Spatial resolution in vector potential photoelectron microscopy R. Browning Citation: Review of Scientific Instruments 85, 033705 (2014); doi: 10.1063/1.4868559 View online: http://dx.doi.org/10.1063/1.4868559 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tuning donut profile for spatial resolution in stimulated emission depletion microscopy Rev. Sci. Instrum. 84, 043701 (2013); 10.1063/1.4799665 The electron spectro-microscopy beamline at National Synchrotron Light Source II: A wide photon energy range, micro-focusing beamline for photoelectron spectro-microscopies Rev. Sci. Instrum. 83, 023102 (2012); 10.1063/1.3681440 Vector potential photoelectron microscopy Rev. Sci. Instrum. 82, 103703 (2011); 10.1063/1.3648135 Large-grazing-angle, multi-image Kirkpatrick–Baez microscope as the front end to a high-resolution streak camera for OMEGA Rev. Sci. Instrum. 74, 5065 (2003); 10.1063/1.1623621 Spatial-domain cavity ringdown from a high-finesse plane Fabry–Perot cavity J. Appl. Phys. 91, 582 (2002); 10.1063/1.1425443
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 033705 (2014)
Spatial resolution in vector potential photoelectron microscopy R. Browning R. Browning Consultants, 1 Barnhart Place, Shoreham, New York 11786, USA
(Received 18 January 2014; accepted 3 March 2014; published online 19 March 2014) The experimental spatial resolution of vector potential photoelectron microscopy is found to be much higher than expected because of the cancellation of one of the expected contributions to the point spread function. We present a new calculation of the spatial resolution with support from finite element ray tracing, and experimental results. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4868559] I. INTRODUCTION
Vector potential photoelectron microscopy (VPPEM) is a new technique. It is only recently that the experimental equipment used in the proof-of-principle demonstration1 has been developed sufficiently to make a good test of the spatial resolution. We find experimentally that the VPPEM spatial resolution is much higher than our previous theoretical estimate. It appears that conservation of angular momentum as the cyclotron electron orbits exit the magnetic field cancels out the expected contribution to image broadening from the off-axis orbital momentum. This cancellation not only removes a large part of the expected broadening, but also implies that the point spread function (PSF) has a high spatial frequency response. The form of the VPPEM PSF is identical to that of magnetic projection lens microscopes.2–4 While the basis of the VPPEM PSF is the same as that of the magnetic projection microscope, the method of forming an image is distinct. In the magnetic projection microscope photoelectrons from a sample are trapped in cyclotron orbits along magnetic field lines in a decreasing field. The photoelectron orbits are collimated by the decrease in field, and magnified by the expanding magnetic field lines. The real projected image is detected within the magnetic field by image detector with a high pass energy filter. In contrast, the VPPEM image exits the vector potential field as an angular image. The VPPEM angular image is energy analyzed, with a band pass analyzer, magnified using electrostatic elements, and projected onto a detector in field free space. II. DISCUSSION OF THE VPPEM METHOD
VPPEM uses the magnetic vector potential field as a spatially resolved reference for image formation. A magnet with a uniform field region, such as a solenoid, is used to create a cylindrically symmetric vector potential field which acts as a two-dimensional reference. The sample to be imaged is immersed in the center of the field, and illuminated by a UV or X-ray photon beam. The resultant photoelectrons are emitted into the vector potential field, and travel down magnetic field lines in cyclotron orbits towards a ferromagnetic shield with an aperture. When the photoelectrons leave the vector potential field by passing through the aperture in the ferromagnetic shield, they gain off-axis momentum in the radial, and azimuthal directions that depends on their position in the 0034-6748/2014/85(3)/033705/4/$30.00
vector potential field at the sample. The momentum distribution gained from the sudden change in the vector potential field at the aperture forms an angular image of the electron emission. The vector potential field is a momentum field with dimensions of momentum per unit charge. When an electron leaves the field suddenly, it gains momentum due to its charge: p = −eA,
(1)
where e is the charge on the electron, A is the vector potential at the aperture, and the off-axis momentum is p. The off-axis gain in momentum creates the angular image. The photoelectrons have chemical information from the sample in the electron energy spectrum, and two-dimensional spatial information from the magnetic vector potential field. The resulting angular electron image can be passed through an electron energy analyzer to produce a photoelectron spectroscopic (PES) image. We can illustrate the mechanics of VPPEM image formation with a simulation of the exit of electron trajectories from a simple magnetic circuit shown in Fig. 1. In this example, we have used Tricomp simulation code from Field Precision Corp. Figure 1 shows a magnet set in a soft iron yoke with a tapered aperture for the exit of electrons from the field. The magnetic axis is rotationally symmetric around the horizontal axis. The volume near the magnet face has a constant magnetic field, and therefore the vector potential field is a radially increasing circular vector field. Figure 1 shows a set of electron trajectories which begin at equally spaced radial points just off the axis near the pole piece. The electron trajectories initially travel along parallel to the optical axis, and are constrained into a narrow beam by the magnetic field lines near the axis. There is a small magnification of the radial distances as the magnetic field weakens moving away from the pole piece. When the electrons exit the field through the aperture, they leave the field lines, and become deflected away from the axis by the change in the vector potential. The deflection produces an angular distribution dependent on the initial radial distribution in the vector potential field. Only the radial distances are plotted in Fig. 1, the electron trajectories are rotated out of the plane of the diagram. The deflection angles agree with Eq. (1) to a few percent, and are slightly dependent on the details of the aperture. The
85, 033705-1
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033705-2
R. Browning
Rev. Sci. Instrum. 85, 033705 (2014)
FIG. 1. Electron trajectories in a magnetic circuit composed of a permanent magnet in a ferromagnetic yoke with a tapered aperture.
simulation shows the deflection angles are proportional to the radial positions at the origin for small, and moderate angles. If the electron trajectories are emitted at to an angle to the magnetic axis they follow cyclotron orbits until they exit the aperture where they are deflected into a final direction. This is illustrated in Fig. 2. Figure 2 shows a simulated trajectory of a 500 eV electron from a 1 T field emitted at 45◦ to the magnetic axis, and at a small distance off-axis in the x direction. In Fig. 2, we are looking down the magnetic axis. We see the cyclotron orbit moves away from the y axis, and expands as the field decreases. As the trajectory exits the field, it undergoes a deflection into its final direction. Clearly, the final direction will depend on exactly where the orbit crosses the aperture as the vector potential varies across the aperture. If we simulate multiple starting points along the axis, the direction changes with the off-axis trajectories forming an angular cone. If we start the simulation of Fig. 2 with a range of off axis angles from 0◦ to 90◦ , and with a weighting for a cosine emission distribution, if there are no other off-axis effects, we would expect to get a cone of confusion equivalent to the disk of confusion
FIG. 3. Schematic of the VPPEM electrostatic focus and energy analysis elements.
from a magnetic projection microscope. This is what we find. By ray tracing over the full range of angles, we can see the VPPEM cone of confusion is simply caused by the variation of the exit point in the vector potential field at the aperture. In the VPPEM, electrons of all energies from a sample are deflected into an angular image by the change in vector potential at the aperture. The angular image can be converted into a monochromatic real image focused at an image plane. Figure 3 illustrates the electrostatic elements needed. Diverging electron trajectories exiting the ferromagnetic aperture are focused into a concentric hemispherical analyzer (CHA) by a condenser lens. After energy analysis by the CHA, which is double focusing, the diverging monochromatic image is focused onto an image plane, and an image detector. It has been the upgrading of the lens system, and the image detector that has allowed us to measure the VPPEM image properties with an order of magnitude greater precision compared with the proof of principle results previously reported.1 In practice, emitted electrons can be accelerated, or decelerated within the magnetic field without distorting the image. Very low energy electrons, with energies below 1 eV, can be imaged by accelerating them off the sample to several tens of electron volts.
III. SPATIAL RESOLUTION
We had previously argued1 that the spatial resolution of the VPPEM technique depends on two different factors from the geometry of the electron cyclotron orbits. These factors are the radius of the cyclotron orbits, and an additional contribution from the off-axis momentum of the cyclotron orbit. Our argument was that the PSF was determined by the combination of these effects. However, that argument was incorrect. As the electron leaves the field line, it still retains angular momentum L, FIG. 2. An electron orbit in the magnetic circuit of Fig. 1 looking down the magnetic axis.
L = r × mv,
(2)
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033705-3
R. Browning
Rev. Sci. Instrum. 85, 033705 (2014)
where r is the orbit radius, m the electron mass, and v the orbital velocity. As the cyclotron orbit unwinds to an infinite radius conservation of angular momentum implies that the off-axis momentum due to the velocity of the electrons in the obit must go to zero. Using a full simulation of the electron orbits, the off-axis motion due to the orbital velocity at the aperture can be shown to go to zero. Although it is difficult to observe in a trajectory such shown in Fig. 2. Then the only contribution to the off-axis momentum in the image is the vector potential at the aperture. Because the distance across the cyclotron orbit is the same as that at a magnetic projection microscope detector, the disk of confusion has the same form as in the magnetic projection microscope, but translated into an angular disk of confusion. The disk of confusion is created by electrons being emitted at all angles up to 90◦ from the magnetic axis. The maximum off-axis distance is twice the maximum radius. The helical trajectories have a maximum radius that depend on the energy of the electrons E, and the magnetic field B, in the following relationship:2 √ 2mE . (3) rmax = eB The probability of being at a distance x from the point of emission can be calculated using the geometry of Fig. 4. The cyclotron orbit is illustrated by the circle of radius r. The point the electron crosses the image plane will be a distance x from the point of emission where x is given by the cosine law: x 2 = 2r 2 − 2r 2 cos(θ ),
(4)
x2 θ = cos−1 1 − 2 . 2r
(5)
During one cyclotron cycle, the probability of being between x and x + dx will depend on the time, t, spent in the interval dx over the cycle. If the cycle time is T, and the velocity in x is dx/dt then, as time is distance/velocity we have the probability P(x): 1 dt 1 dt dθ t = dx = dx. T T dx T dθ dx The angular speed is constant: P (x) =
dθ 2π = . dt T
FIG. 4. Construction to calculate the VPPEM PSF.
(6)
(7)
FIG. 5. Two-dimensional PSF for a VPPEM in units of the maximum cyclotron radius.
Differentiating Eq. (5) to find dθ /dx, and substituting with Eq. (7) into Eq. (6) the probability of an electron crossing the image plane between x and x + dx is: P (x) =
1
x2 2π 1 − 2 4r
1/2 dx.
(8)
Equation (8) is for a single radius r. The single electron radius must be convolved with distribution of emitted radii which we assume is a cosine distribution. This gives the one dimensional PSF. Dividing the result by 2π r, the area of the PSF at a radius r, gives the cross section of the two-dimensional PSF as shown in Fig. 5. While the two-dimensional PSF is sharply peaked at the center, nearly 50% of the integrated intensity is at a radius greater than 1.0 rmax . The spatial resolution is determined by the energy of the electrons at the sample, and the strength of the vector field. We can convolve the PSF of Fig. 5 with a step function to simulate the edge response. A good approximation√ of the edge resolution defined as the 20% to 80% change is 3 E/B μm with E in electron volts, and B in Tesla. Electrons emitted with 1 eV energy in a 2 T field can be imaged with an edge resolution of 1.5 μm. The energy of the electrons can be changed along the trajectories within the field by biasing the sample. This change of energy does not change the spatial resolution which is determined solely by the electron emission energy. The angle of deflection is inversely proportional to the square root of the energy of the electrons when they exit the field. However, the angles of the cyclotron orbits also change in the same way as the electrons are accelerated or decelerated. We can change the electron energy that we image from the sample without having to change the energy of the electrons exiting the vector potential field. This implies that the magnification of the
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033705-4
R. Browning
Rev. Sci. Instrum. 85, 033705 (2014)
600 mesh Au grid. The plot is of the 20%–80% edge resolution versus the square root of the electron energy at the sample in electron volts. The slope of the plot is 2.93 very close to the simulated estimate. IV. CONCLUSION
We find the VPPEM PSF is identical to that of the magnetic projection electron microscope. A good approximation for √ edge resolution of the VPPEM in micrometers equals 3 E/B with E in electron volts and B in Tesla, and this has been experimentally confirmed.
FIG. 6. Experimental measurement of the 20%–80% edge resolution for electron emission energies between 0.5 eV and 50 eV.
system is the same for all energies, and comparing the spatial resolution at different emission energies is straightforward. Using a fixed photon energy of 80 eV, we scanned across the edge of a cleaved Si wafer, and measured the 20%–80% edge resolution for different emitted electron energies. Figure 6 shows the resolution measured across the cleaved Si wafer edge for different energies from 0.5 eV to 50 eV, and a field at the sample of 1 T. The energy resolution is 0.25 eV, and the length scale was established using a
ACKNOWLEDGMENTS
My thanks to Daniel Fischer of NIST for his support for this project. This work has been supported under NIST small business innovation, and research (SBIR) awards SB134112CN0075 and SB134113SU0637. 1 R.
Browning, Rev. Sci. Instrum. 82, 103703 (2011). Beamson, Q. Porter, and D. W. Turner, J. Phys. E: Sci. Instrum. 13, 64–66 (1980). 3 G. Beamson, Q. Porter, and D. W. Turner, Nature (London) 290, 556 (1981). 4 P. Kruit and F. H. Reed, J. Phys. E: Sci. Instrum. 16, 313 (1983). 2 G.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 142.157.129.67 On: Tue, 16 Dec 2014 19:46:35
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 033705 (2014)
Spatial resolution in vector potential photoelectron microscopy R. Browning R. Browning Consultants, 1 Barnhart Place, Shoreham, New York 11786, USA
(Received 18 January 2014; accepted 3 March 2014; published online 19 March 2014) The experimental spatial resolution of vector potential photoelectron microscopy is found to be much higher than expected because of the cancellation of one of the expected contributions to the point spread function. We present a new calculation of the spatial resolution with support from finite element ray tracing, and experimental results. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4868559] I. INTRODUCTION
Vector potential photoelectron microscopy (VPPEM) is a new technique. It is only recently that the experimental equipment used in the proof-of-principle demonstration1 has been developed sufficiently to make a good test of the spatial resolution. We find experimentally that the VPPEM spatial resolution is much higher than our previous theoretical estimate. It appears that conservation of angular momentum as the cyclotron electron orbits exit the magnetic field cancels out the expected contribution to image broadening from the off-axis orbital momentum. This cancellation not only removes a large part of the expected broadening, but also implies that the point spread function (PSF) has a high spatial frequency response. The form of the VPPEM PSF is identical to that of magnetic projection lens microscopes.2–4 While the basis of the VPPEM PSF is the same as that of the magnetic projection microscope, the method of forming an image is distinct. In the magnetic projection microscope photoelectrons from a sample are trapped in cyclotron orbits along magnetic field lines in a decreasing field. The photoelectron orbits are collimated by the decrease in field, and magnified by the expanding magnetic field lines. The real projected image is detected within the magnetic field by image detector with a high pass energy filter. In contrast, the VPPEM image exits the vector potential field as an angular image. The VPPEM angular image is energy analyzed, with a band pass analyzer, magnified using electrostatic elements, and projected onto a detector in field free space. II. DISCUSSION OF THE VPPEM METHOD
VPPEM uses the magnetic vector potential field as a spatially resolved reference for image formation. A magnet with a uniform field region, such as a solenoid, is used to create a cylindrically symmetric vector potential field which acts as a two-dimensional reference. The sample to be imaged is immersed in the center of the field, and illuminated by a UV or X-ray photon beam. The resultant photoelectrons are emitted into the vector potential field, and travel down magnetic field lines in cyclotron orbits towards a ferromagnetic shield with an aperture. When the photoelectrons leave the vector potential field by passing through the aperture in the ferromagnetic shield, they gain off-axis momentum in the radial, and azimuthal directions that depends on their position in the 0034-6748/2014/85(3)/033705/4/$30.00
vector potential field at the sample. The momentum distribution gained from the sudden change in the vector potential field at the aperture forms an angular image of the electron emission. The vector potential field is a momentum field with dimensions of momentum per unit charge. When an electron leaves the field suddenly, it gains momentum due to its charge: p = −eA,
(1)
where e is the charge on the electron, A is the vector potential at the aperture, and the off-axis momentum is p. The off-axis gain in momentum creates the angular image. The photoelectrons have chemical information from the sample in the electron energy spectrum, and two-dimensional spatial information from the magnetic vector potential field. The resulting angular electron image can be passed through an electron energy analyzer to produce a photoelectron spectroscopic (PES) image. We can illustrate the mechanics of VPPEM image formation with a simulation of the exit of electron trajectories from a simple magnetic circuit shown in Fig. 1. In this example, we have used Tricomp simulation code from Field Precision Corp. Figure 1 shows a magnet set in a soft iron yoke with a tapered aperture for the exit of electrons from the field. The magnetic axis is rotationally symmetric around the horizontal axis. The volume near the magnet face has a constant magnetic field, and therefore the vector potential field is a radially increasing circular vector field. Figure 1 shows a set of electron trajectories which begin at equally spaced radial points just off the axis near the pole piece. The electron trajectories initially travel along parallel to the optical axis, and are constrained into a narrow beam by the magnetic field lines near the axis. There is a small magnification of the radial distances as the magnetic field weakens moving away from the pole piece. When the electrons exit the field through the aperture, they leave the field lines, and become deflected away from the axis by the change in the vector potential. The deflection produces an angular distribution dependent on the initial radial distribution in the vector potential field. Only the radial distances are plotted in Fig. 1, the electron trajectories are rotated out of the plane of the diagram. The deflection angles agree with Eq. (1) to a few percent, and are slightly dependent on the details of the aperture. The
85, 033705-1
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033705-2
R. Browning
Rev. Sci. Instrum. 85, 033705 (2014)
FIG. 1. Electron trajectories in a magnetic circuit composed of a permanent magnet in a ferromagnetic yoke with a tapered aperture.
simulation shows the deflection angles are proportional to the radial positions at the origin for small, and moderate angles. If the electron trajectories are emitted at to an angle to the magnetic axis they follow cyclotron orbits until they exit the aperture where they are deflected into a final direction. This is illustrated in Fig. 2. Figure 2 shows a simulated trajectory of a 500 eV electron from a 1 T field emitted at 45◦ to the magnetic axis, and at a small distance off-axis in the x direction. In Fig. 2, we are looking down the magnetic axis. We see the cyclotron orbit moves away from the y axis, and expands as the field decreases. As the trajectory exits the field, it undergoes a deflection into its final direction. Clearly, the final direction will depend on exactly where the orbit crosses the aperture as the vector potential varies across the aperture. If we simulate multiple starting points along the axis, the direction changes with the off-axis trajectories forming an angular cone. If we start the simulation of Fig. 2 with a range of off axis angles from 0◦ to 90◦ , and with a weighting for a cosine emission distribution, if there are no other off-axis effects, we would expect to get a cone of confusion equivalent to the disk of confusion
FIG. 3. Schematic of the VPPEM electrostatic focus and energy analysis elements.
from a magnetic projection microscope. This is what we find. By ray tracing over the full range of angles, we can see the VPPEM cone of confusion is simply caused by the variation of the exit point in the vector potential field at the aperture. In the VPPEM, electrons of all energies from a sample are deflected into an angular image by the change in vector potential at the aperture. The angular image can be converted into a monochromatic real image focused at an image plane. Figure 3 illustrates the electrostatic elements needed. Diverging electron trajectories exiting the ferromagnetic aperture are focused into a concentric hemispherical analyzer (CHA) by a condenser lens. After energy analysis by the CHA, which is double focusing, the diverging monochromatic image is focused onto an image plane, and an image detector. It has been the upgrading of the lens system, and the image detector that has allowed us to measure the VPPEM image properties with an order of magnitude greater precision compared with the proof of principle results previously reported.1 In practice, emitted electrons can be accelerated, or decelerated within the magnetic field without distorting the image. Very low energy electrons, with energies below 1 eV, can be imaged by accelerating them off the sample to several tens of electron volts.
III. SPATIAL RESOLUTION
We had previously argued1 that the spatial resolution of the VPPEM technique depends on two different factors from the geometry of the electron cyclotron orbits. These factors are the radius of the cyclotron orbits, and an additional contribution from the off-axis momentum of the cyclotron orbit. Our argument was that the PSF was determined by the combination of these effects. However, that argument was incorrect. As the electron leaves the field line, it still retains angular momentum L, FIG. 2. An electron orbit in the magnetic circuit of Fig. 1 looking down the magnetic axis.
L = r × mv,
(2)
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033705-3
R. Browning
Rev. Sci. Instrum. 85, 033705 (2014)
where r is the orbit radius, m the electron mass, and v the orbital velocity. As the cyclotron orbit unwinds to an infinite radius conservation of angular momentum implies that the off-axis momentum due to the velocity of the electrons in the obit must go to zero. Using a full simulation of the electron orbits, the off-axis motion due to the orbital velocity at the aperture can be shown to go to zero. Although it is difficult to observe in a trajectory such shown in Fig. 2. Then the only contribution to the off-axis momentum in the image is the vector potential at the aperture. Because the distance across the cyclotron orbit is the same as that at a magnetic projection microscope detector, the disk of confusion has the same form as in the magnetic projection microscope, but translated into an angular disk of confusion. The disk of confusion is created by electrons being emitted at all angles up to 90◦ from the magnetic axis. The maximum off-axis distance is twice the maximum radius. The helical trajectories have a maximum radius that depend on the energy of the electrons E, and the magnetic field B, in the following relationship:2 √ 2mE . (3) rmax = eB The probability of being at a distance x from the point of emission can be calculated using the geometry of Fig. 4. The cyclotron orbit is illustrated by the circle of radius r. The point the electron crosses the image plane will be a distance x from the point of emission where x is given by the cosine law: x 2 = 2r 2 − 2r 2 cos(θ ),
(4)
x2 θ = cos−1 1 − 2 . 2r
(5)
During one cyclotron cycle, the probability of being between x and x + dx will depend on the time, t, spent in the interval dx over the cycle. If the cycle time is T, and the velocity in x is dx/dt then, as time is distance/velocity we have the probability P(x): 1 dt 1 dt dθ t = dx = dx. T T dx T dθ dx The angular speed is constant: P (x) =
dθ 2π = . dt T
FIG. 4. Construction to calculate the VPPEM PSF.
(6)
(7)
FIG. 5. Two-dimensional PSF for a VPPEM in units of the maximum cyclotron radius.
Differentiating Eq. (5) to find dθ /dx, and substituting with Eq. (7) into Eq. (6) the probability of an electron crossing the image plane between x and x + dx is: P (x) =
1
x2 2π 1 − 2 4r
1/2 dx.
(8)
Equation (8) is for a single radius r. The single electron radius must be convolved with distribution of emitted radii which we assume is a cosine distribution. This gives the one dimensional PSF. Dividing the result by 2π r, the area of the PSF at a radius r, gives the cross section of the two-dimensional PSF as shown in Fig. 5. While the two-dimensional PSF is sharply peaked at the center, nearly 50% of the integrated intensity is at a radius greater than 1.0 rmax . The spatial resolution is determined by the energy of the electrons at the sample, and the strength of the vector field. We can convolve the PSF of Fig. 5 with a step function to simulate the edge response. A good approximation√ of the edge resolution defined as the 20% to 80% change is 3 E/B μm with E in electron volts, and B in Tesla. Electrons emitted with 1 eV energy in a 2 T field can be imaged with an edge resolution of 1.5 μm. The energy of the electrons can be changed along the trajectories within the field by biasing the sample. This change of energy does not change the spatial resolution which is determined solely by the electron emission energy. The angle of deflection is inversely proportional to the square root of the energy of the electrons when they exit the field. However, the angles of the cyclotron orbits also change in the same way as the electrons are accelerated or decelerated. We can change the electron energy that we image from the sample without having to change the energy of the electrons exiting the vector potential field. This implies that the magnification of the
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 142.157.129.67 On: Tue, 16 Dec 2014 19:46:35
033705-4
R. Browning
Rev. Sci. Instrum. 85, 033705 (2014)
600 mesh Au grid. The plot is of the 20%–80% edge resolution versus the square root of the electron energy at the sample in electron volts. The slope of the plot is 2.93 very close to the simulated estimate. IV. CONCLUSION
We find the VPPEM PSF is identical to that of the magnetic projection electron microscope. A good approximation for √ edge resolution of the VPPEM in micrometers equals 3 E/B with E in electron volts and B in Tesla, and this has been experimentally confirmed.
FIG. 6. Experimental measurement of the 20%–80% edge resolution for electron emission energies between 0.5 eV and 50 eV.
system is the same for all energies, and comparing the spatial resolution at different emission energies is straightforward. Using a fixed photon energy of 80 eV, we scanned across the edge of a cleaved Si wafer, and measured the 20%–80% edge resolution for different emitted electron energies. Figure 6 shows the resolution measured across the cleaved Si wafer edge for different energies from 0.5 eV to 50 eV, and a field at the sample of 1 T. The energy resolution is 0.25 eV, and the length scale was established using a
ACKNOWLEDGMENTS
My thanks to Daniel Fischer of NIST for his support for this project. This work has been supported under NIST small business innovation, and research (SBIR) awards SB134112CN0075 and SB134113SU0637. 1 R.
Browning, Rev. Sci. Instrum. 82, 103703 (2011). Beamson, Q. Porter, and D. W. Turner, J. Phys. E: Sci. Instrum. 13, 64–66 (1980). 3 G. Beamson, Q. Porter, and D. W. Turner, Nature (London) 290, 556 (1981). 4 P. Kruit and F. H. Reed, J. Phys. E: Sci. Instrum. 16, 313 (1983). 2 G.
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