Brightness calculation formula for Kohler illumination beam Open Access

The Langmuir limit, defined by Langmuir, was first obtained from¹ 

where A is the emission area, r is the distance from A, and Θ is the angle between r and the optical axis. This equation was derived from Lambert’s law. We showed² that for a convex cathode or curved trajectories, beams with brightness higher than the Langmuir limit were obtainable. Measured brightness varied dramatically when we varied the excitation in the first lens of a system consisting of an electron gun and two magnetic lenses. The ratios of maximum to minimum brightness were measured as 717, 403, and 310 for beam energies of 10 keV, 20 keV, and 30 keV, respectively. These results indicate that the brightness was not conserved when lens excitation was varied. The maximum brightness values for beam energies of 10 keV, 20 keV, and 30 keV were observed to be, respectively, 143, 208, and 92.1 times higher than the Langmuir limit, and these experiments were performed using a cathode with a large curvature. Furthermore, there was no explanation regarding why the high brightness was obtained.

The beam current variation as a function of the excitation corresponding to the first lens variation showed very steep variation. Furthermore, the variation in brightness B may be caused by the variation in beam current I_b. To explain this steep variation, the radial intensity distribution of the source image at the numerical aperture (NA) of the lens was assumed as a double Gaussian distribution.² However, this assumption cannot fully explain the I_b variation.

Figure 1 displays the electron optics for explaining why the obtained brightness is higher than the Langmuir limit. For a particular first lens excitation, after adjusting the final lens, two focused beams are generated: one is the crossover beam and the other is the image of the beam at the first lens. When the crossover image of the first lens is focused near the principal plane of the final lens, the latter beam becomes similar to a Kohler illumination beam; thus, this latter beam is referred to as a Kohler illumination beam in this paper for simplicity.

The blue lines represent the beam trajectories for the crossover image. The image of the source is formed at the NA and at the target by the final lens. The beam radius at the target is denoted by R_co. The semi-angle (θ) is determined by the blue line from R_b, as shown in Fig. 1. The broken line determines the semi-angle κ. Particularly, the broken line is formed from the center of the first lens and the NA edge, and it extends to the source and final lens. The point of intersection of the broken line and the principal plane of the final lens is represented as i. A critical point C is defined as the point between the optical axis and the line connecting i and R_b, as depicted in Fig. 1. When b is greater than C or is negative, κ becomes greater than θ, and, then, the beam semi-angle (α) is decided only by κ.

For each condition of b, two images, i.e., the crossover image and the Kohler illumination image, can be obtained by adjusting the final lens excitation. The terms I_b and α remain the same for both the images; therefore, the ratio of the brightness of the first beam (B_co) to that of the other beam, (B/B_co) is studied as follows: B was defined such that I_b was divided by the beam area [π(ϕ/2)²] and the beam solid angle (π sin² α).

When α ≪ 1, sin² α = α²; therefore,² 

and

where B_co and ϕ_co are the brightness and beam diameter for the crossover image, respectively, and ϕ and ϕ_co are the beam sizes for the same first lens excitation. From Eq. (3), we obtain

When b is sufficiently low, the purple lines and radius R_s are determined as shown in Fig. 1. For the other beam, the radius of the object is not R_b but R_s because the beam from the outer part of R_s is captured by the NA. Notably, R_s is again reduced by the final lens and becomes smaller than the resolution limit of the final lens for the small-b case. Therefore, only the crossover image is observed. When b is increased, α remains nearly constant, while the beam size, E, and beam current increase so that B_co is conserved for the crossover image. The term B_co is constant for all the first lens excitations, and its value is the brightness measured for a sufficiently large first lens excitation. When b reaches the principal plane of the final lens position, ϕ_co and B approach infinity. However, because the crossover image entails a spherical aberration, each beam with varied semi-angles focuses at different positions. Because B denotes the mean value of the beam corresponding to different semi-angles, it may not approach infinity.

From Eq. (2), the following equation is obtained:

Accordingly,

For each Kohler illumination image, the distance between the object and the final lens and that between the final lens and the target are equal, respectively. However, each beam size at the target shows a different value. The red, orange, green, and blue lines indicate the beam radius for the cases wherein the crossover image is formed at the NA, at the principal plane of the final lens, the case wherein the first lens forms parallel beams, and wherein the crossover image is formed at the target, respectively. Using the real optical length and the thin lens equation, these radii can be calculated, and the results are R_co ≫ R_N > R_K > R_p.

B measurements were performed using a JEOL JSM-5400 scanning electron microscope (SEM). The calculation methods for the relation between the lens excitation and b are the same as those used by Nakasuji et al.² and only the results are depicted in Figs. 2 and 3.

Figure 4 is an SEM image of the cathode used in this experiment, where the cathode radius of curvature and cone angle were 30 µm and 60°, respectively. The cathode tip was welded to a tungsten hairpin, as shown in Fig. 4. Both the cathode and tungsten-filament diameters were 200 µm. A simulation model with this cathode tip for calculating J_c was designed, where the distances from the cathode to the anode and to the Wehnelt were 10 mm and 0 mm, respectively.

When the beam energy and emission current were 20 keV and 54 µA, respectively, the cathode temperature was 3000 K. The work function of the tungsten filament was 4.46 eV, and the simulated J_c was 16.7 A/cm². The Langmuir limit is calculated as 4.11 × 10⁵ A/cm² sr. The Langmuir limit values are shown in Fig. 5 as lines that run parallel to the x-axis.

The measurements were performed using an emission current (I_e) of 54 µA. For the B measurement in Eq. (3), I_b and ϕ were measured directly for each lens current. ϕ was defined as the distance between the readings³ corresponding to 12% and 88%⁴ of the secondary electron (SE) integrated intensity distribution during the scan at a knife edge made of silicon. I_b and ϕ were measured as functions of the first lens current (I_l). Each measured result for I_b and ϕ, at a beam energy of 20 keV, is shown in Fig. 5 as blue and green curves, respectively, where I_l changes with the lens excitation.² Using each lens excitation, b was determined from Fig. 2 for lens excitations greater than 214 AT. Additionally, −b was determined from Fig. 3 for excitations less than 214 AT.

The calculation method of α could be referred to from a previous report.² The result is shown as a green curve in Fig. 5. B was calculated using Eq. (2). The B/10⁵ (A/cm² sr) values are shown in Fig. 5. A maximum value of B = 2.05 × 10⁸ A/cm² sr was observed. This value is 499 times higher than the Langmuir limit of 4.11 × 10⁵ A/cm² sr, indicating that it is possible to surpass the Langmuir limit. We observed the maximum and minimum B ratios to be 2050/3.6 = 569.

In the case where the cathode is a tungsten filament, the same measurement was performed. The results are shown in Fig. 6. Although the cathode was not axially symmetric, a maximum-to-minimum brightness ratio of 200:1 was observed.

As stated in the theory, ϕ is the crossover image when the excitation is sufficiently large, i.e., for minimum b. The source size (Φ_g) is reduced by the first and final lenses. Φ_g can be calculated² as follows by referring to Fig. 1,

When b and ϕ are 11.2 mm and 0.151 µm, respectively, Φ_g was 15.7 µm. Using this value, ϕ_co is calculated² as follows:

ϕ_co as a function of the excitation is shown as the purple curve in Fig. 5. A comparison between the purple curves revealed that ϕ_co was always 1–20.3 times greater than ϕ. The pink curve (ϕ_co/ϕ)² is similar to the brightness curve. The red curve, which is the measured B, and the light blue curve, which corresponds to B, calculated using Eq. (4), are similar. Therefore, Eq. (4) was validated.

As given in Fig. 5, when the excitation of the first lens is maximum, the measured brightness is 5 × 10⁵ A/cm² sr. Therefore, B_co is 5 × 10⁵ A/cm² sr. When we varied the magnetic lens excitation from the minimum to a larger value, the crossover image changed into the Kohler illumination image, and the brightness increased to its maximum value. On achieving this condition, the brightness was reduced because (ϕ_co/ϕ)² decreased.

In Fig. 5, near the Kohler illumination condition, ϕ is slightly decreasing as a function of the lens excitation. This result is the same as that obtained theoretically, i.e., R_N > R_K > R_p, where R_N, R_K, and R_p are the beam radii for the cases where the crossover image is formed at the NA, the Kohler illumination condition, and the case where the first lens forms a parallel beam, respectively.

The same comparison was made for the tungsten filament cathode case. The results are shown in Fig. 6. The purple curve is given by (ϕ_co/ϕ). The Kohler illumination image is the reduced beam of the beam at the first lens. The beams at the first lens may be nearly equal to the cathode image, i.e., the tungsten filament image. The resultant image is not circular but elliptical. Upon varying the lens excitation, the elliptical beam rotated. Because of the beam rotation, the angle between the knife edge direction and the longer axis of the ellipse was validated, and the measured value of ϕ had a small ripple. Therefore, the pink curve is not smooth. The first lens excitation, which gives the maximum value of B, and that which gives the maximum (ϕ_co/ϕ)² are not same, as depicted in Fig. 6. However, the red curve and the pale curve showed a similar profile, and, thus, Eq. (4) was confirmed.

Regarding the reason for the sharp distribution profile of I_b, we stated that² the beam intensity distribution was assumed to be double Gaussian. Unfortunately, this assumption cannot fully explain the I_b. The real reason was revealed from a comparison between the blue and orange curves in Fig. 6. The former curve corresponds to the measured I_b, and the latter one is calculated from Eq. (5) for B_co of 3.6 × 10⁴ A/cm² sr. Both curves coincided for the full measured range. Therefore, Eq. (5) was confirmed. A small discrepancy was caused because the NA was not positioned at the principal plane of the final lens. The latter curve exhibited its maxima at the Kohler illumination condition, and the former curve shifted toward the condition wherein the crossover image was formed at the NA.

As the Langmuir limit was surpassed, we believe that the high-current-density cathodes⁵ may be replaced by large-area cathodes⁶ in high-brightness electron guns, as the beam from such a cathode exhibits a fine crossover. Furthermore, Pierce guns⁷ may be redesigned based on these results.

In a two-lens optical system, there are two beams: a crossover beam and a Kohler illumination beam. Upon varying the first lens excitation from large to small values, the crossover beam changed to a Kohler illumination beam with increased brightness. We derive new calculation formulas for the Kohler illumination beam. The brightness B and beam current I_b can be expressed as

where B_co, ϕ_co, ϕ, and α denote the crossover beam brightness, crossover beam size, Kohler illumination beam size, and beam semi-angle, respectively; additionally, ϕ_co and ϕ are formed at the same first lens excitation. These two equations were experimentally proven. We obtained a brightness of 2.05 × 10⁸ A/cm² sr using a beam energy and an emission current of 20 keV and 54 µA, respectively. This obtained brightness surpasses the Langmuir limit of 4.11 × 10⁵ A/cm² sr by 499 times. As the Langmuir limit was overcome, we believe that high-brightness electron guns may employ large-area cathodes instead of high-current-density cathodes, when beams from the former exhibit a fine crossover. Thus, Pierce guns may also be redesigned.

The authors wish to thank Mr. Syuuichi Goto for access to his measurement system; Mr. Masanori Watanabe, Mr. Shinji Nakashima, and Mr. Keiya Yamagishi for their assembly of the cathode; Mr. Tukasa Watanabe for his calculation of the definite double integration; and Editage (www.editage.com) for English language editing.

The data that support the findings of this study are available from the corresponding author upon reasonable request.