the magnetic field strength, m is the mass of the electron, and e is the charge of the electron. This can be written: ( θ ) / 3.375E sin R () B Where R is in microns, E is in electron volts, and B in Tesla. Page of 9

2 As illustrated in FIG : The photoelectrons are emitted at all angles typically with Lambert s cosine law, and form helical trajectories leaving the surface. Photons Electron trajectories Magnetic field direction Cyclotron orbit diameter Sample FIG The effect of emission angle on cyclotron orbit radius The PSF is determined by the combination of cyclotron electron orbits in all directions around an emission point. R max Image spatial response 4 R max FIG The spatial response of a projection lens In FIG, four circles represent in plane view helical electron trajectories being emitted from a point on a sample. When the helical electron trajectories are projected forward to an image plane, they can intersect the image plane at any point in their cycle. Therefore, Page of 9

3 a disk of confusion is formed. The helical trajectories have a maximum radius that is dependent on the energy of the electrons E, and the magnetic field B, in the following relationship: R max ( me) / (3) eb FIG shows that the maximum spread of the electrons from a single sample point is four times the maximum radius. The probability of being at a distance x from the point of emission can be calculated using the geometry of FIG 3. r x θ r Point of emission FIG 3 The distance x from an emission point with a random angle θ and a radius r The electron can cross the image plane at any point on the circumference of the cyclotron orbit radius r. This point will be a distance x from the point of emission where x is given by the cosine law: x r r cosθ (4) ( cosθ ) x r (5) x θ cos (6) r The angle θ will be randomly distributed from 0 to π. Therefore, the probability, dθ), of being between 0 and θ, will be proportional to: Page 3 of 9

4 θ P ( θ ) (7) π Substituting using equation 6: x) cos π x r (9) Differentiating to find the probability of an electron crossing the image plane between x and x + dx: dx) x x r π ( ) r dx (0) Equation 0 is for a single radius r. For a uniform distribution of radii from 0 to r, any contribution to flux at any x will come from those electrons with radii greater than x/ out to r max. The single electron radius must be convolved with distribution of emitted radii. For a uniform distribution of radii from 0 to r max : dx) rmax x / r x π ( r xdx ) dr () The integral can be integrated numerically and dx) for x between 0 and r max is shown in FIG 4. Page 4 of 9

5 0.75 Probability Distance from origin FIG 4 Probability distribution (arbitrary units) for an electron crossing image plane with distance (units r max ) from origin. This distribution is all the electrons in a circle of diameter 4r max. So that it can be seen that while the distribution is peaked in the center just over half the electrons from a point source are greater than 0.5r max from the origin. When we introduce the CORF grid into the system we remove the electrons that are emitted with large angles to the sample normal. The action of the filter leaves only those electrons which have small energies in the radial direction. This implies that only top of the cosine distribution from Lambert s law will be sampled. This implies the CORF is effectively sampling a uniform distribution for the electrons that can pass. Given a uniform incident distribution, the distribution of radii passed through the CORF will be peaked towards the smaller radii. The CORF grid defines the maximum radius of the cyclotron orbits which by definition from the action of the CORF is the physical radius of the apertures for a grid of circular apertures. The cyclotron electron orbits of radius r, with a maximum radius of the radius of the CORF grid aperture r max, are passed with the probability: Page 5 of 9

6 ( r r) max rmax r) () The integral Equation 4 then becomes: dx, r max ) rmax x / ( r r) max rmax r x π ( r xdx ) dr (3) The result of numerically integrating Equation 6 for x between 0 and r max is shown in FIG Probability Distance from origin FIG 5 Probability distribution (arbitrary units) for a magnetic projection lens and a CORF to the maximum aperture diameter r max. The probability distribution for the combined projection lens/corf is sharper than the projection lens and half the electrons are now within 0.5 r max. The distribution shown in FIG 5 is for all the electrons around in a circle of diameter 4r max. This is a D distribution but the PSF is usually defined as a D distribution, or the Page 6 of 9

7 density function. To get the density function the D distribution is divided by the area of the element dx or by πx. The density function is therefore dramatically peaked in the center Density function Distance from origin FIG 5 Density distribution function of a magnetic projection lens and a CORF out to /0 of the aperture diameter r max. Conventionally, to get the spatial resolution of the instrument we can define either the edge response or the resolving power between two point sources. Typically the edge response is used in microscopy as this can be readily measured. The problem with measuring the point response in microscopy is finding a distribution of small particles with just sufficient area to be detected. However, with a digital scan it is worth defining the pixel to pixel distance in terms of the point response. By defining that two pixels have different information in them then the image is not over magnified and the magnification is not empty. The response to an edge is shown in FIG. Page 7 of 9

8 0.8 Edge response Distance from edge FIG 7 Edge response for projection lens/corf combination. Distance from edge in units of r max. Clearly, the edge response of FIG 7 does not look Gaussian, it is sharper in the center with more extended tails. The resolution defined as the 0%-80% distance is 0.3 r max. The PSF for the projection lens/corf combination is very different to a conventional electron microscope and we believe this is important. The modulation function (the Fourier transform of the PSF) will be very rich in high frequency components. With sufficient signal, high pass filtering of an image can bring out very high resolution detail. This is in contrast to Gaussian optics which often produces artifacts after high pass filtering. This suggests that more information is available in the image, and we should define the image resolution differently. We also need a definition to give us some measure of the pixel to pixel spacing we should use for optimizing the magnification. However, the half height of the PSF is difficult to define as its mathematical height is infinite at the center. This implies a simple definition of a 50% overlap between neighboring points does not work. At this point in time it would appear that the actual resolution will be limited by the signal to noise available rather than the mathematics of Page 8 of 9

9 the PSF. A factor x over the edge resolution does not seem unreasonable for a design rule, in this case it is 0.08D aperture as the definition of minimum useful pixel size. But this could be pushed downwards by a factor x0 with good statistics and image processing. In other classes of electron microscopy, the edge resolution is probably the most useful definition of imaging acuity, but the high resolution response of the Projection Lens/ CORF PSF will be especially valuable on samples with small grain distributions, or repetitive structures, and may thus define its advantage. In these cases the signal from all the structures can be effectively summed to give a high signal to noise ratio in certain frequency bands. To calculate the spatial response at the sample given a response at the CORF, we must divide the response at the sample with the magnification of the projection lens at the CORF. The magnification M is: Bsample M (4) B CORF Where B sample is the field at the sample, and B CORF is the field at the CORF grid. Thus, the spatial resolution S 0-80 is: 0.6 Daperture S 0 80 (5) M And the point spatial resolution, or minimum pixel size D pixel is: D pixel Daperture (6) M Where D aperture R max, and R max is the maximum radius accepted by the CORF grid. Page 9 of 9