Astigmatism exists in a focused-ion-beam (FIB) system and causes the shape of a beam spot to change from a normal circle to an ellipse. This variation influences the fabrication of diffractive structures by use of programmable controlled milling of a FIB. It is analyzed combined with the fabrication of blazed gratings and Fresnel diffractive lenses. Fabrication errors caused by a beam spot with astigmatism is discussed in detail for four cases of the long axis of an ellipse (a) in accordance with the X axis, (b) in accordance with the Y axis, (c) at 45° with the X axis, and (d) at -45° with the X axis. Finally, a method is given for correction of the astigmatism and how to determine the circularity of the beam spot qualitatively.
© 2004 Optical Society of America
Focused-ion-beam (FIB) milling is one of the practical approaches for fabrication of diffractive lenses and gratings [1–6]. Compared with other conventional techniques, e.g., laser direct writing and electron-beam direct lithography, it has the advantages of one-step fabrication, any pattern transfer steps are unnecessary, and substrate material selectivity is not required. Focusing an ion beam in an ion column is similar to the principles of geometrical optics. The lenses in an ion column consist of electrodes instead of conventional glass optical lenses and also have image aberrations. Therefore, the theory of image aberration is still applicable to the focusing of an ion beam. Many aberrations, such as spherical aberration, chromatic aberration, coma, and astigmatism, still exist during focusing. For a commercial FIB machine, the former three kinds of aberration are fixed after the machine is ready for use and barely need to be changed by the operators. We assume that the aberrations are small enough so that we can neglect their influence in our discussion. The important factor during fabrication of diffractive structures is astigmatism, which does affect the quality of fabricated structures. Astigmatism causes nonuniformity of the relief depth in diffractive structures and degrades optical properties such as diffractive efficiency. This influence is discussed in detail in Section 3. In Section 4 we discuss a qualitative method for correction of astigmatism and how to determine the circularity of a beam spot qualitatively.
2. Focused-ion-beam milling experimental setup
We conducted experiments for fabrication of diffractive structures by using our FIB machine (Micrion 9500EX) integrated with an ion column with an ion source of liquid gallium, a scanning electron microscope (SEM), energy dispersion x-ray spectrometer facilities, and gas-assisted etching functions. This machine uses a focused Ga+ ion beam with 50-keV energy, a probe current ranging from 4 pA to 19.7 nA, and a beam-limiting aperture size ranging from 25 to 350 µm. For the smallest beam currents, the beam can be focused to as small as 7-nm diameter full width at half-maximum (FWHM). The milling process is performed by program control, that is, by means of varying the ion dose for different relief depths. The ion column uses two focusing lenses and octapole deflection assemblies that can be used to achieve scanning to a resolution as small as several nanometers. The stage is commercially supplied with a motion range of ±100 mm at the X and Y axes and ±1-µm positioning accuracy. The stage can be rotated to 360° and tilted to a maximum angle of 60°. Loading is done through a vacuum load lock, and the load chamber, work chamber, and ion gun column are all isolated by pneumatic valves.
The diffractive structures can be FIB milled by use of its programming function [1–5]. A computer program can be written (by use of the machine-provided commands instead of the commonly used computer languages) in terms of the designed parameters and discrete data. Then the program can run in a defined milling area. The manufacturing procedures are as follows:
• Discretizing the relief curves according to a select ion-beam spot size and feature size of the diffractive lenses.
• Selecting suitable ion energy (ion dose) and dwell time in terms of discrete relief height and curvature of a spherical lens so as to determine the total milling depth for each step [see Figs. 1(a) and 1(b)].
• Programming and optimizing the parameters.
• Milling the relief microstructures by running the designed computer programs.
• Measuring the accuracy of the relief profile and form by interferometers or atomic force microscopy (AFM).
• Revising machining parameters and recycling the above procedures until fabrication accuracy of the diffractive lenses reaches the appropriate design.
The FIB machine mills the circles (straight lines for a blazed grating) with different radii according to the discrete data in step 2 above, one by one sequentially (from edge to center).
We used these discrete circles with small 0.5-µm interval steps to approach the continuous relief. To reduce the effect of variations in the interscan distance and other tolerances of the scanning stages, a large spot size is desired. But a large beam spot limits the resolution of the writing process. Mathematically, the profile obtained by the writing process is given by the convolution of the input surface relief with a point-spread function of the spot size. The convolution causes round corners at both the top and the bottom of the relief .
3. Influence of astigmatism on fabrication of diffractive structures
When astigmatism exists, the shape of the beam spot changes from a circle to an ellipse, as shown in Figs. 2(a), 2(b), and 2(c). There are four different orientations of the ellipse that are due to the imbalance of electrostatic forces that originate from octopoles: the long axis of the ellipse (a) in accordance with the X axis, (b) in accordance with the Yaxis, (c) at 45° with the X axis, and (d) at -45° with the X axis, respectively, as shown in Figs. 3(a), 3(b), 3(c), and 3(d). The beam spot sizes are 2a and 2b at FWHM for both the long and the short axes, respectively. From a theoretical point of view, the astigmatism can be written as 
where B is the magnetic field, q the velocity of an ion of charge, M is the mass, V is the accelerating voltage, and Z is the mass selection size. For a commercial FIB machine, the Z, B, and M are fixed. The q is determined by accelerating voltage V, which can be changed by the operator in the setting window of the process parameters.
It was caused by the nonuniform fringe fields at the entrance and exit of the analyzer, as well as more or less fluctuation of the beam current. Some degree of astigmatism is also introduced by the act of filtering itself, so that the beam of selected ions is stretched asymmetrically because of the spread in energies.
The astigmatism causes depth error during FIB milling. A milling error can be interpreted as a wavelength error. If depth error ε is small, the new design wavelength is approximately equal to d(n -1), and the diffraction efficiency is approximately equal to
If the error is 0.2%, the diffractive efficiency of the diffractive optical elements with continuous relief can be degraded from 100% (in scalar theory for the period of the diffractive structures, p≥5λ, where λ is the working wavelength) to 87.5%.
To have a uniformly scanned ion flux in channel milling, the normal pixel spacing (p x/σ, and p y/σ) should be equal to or less than 1.5, where σ is the standard deviation of the Gaussian distribution, px and py are pixel spaces in the X and Y directions, respectively, and for the beam without stigmatism, σ=σx=σy. The scanned ion flux is steady and unwavering when the normal pixel spacing is less than this value . Figures 4(a) and 4(b) and 5(a), 5(b), and 5(c) show simulation results of FIB milling with different pixel spaces in the X and Y directions. It is obvious that the top surface is smooth for a pixel spacing of less than 1.5. But the premise is that the beam spot is circular (without astigmatism). For a beam with astigmatism, the milling results will be similar to the cases shown in Figs. 4(b) and 5(a). For a clear statement, we discuss the fabrication errors for blazed gratings and Fresnel diffractive lenses separately.
3.1 Blazed grating for dispersion
If a beam spot changes from a circle to an ellipse, the spot resolution differs in the directions of the long axis (low resolution) and the short axis (high resolution). The change causes redistribution of the substrate material because of the variation in resolutions in both directions. The spot with higher resolution produces a larger depth and vice versa, but this does not mean that the largest depth can be obtained by choosing the smallest spot size (achieved by choosing the smallest aperture size) because beam current degrades significantly as well. The milling depth is relevant to the aspect ratio of the microstructures, which varies for different spot sizes and beam currents. Smaller exposure pixel sizes unnecessarily increase only the amount of data to be handled but significantly increase the milling time, which increases exponentially with a decrease in pixel sizes. If the long axis of the elliptical spot coincides with the Y axis [as in Fig. 3(b)], it causes a relative increase in the grating depth, as shown in Fig. 6(a). Similarly, when the long axis of the elliptical spot coincides with the X axis [as in Fig. 3(a)], the depth decreases, as shown in Fig. 6(b). The designed depth of the grating is 633 nm. Figure 6 shows that the depth is still less than the designed value because of the redeposition effect during the milling process. The depth can be increased by tilting the stage to increase the incident angle of the ion beam. For the beam shape shown in Figs. 3(c) and 3(d), the roughness of the relief surface degrades because of the increase in pixel spacing.
3.2 Fresnel diffractive lenses for focusing or collimating
We fabricated diffractive lenses on diamond thin film that was coated on a 1.5-µm-thick substrate of Si (100). We used a 0.54-µm3/s milling rate with a raster scan. The depth variation in the lenses was caused by circular symmetric structures. The relief depth differs in different directions. If the long axis of an elliptical spot coincides with the X axis [as in Fig. 3 (a)], it would cause the grating depth to decrease in the horizontal direction and to increase in the vertical direction, as shown in Fig. 7. Similarly, the depth would increase and decrease in the -45° and 45° directions with the X axis, respectively, for the beam shape shown in Fig. 3 (c) and vice versa for the case in Fig. 3(d). Figure 8 is an example of the fabrication mold for diffractive lenses. The variation rule is the same as that for the lens shown in Fig. 7. We paid particular attention to the milling depth at central areas of both diffractive lenses and observed that its mold was not affected by astigmatism. The correct depth was reached with or without astigmatism.
It is obvious that the milling depth is affected by the aspect ratio of the microstructures. The aspect ratio degrades rapidly when the beam currents or spot sizes increase because the influence of the redeposition effect of the sputtered materials is stronger than that of the structures with a larger depth. The sputtered materials from the bottom of the grooves redeposited onto a sidewall at height h above the bottom is given by Eq. (3) 
where F 0 is the total sputtered material from the bottom.
Figure 9 shows the aspect ratio of the ion flux reached versus normalized redeposited material onto the sidewall. It can be seen that the more volume of redeposited material, the smaller the aspect ratio and vice versa. For our FIB machine, the maximum aspect ratio it can reach is 5:1 with pure milling and 10:1 with chemical gas-assisted etching. Therefore, for the beam spot with astigmatism, the smaller spot size on the short axis corresponds to a higher aspect ratio than that of the spot size on the long axis. However, the same beam current was used during the whole milling process, which means that the aspect ratio should be the same for the depth relief in the diffractive structures. Lateral dimension d at the central area is larger than the other dimensions and so corresponds to a larger depth for the same aspect ratio because the redeposition effect has less influence.
This is why the milling depth is not sensitive to the variation of spot sizes in both the long and the short axes for milling at the central area. It could also be the reason for the unchanged milling depth at the central area for the fabrication of both the diffractive lens and its mold.
4. Correction of astigmatism
The FIB machine has the capability to correct astigmatism, which is the same for other commercial FIB machines such as the FEI Quanta 200 3D. In practice, this is sufficiently small to be removed by stigmators, also known as upper and lower octopoles and located above the assembly of lens1 and below lens2 in the ion column, that correct beams for astigmatism and whose cross section is elliptical instead of circular. This cross section, known as astigmatism, is necessary because an astigmatic beam cannot be finely focused. The stigmator consists of eight radially oriented electrostatic deflection elements. By applying combinations of voltages to these deflectors, asymmetrical forces can be exerted on the beam to change its cross-sectional shape. The beam shape becomes circular when stigmating forces are exerted by the octoples. By changing the polarity and magnitude of the set of voltages applied to opposite poles of the stigmator, the beam can be pushed inward or pulled outward. Minor differences in the forces exerted by a given voltage on opposite poles of the stigmator (because of manufacturing tolerances) can be corrected by the introduction of voltage offsets (it might differ for focusing with different beam currents) through a procedure known as quad balance calibration. Unfortunately, the voltage offsets cannot be displayed on the screen. This also applies to the FIB machine with the FEI Quanta 200 3D. Assuming the value of the voltage offset can be displayed, it should differ for the different aperture sizes and beam currents. We can check only the shape of the beam spot qualitatively by the FIB images with a single scan or AFM images after FIB milling with a single scan, instead of quantitatively to determine the current status of the beam spot. It is also necessary to do the quad balance calibration after the machine is restarted or change the setting of the beam current at any time.
The key issue before fabrication is how to judge the quality of the beam spot by the operator after quad balance calibration. Our method is to check the beam shape when a bombed mark remains on the substrate surface that was bombed by a single scan in a defined area before fabrication. In other words, the beam shape is reflected by the bombed mark imprinted on the substrate surface at the location of each separate pixel. First, an area with a length and width of L×W was defined, as shown in Fig. 10. Then the pixel space settings in both the X and the Y directions were changed from a normal overlap of between 50% and 60% to 0, and the pixel spaces increased to ten times their normal value. The dwell time setting also changed from a previous normal value of 3 µs to the current value of 50 ms. Each beam spot scanned independently in its pixel location without overlap. After a single scan, the quality of the beam spot can be checked with the FIB images (see the insets). Figure 11 is a flow chart that shows the procedures to correct astigmatism. As can be seen in Figs. 12(a)–12(d), we enlarged the images by using an AFM. It can be seen from the images that the shape of the beam spot is apparently imprinted The astigmatism existed for the images in Figs. 11(a)–11(c) but do not exist when the shape of the beam spot is adjusted to be circular. In this case, the mark imprinted by the beam spot also looks circular, as is obvious in Fig. 11(d).
Figures 13(a)–13(c) show diffractive lenses with a designed depth of 1.06 µm fabricated on silicon after correction of astigmatism. It can be seen that the milling depth is uniform in the horizontal and vertical directions.
Astigmatism of a beam spot strongly influences fabrication error and milling depth. The astigmatism-determined orientation of a beam spot proves that the milling depth differs in different directions because of the various resolutions for the long and the short axes and leads to redistribution of the substrate material. Astigmatism can be corrected by performing a quad balance calibration manually. This is a general scheme and is applicable for use with other commercial FIB machines. Similar phenomena exist in laser writing systems as well as electron-beam writing systems. The correction method for astigmatism should be applicable for both, especially for the electron-beam system because the working principle in the electron column and operation system is exactly the same as that for FIBs; only the emission sources differ: one is an electron and the other is an ion. In addition, the method of qualitative judgment can help the operators know the current status of a beam spot.
This research was supported in part by the Funding for Strategic Research Program on Ultraprecision Engineering from the National Science and Technology Board (NSTB), Singapore, and Innovation in Manufacturing Systems and Technology (IMST) Singapore–Massachusetts Institute of Technology (MIT) Alliance. The authors thank Ampere A. Tseng, Department of Mechanical and Aerospace Engineering, Arizona State University, for his enthusiastic help.
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