The computer-controlled optical surfacing (CCOS) method is a deterministic optical fabricating method based on the known figuring process model. Its ideal figuring process is based on the following several assumptions:

1. Material removal is linear and proportional to the dwell time.
2. Respective material removal rate is constant over the entire surface.
3. Material removal function is non-variant with time.
4. Material removal function is insensitive to position on the optical surface.

Analyzing these assumptions, the core is to gain a desired non-variant material removal function or to know its varied property. For conventional deterministic figuring processes, it is very difficult to know a precise material removal function because of some of the physical constraints, such as optic geometry and construction, aspheric departure, tool wear, edge roll-off effect, and loading force on optical component. These constraints limit the fabricated error convergence rate and the final fabricated contour accuracy. With IBF, it is nearly independent of the above constraints since the “figuring tool” is a noncontact and controllable compliant tool (Shengyi and Xuhui 2010). It is can be said that the IBF is reasonably valid to these assumptions in a wide range of material applications which is proved by many actual applications. It is acknowledged as the most deterministic optical figuring method.

Removal Function Modeling and Analyzing of IBF

In IBF, a typical material removal function (or called beam removal function, BRF) b(x, y) shown in Fig. 5 is defined as the material removal profile or “footprint” on optical surface generated by the projected ion beam per unit time. In most cases, the removal function distribution is a circular Gaussian shape, which is dependent of the ion beam size and intensity, the bombarded material. When these are determined, the removal function can be described as (Lin et al. 2007)

where A is the peak removal rate and σ is the Gaussian parameter. It can be also expressed as

where B is the volumetric removal rate; it can be calculated by

B is in unit of μm · mm2  · min-1 . It means the volumetric removal of the BRF, and

There are two kinds of method to determine a BRF. One is an empirical method by optical or electro-probing. The electro-probing method is used to measure the ion beam current distribution by Faraday cup to get a BRF profile; its drawback is not to contain the correlation with the actual removed material, but it is a good method to regularly monitor the stability of BRF in the actual figuring process. Compared with this method, that of the optical probing is more efficient because it directly uses the ion beam to locate and bombard an optical surface in a given time to generate a “footprint.” The BRF mathematical expression is obtained from the real interferometric evaluations to this “footprint.” The other is mathematical method based on the ion sputtering theory.

With the optical probing method, the key problem is to correctively evaluate the parameters of Gaussian-type BRFs: A, σ (or d), and B, as shown in Fig. 6a. There are two ways to do this. The common way is to hit 4–8 “footprints” on an optical planar surface and then to evaluate the BRF Gaussian-type function with the average of these “footprints.” Another way is to scan a trench on planar optical surface with travel speed v (mm · min-1) and hold the ion source parameters constant; the transverse material removal profile along the trench can be approximated as a Gaussian function rs( y) as shown in Fig. 6b:

where it is assumed that the ion beam scanning direction is x and its transverse direction is y. σs is the standard deviation; it equals to σ:

The peak removal rate As is

Therefore, the BRF key parameters A and σ can be estimated by the trench transverse profile parameters As and σs in Eqs. 6–9.

The volumetric removal rate B can also be estimated as

However, the estimated B is less believable because of the estimated errors of As and σs. So, a better method to estimate the volumetric removal rate B is to define rs(y) as

The both sides integral form of Eq. 11 is

Then, B can be estimated by

Here, rs(y) is the interferometric data of the trench transverse profile. It is proved that it is more precious to calculate B with Eq. 13 than that of Eq. 10. The other advantage with this method is that the uncertainty of RBF can be evaluated at the same time by analyzing the scatters on the width and depth of the profile of trench.
The mathematical method to model the material removal function is based on the Sigmund sputtering theory (Sigmund 1973). According to this theory, the energy deposition of incident ions follows a Gaussian distribution as shown in Fig. 7. The energy deposition at point O as the result of cascade collision when ions hit the surface at point A and travel along the z-axis can be written as

where (x, y, z) is a coordinate system with origin at the energy deposition center and the z-axis along the projected path, ε is the total energy deposited, a is the average incidence depth of the ions, and σ and μ are the Gaussian distribution parameters along and transverse to the beam direction, respectively.

It is known that the material removal rate and profile relate to the parameters of the IBF process. Besides these, they also are influenced by the curvature of optical surface and the beam incidence angle which would be paid attention in the fine figuring process. With Sigmund sputtering theory, Bradley and Harper deduced the etching rate on an arbitrary surface z = h(x, y) (Bradley and Harper 1988). When a uniform ion beam bombards a surface at incidence angle ϕ, its etching rate is

where Y0(ϕ) is the sputtering yield for a flat surface

where f is the ion flux, n is the amount of atoms at unit volume in the amorphous solid, ϕ is the incidence angle, and p is the proportional constant relating the power deposited at the bombarded point to the rate of erosion. The other coefficients are defined as

From Eq. 17, it can be seen that the coefficients c1 and c2 depend on the curvature of the surface and the coefficients Γ1(ϕ), Γ2(ϕ), and Y0(ϕ) depend on the incident angle ϕ. Therefore, the relationship of etching rate function with incidence angle and bombarded surface curvature can be obtained using Eqs. 15–17.

Although Eq. 15 is the etching rate of uniform ion flux deduced in the micro area, it is also practicable in macro area with ion beam diameter in millimeter or centimeter scale since the local point flux could be considered identical. In Fig. 8, an ion beam with flux f bombards the surface at point O by the incidence angle ϕ; the distribution of flux is approximately f(x cos ϕ, y) in workpiece reference frame (x, y, z) according to the projection theory. Thus the theoretical model of the beam removal function is

where c1 and c2 are

For most of the surfaces (especially spherical surfaces), the unit of a is A° which is not comparable in magnitude with r0x and r0y. Hence c1 and c2 are infinitesimals, but Γ1(ϕ) and Γ2(ϕ) are finite coefficients, so Eq. 18 can be approximated by

Equation 19 shows that the contribution of the curvature of surface to the beam removal function can be neglected. It can be concluded that the ion beam removal function varies with the varying of ion beam incidence angle and its footprint on the oxy plane transforms from circle to ellipse with the increasing of its incidence angle. In the following of this section simulation and experiment on the BRF will be done.

Assuming that 8,000 vertically incident Ar+ with the energy of 1,200 eV were simulated to bombard fused silica by the SRIM program (http www srim org), then the parameters of ion energy distribution in Eq. 14 can be obtained. In Fig. 9, the simulation result shows that the average incidence depth of ions is a = 48 A° and Gaussian distributing parameters are σ = 20 A° and μ = 13 A°.
From Eqs. 16 and 19, it is known that a theoretical beam removal function with arbitrary incident angle is difficult to be gained since the coefficients n, f and p in these equations are unobtainable. But a beam removal function with vertical incident angle is usually easily obtained by experiment. It is enlightened that to an arbitrary incident angle theoretical beam removal function, a valid way is to estimate it by that of vertical incident angle case which is unnecessarily known all above coefficients.
To gain a theoretical beam removal function model with arbitrary incident angle by this experimental method, the normalized peak etching rate is introduced:

Figure 10 shows the simulation result of the beam removal function with different incident angles with Eq. 21. It is apparently shown that the incident angle influences the footprint shape of the beam removal function, and their footprints on the oxy plane change from circle to ellipse and expanse their active zone with the increasing incident angles. However the length of semi-minor axis a is invariable, and the semi-major axis b is varying with a/cos ϕ.
Figure 11 also shows that the normalized peak etching rates depend strongly on the incident angle. It can be seen that the etching rate increases firstly and then decreases with the increasing incident angle. At about 70o, it reaches the maximum etching rate which is about 4.5 times than that at 0o.
To testify this theoretical model by experiment, an experimental equipment is set up in Fig. 12. In order to obtain removal functions with different incident angles, the experiment was done on six small fused silica samples (the size is 10 mm x 10 mm x 15 mm) with a 2 mm diameter ion beam. In this experiment, the samples are fixed at angles from 10o to 60o with 10o interval relative to the experimental planar board. The ion beam respectively bombards the surfaces of these samples along the vertical direction of experimental planar board. Every sample is bombarded 3 minuts, and the process parameters were set according to the modeling simulation. By this experiment, there are six footprints gained to estimate their beam removal functions. The one of footprints is shown in Fig. 13. All removal functions of this experiment are shown in Fig. 14, which validate the result of the theoretical analysis that the etching rates and footprints vary with incident angles.

With Gauss–Newton method to fit these experimental beam removal functions, their beam removal functions are obtained as shown in Fig. 15. For example, the fitted model at 40 is shown in Fig. 16a, and its fitted residual error is acceptable (Fig. 16b). The footprint eigenvalues of these removal functions are estimated by 6σ method in mathematical statistics, which are shown in Fig. 17a with the semi-minor axis a and the semi-major axis b. In Fig. 17b, c, they show that the eigenvalues and the etching rates of this experiment are close to the theoretical values. These results indicate that the length of semi-minor axis is approximately invariable and the semi-major axis is inversely proportional to the cosine of the incident angle.

From the above analysis, the following conclusions can be gained:

1. In IBF, the beam removal function varies with the ion beam incident angle.
2. To figure optical surface, the best way is that the ion beam bombards the surface along its normal direction, which can well constantly hold the etching rate and footprint of beam removal function.
3. For spherical and aspherical optical surface figuring, the good suggestion is that the ion source hold and motion mechanism is designed 5 axes motion mechanism to hold the ion beam moved along the normal direction of optical surface. For a 3-axis motion mechanism, the beam removal function must be real-time correct to compensate etching rate and footprint varying with the ion beam incident angle.

Contouring Algorithm for IBF

The figuring process is represented in Eq. 1. A significant step in the process is the calculation of dwell-time function that controls the correction of optical surface error. It is known that the ion beam figuring process is a convolution process and the calculation of dwell-time function is a deconvolution process.
To obtain good dwell-time function, there are three important problems that should be paid more attention which are (1) calculation algorithm, which calculates the dwell-time function distribution that will yield a desired material removal distribution when a specific beam removal function is applied; (2) the data process of optical surface edge, which influences the calculated dwell-time precision at the neighbor of surface edge; and (3) determining or predicting the optimal removed material amount or residual error result in every step of iteration, which is very important to gain the efficient dwell-time function and decrease the times of iteration and is very useful to control the middle- and high-spatial-frequency errors. About the third problem, it will be discussed in the next section.
About the algorithms to calculate the dwell time, they can be summed up to four kinds: (1) Fourier transform method (the early work was done by Wilson et al.) (1987); (2) iterative method for finding a solution to the dwell-time function (Allen and Roming 1990); (3) algebra method, such as wavelet algorithm (Shanbhag et al. 2000) and Bayesian algorithm (Changjun et al. 2009); and (4) matrix-based method, such as in Carnal et al. (1992) and Zhou lin et al. (2007).

Rewrite Eq. 1 as

Here, T is the dwell-time function. Assuming that the IBF scanning is a raster routine as shown in Fig. 2, there are two figuring techniques to realize the dwell time in the actual IBF: (1) position mode, where T represents the time that the ion beam spends at a location on the optical component per unit area, and (2) velocity mode, where T represents the ion beam raster scanning speed of a strip unit width.

In the position mode, the dwell-time function can either be broken into areas or interpreted as a time. So, the amount of removed material is proportional to the amount of time the ion beam must be positioned in the area which is equal to the integral of the dwell-time function over the area. For example, if the total optical component is broken into a square grid as shown in Fig. 18, the ion beam is centered at each square an mount of time approximately equal to the value of the dwell-time function at the center times the area of square, which discretizes Eq. 22 as

This discretization of the removal function suggests that the figuring process can also be discretized in a similar way. Thereby the IBF process is represented as a discrete two-dimensional (2D) convolution. In actual figuring process, the optical component surface profile and the material removal function are provided as an x–y grid array from the profile interferometer, so the position mode process can be exactly realized with a square discrete convolution by maintaining the same discretization as the profile interferometer.
Alternatively, in the velocity mode, the optical component is broken in strips, where each strip has a width and a velocity function associated with it. The velocity function is the scanning speed that the ion beam moves along the strip and is equal to the inverse of the dwell-time function integrated over the strip width W:

Then Eq. 22 can be approximated as

The IBF process is represented as a discrete one-dimensional (1D) convolution. Each one-dimensional strip has a raster interval. For example, if the optical component is broken into strips parallel to the x-axis, the raster interval width is Δy which is the y separation between the boundaries of the strips 

In this mode, the calculation of dwell time is more complicated than that of position mode since the precious material removed is related to the velocity and acceleration performance of the machine tool. This mode provides a partial discretization that has some advantages over the position mode. For the position mode, the ion beam remains on while it traverses from one grid point to another, which may cause unwanted material removed if the position time is not negligible compared to the dwell time in this grid area. In the velocity mode, the strip scanning is continuous which can avoid the unwanted material removed. In most cases, the velocity mode is chosen since the machine tool moves smoother which can decrease the generated middle or high-spatial-frequency error in the actual figuring process.
In the following of this section, an example is given about how to realize dwell time based on the Bayesian algorithm (Changjun et al. 2009).
Assuming that the optical surface error E(x, y) and dwell-time function T(x, y) are both random, the solution to dwell-time function T(x, y) to maximize the posterior probability function P(TǀE) of the dwell-time function according to the maximum likelihood method (Molina et al. 2001; Fang and Xiao 1998). With the Bayesian principle, the relation among the posterior probability function P(TǀE), the prior probability function P(T), and the probability function P(EǀT) of the simulated removal error E if T were the true dwell time is

In the IBF process, the ion beam density is the result of statistical average. Assume that it is of Poisson distribution and P(EǀT) follows the Poisson distribution with parameter B ⊗ T. In this case, maximizing Eq. 27 could be transferred into the following minimization problem:

where the performance function J1(T) is

With calculus of variation, the optimization condition for Eq. 28 can be deduced as

With a multiplicative algorithm, the Bayesian-based iterative algorithm can be deduced from Eq. 30:

Equation 31 is the generalized form of the traditional Richardson–Lucy algorithm (Lucy 1974; Richardson 1972) with respect to the nonnormalized removal function. This algorithm has an interesting property of nonnegativity: if the first estimate T0 is nonnegative, none of the further estimate will be negative. In dwelltime iteration, the initial value is usually the offset nonnegative surface error. With this property, the nonnegativity demand of dwell time is satisfied.

In the beginning of this section, the data process of optical surface edge has been simply discussed. About how to solve this problem, it will be discussed. Since the optical shape usually is circular or non-regular, the discretized matrix of surface error function would not be filled completely, which may make the property of the points at the edge of the optical surface differ from that of the points in its inner. This difference can induce an algorithm edge effect which affects the convergence and calculated accuracy of algorithm at the edge of surface. So, it must be the edge extension which can weaken or even eliminate this effect. For example, in a circular optical component as shown in Fig. 19, the diameter of the component is Dw, and the radius of the removal function is Bt. The width of the extended rectangular area as shown in Fig. 19 is (Dw + 2Bt). With a Gaussian algorithm, the data of any point f in the extended area of the surface error can be expressed as

 where σ is the Gaussian extension parameter, generally σ ≥ Bt/3. With the extended matrix above, only four fast Fourier transform computations are needed in one iteration of Bayesian-based algorithm in Eq. 31.

Analysis of Correcting Ability of IBF

In ion beam figuring, the ability to “correct” the surface errors of specific spatial wavelength or frequency depends on the ratio of the ion beam diameter d to the error wavelength λ, i.e., d/λ. The evalute of this correcting ability is the material removal efficiency ε, which is defined as the ratio of the volume of desired material removal to the volume of the predicted (or real) material removal. The Kodak had estimated that the ratio of the beam diameter to the error wavelength must be  ≤ 0.5 to achieve 90% of material removal efficiency (Allen and Keim 1989). It is obvious that the ion beam diameter is a key parameter in ion beam figuring process. In order to find an optimal ion beam diameter, Kodak used four ion beam diameters of 2.5, 5.1, 10.2, and 12.7 cm in their simulations to correct the presented error of wavelength of 10.5 and 9.5 cm. They found that the best choice is the 5.1 cm diameter which indicated an excellent correcting ability, while the total dwell time was favorable. Besides Kodak, the IOM in Germany also researched the correcting ability of different ion beam diameter. They used a smaller diameter down to 0.5 mm to meet the demanding requirements for correcting the long spatial wavelength part of the so-called mid-spatial-frequency roughness (MSFR) down to the sub-nanometer RMS level (Haensel et al. 2006). Further work has been done by Lin Zhou et al. in China (Lin et al. 2008, 2009). Assume that Eq. 25 is normalized as

And assume that the optical surface has the spatial frequency error Rλ

where λ is the wavelength of the surface error and δλ is the amplitude of the error.

The circular Gaussian-shaped ion beam removal function is

Combining Eqs. 33–35, the dwell-time function can be calculated as

Since the dwell-time function distribution must be nonnegative, the actual dwell-time function distribution is

Based on Eqs. 33 and 37, the actual material removed can be calculated as

Equation 38 shows that in any ion beam figuring process, there are always extra materials removed. The extra removal material γ(x) is

where d = 6σ is the ion beam diameter. Equation 39 shows that the extra removal material γ(x) is independent of x; therefore, γ(x) can be shortly written as γ.

The material removal efficiency ε can be described as

Equation 40 shows that the material removal efficiency ε is a negative exponential function of d/λ. For a known spatial frequency error with wavelength λ, the more the beam diameter is, the less the material removal efficiency. For a given beam diameter, the smaller the wavelength is, the less the material removal efficiency. In this sense, the smaller ion beam diameter is always a best choice in ion beam figuring. However smaller ion beam diameter always result in more process time, which usually decreases the process reliability. To balance the material removal efficiency and process stability, a suitable ion beam diameter should be chosen.

Simulating the material removal efficiency ε at different conditions of d/λ and comparing the results with the theoretical results achieved by Eq. 40, based on the result shown in Fig. 20, it can be seen that the simulation-predicted material removal efficiencies are in satisfactory agreement with the theoretical ones. From Fig. 20, it can be seen that the material removal efficiency is 87 % when d/λ is 0.5. This value approximately corresponds to the Allen’s estimated material removal efficiency, 90 % (Allen and Keim 1989), which is acceptable in ion beam figuring process. When the ratio of d/λ is up to 1, the ε rapidly decreases to 58 % which is unacceptable, and it extremely degrades to only 11 % when the ratio is up to 2.
According to theoretical analysis and simulation results, it is recommended that the ratio of d/λ should be less than 0.5 in order to obtain acceptable material removal efficiency in ion beam figuring process.

The Optimum Material Removal of IBF

It has been known that if the less desired material removal is specified, the calculated dwell function is consequently small for a given initial surface figure error, which means a short IBF process time, but as a result, the actual postmachined residual surface figure error is great. If a larger desired material removal is specified, although the resulted residual figure error is smaller, the calculated dwell function is greater, which means a longer process time.
Therefore, in order to balance the process time, which is determined by the dwell function and the resulted residual error, the desired material removal specified to the contour algorithms should be considered and optimized. In this section, it will be discussed how to determine an optimum material removal. Firstly the conventional method to determine the removal is discussed in Lin Zhou et al. (2010).
Since the real removal in an IBF process is always nonnegative, the specified removal should be nonnegative too. However, the data of a surface figure error from metrology usually contain negative elements. Therefore, in order to obtain a nonnegative removal, the conventional method is that the error data is simply offset to be nonnegative:

where R is the desired material removal and E is the surface figure error and min(E) is the minimum of E. This method is illustrated in Fig. 21. Due to the original surface figure error from metrology that inevitably contains noises, E should be smoothed to reduce the influence of noises on the magnitude of min(E). With a smoothed figure error, the removal determined by Eq. 41 is reasonable.

However, for ultraprecision optics, in order to keep more details or higherspatial-frequency information about the figure error, the original figure error from metrology is often used instead of the smoothed figure error. For original errors, since they contain more noises, and more edge fall and pits, as shown in Figs. 22 and 23, the material removal determined by Eq. 41 are not reasonable and tend to be greater, consequently causing longer IBF process time.

It is concluded that, in the conventional method, the material removal is the sum of the figure error and an invariable uniform material removal. To avoid the drawback of the conventional method, Lin Zhou et al. proposed an optimum material removal method which determines the material removal flexibly (Lin Zhou et al. 2010). In this method, an adjustable uniform removal U is introduced to substitute the invariable [min(E)]. The formula is

In Eq. 42, the adjustable uniform removal U can be broken into two parts, i.e., U = γe, where e is the RMS value of the figure error E and γ is an adjustable parameter to be used to control the magnitude of the material removal. Since e describes the mean deviations of the figure error, it contains the main information of the figure error. Therefore, the adjustable parameter γ is just a simple factor, and for different figure error, the optimum γ values experientially tend to be in the same range from 1 to 4.
Since the iterative dwell-time function algorithm is simple and has met with considerable success in practice, this problem is discussed by one iterative algorithm expressed as (Drueding et al. 1995)

where Tn and En are the dwell time and the residual figure error after n computation iterations, respectively, ξ is the relaxation factor, and B is the beam removal function. The initial T0 is often set to a proportion of the specified removal, usually T0 = R/B0, where B0 is the integration of B. After several computation iterations, the final dwell time and the residual error can be obtained.
For different γ value, the material removal is differently determined by Eq. 42, and consequently the results, which are calculated in Eq. 43, including dwell time and residual figure error, are different too.

A γ curve in a coordinate system of process time (the sum of all dwell time, total dwell time) vs. RMS value of the residual error is shown in Fig. 24, which is called a figuring prediction curve which is usually a monotonically decreasing curve. In this curve, it can be seen that a larger material removal causes a smaller residual error, but consumes a longer process time, and a smaller material removal causes a shorter process time, but induce a larger residual error. This implies it is difficult to determine an optimum material removal for both small residual figure error and short process time. However, fortunately, a figuring prediction curve is often in the shape of the letter “L.” This property is useful to determine the optimum material removal. The optimum material removal should be located at the corner of the letter “L,” since for removals smaller than this point, the residual error decreases rapidly while the process time increases slowly and, for removals greater than this point, the process time increases rapidly without much decrease in residual error.
In addition, the figuring prediction curve can be used to predict the process time and the residual figure error of an IBF process. Since the figuring prediction curve is in the shape of the letter “L,” there exist two significant RMS, the corner’s RMS eC and the smallest RMS eL; both are illustrated in Fig. 24. For an initial optical surface, if the desired residual error is represented by eD RMS, then an IBF process can be classified into one of the following three cases according to the magnitude of eD:

1. eD < eL
In this case, the figuring prediction curve indicates that the desired accuracy cannot be obtained. This implies that the initial figure error is too great to be removed. Therefore the optical component should be returned to the prepolishing process for a smaller initial error.
2. eL < eD < eC
In this case, although the figure-prediction curve indicates that the desired accuracy can be obtained, the process time is too long to be practicable. Therefore, the optical component should also be returned to the prepolishing process for a smaller initial error.
3. eD > eC
In this case, the figuring prediction curve indicates that the desired accuracy can be obtained in a practicable process time and the optimum removal at the corner is preferred. Moreover, in this case, if the desired RMS eD is significantly greater than the corner’s RMS eC, to reduce the process time, a smaller removal may be chosen instead of the corner’s removal.

Although the property of the “L” shape about the figure-prediction curve is found out in the iterative contour algorithm, it also exists in other contour algorithms, such as the matrix algebraic algorithm. Therefore, this method to determine the optimum material removal can be used in other contour algorithms.
A suggestion should be given that a smaller material removal is advisable. According to simulations, a process with larger removal usually induces a smaller residual figure error. However, in a real process, the resulted residual figure error from a process with larger removal is often significantly greater than the prediction. In addition, some experiments have indicated that a process with more removal will likely degrade surface roughness. Therefore, for an IBF process, a smaller material removal is advisable.

Realization of IBF Technique

This section describes the operational steps to realize IBF technique. The procedure used in the process is outlined below:

1. Calculating the removal map
At the beginning of process, an interferometric map of the optical component’s surface height data (or called surface map) is loaded in an array. For aspherical optical surface, this surface map should be calibrated according to the curvature of aspherical surface since the distortion of measure data project. The removal map is equal to the surface map minus the component’s desired surface map. This removal map generally needs to be filtered by Zernike polynomial to avoid influence of the measure noise or high-spatial-frequency error. With this way, it makes the IBF machine move smoothly. Further work in this step is the removal map data extension along the component edge.
2. Selecting beam removal function
It is the most important step to select a suitable beam removal function in an IBF process. It has been known that there are two key parameters for a Gaussian beam removal function, which are beam diameter d evaluated by the full width at half maximum (FWHM) or 6σ and the peak material removal rate (or volume removal rate), which is mainly controlled by the ion beam voltage and current. About how to gain a suitable beam diameter, it will be discussed below.

There are two methods to change the diameter of ion beam. The direct method is to decrease the scale of screen grid of the ion source shown in Fig. 25a. By this way, there is a ratio between the diameter of screen grid and the diameter of ion source inner chamber. This ratio is about 1:4. For example, in a 90 mm ion source, the minimal diameter of screen grid is about 22.5 mm. Based on the ratio, the better practical selectable range of screen grid is 25–90 mm for 90 mm ion source, 15–50 mm for 50 mm ion source, 10–30 mm for 30 mm ion source, and so on. In these ranges, it can gain satisfied material removal rate universally. Otherwise, its peak material removal rate and volume material removal rate are rapidly reduced as the diameter of ion beam decreases. For example, when the diameter of ion beam decreases from 25 mm to 15 mm for 90 mm ion source, its peak material removal rate is rapidly reduced from 0.13 μm/min to 0.025 μm/min, and its volume material removal rate is also reduced from 16.6 x 103 mm3/min to 1.67 x 10-3 mm3/min from 0.13 μm/min to 0.025 μm/min (Xie Xuhui et al. 2009). Because the material removal rate decreases rapidly as the diameter of screen grid decreases, it is difficult to gain very small ion beam by this method. Currently, with this method, the smallest stable ion beam is about 4–5 mm(FWHM) by 3 cm ion beam in IBF. To gain more small ion beam, another method is to put an ion diaphragm (or mask) before the ion beam outlet of the ion source which passes only part of the ion beam through the ion diaphragm as shown in Fig. 25b. Compared with the direct method, this method is a secondary way which can gain about 0.5–4 mm Gaussian-type ion beam (FWHM) with good peak material removal rate. For example, by this way, it is able to gain 1.7 mm diameter (FWHM) of ion beam with a 2 mm diameter of ion diaphragm with peak material removal rate about 200 nm/min (5 cm ion source with 40 mA ion beam current, 1,000 eV ion beam voltage). And the ion beam longtime stability is better than 2 %/3 h. So, it is a conclusion that ion diaphragm is an efficient method to gain more small ion beam which has a good performance to improve IBF figuring ability. Said to this, there is a problem would be paid attention that a match relationship between the diameter of screen grid and the diameter of ion diaphragm is also existed to get full Gaussian-type material removal function.
Based on the above discussion, the following conclusions can be drawn:

1. Different beam removal function can be gained by selecting suitable screen grid and ion beam power parameters.
2. Ion diaphragm is a good secondary method to improve IBF figuring ability.
3. In IBF process, it would select different screen grid and ion diaphragm to control full spatial frequency figuring error.

3. Calculating and Realizing the dwell-time function

About how to calculate dwell-time function, it is has been discussed in detail in the above section. Here, it is discussed how to realize the dwell time which may be seen in the continuation of the section “Contouring Algorithm for IBF.”
Continuing the Bayesian-based algorithm discussion in the section “The Typical Features and Its Purpose of IBF,” assuming that the desired dwell-time function is T(x, y), the discrete intervals are Sx, Sy along the x and y direction, respectively. In the figuring process, the ion beam scans continuously in the x direction and raster moves in the y direction. Omitting the transient (acceleration and deceleration) of the translation system, the velocity distribution in the x direction can be computed as 

The variation of the dwell time along the x direction introduces the transient and induces a realization error of dwell time with velocity given by Eq. 44. As shown in Fig. 26, the relative realization error of dwell time is

where a is the acceleration of the translation system. Equation 45 indicates that the relative realization error is inversely proportional to the acceleration a and to the square of intermittent increment Sy2 and is  proportional to(ꝺxT)2.
Smoothing Eq. 45, it indicates that it can reduce the realization error. In order to smooth dwell time and to de-noise, total variation norm J2(T) is introduced, which is gradient based:

where μ is a weight factor. Adding Eq. 29 to Eq. 46, then the minimization problem is transferred as

With calculus of variation and multiplicative algorithm, the modified Bayesian based algorithm for dwell time can be deduced as

In the iterative process with Eq. 48, a small weight factor μ cannot smooth the dwell time and cannot filter the noise. On the other hand, a large eight factor can make dwell time so smooth as to reduce the precision of dwell-time density function. There is a trade-off.

Another key problem in the calculation of dwell time is the raster pitch (interval). In the raster scanning of ion beam figuring as shown in Fig. 27, the x direction is the scanning direction and the y direction is the raster direction. In the actual figuring process, the “tool trace” can be seen as raster pitch increasing as shown in Fig. 28, which will generate the middle- or high-spatial-frequency residual surface errors on the optical surface. In Fig. 28, the ion beam removal function has 5 mm diameter.
How is a suitable raster pitch to control its figuring residual errors selected? Answering this question, the figuring precision and its efficiency (or figuring time) are needed in actual process firstly. According to the Nyquist sample theorem, the pitch spatial frequency should be at least twice larger than the spatial cutoff frequency of the removal function. Assuming the axial symmetric removal Gaussian function has W width (6σ), its cutoff frequency will be 

So the raster pitch should satisfy

As known in Eq. 49, the small ion beam needs small pitch to satisfy the sample theorem. But in the actual process, an optimized pitch should be considered to match the precision and efficiency comprehensively. By computer simulation of the pitches and its related figuring errors from small to large pitches, a curve of the pitch vs. its figuring error can be drawn. An example shown in Fig. 29, the “dot” curve is for 5 mm diameter of ion beam and “square” one is for 10.6 mm diameter of ion beam. It can be seen that each curve has one turning point – 1 mm pitch for 5 mm diameter of ion beam and 1.8 mm pitch for 10.6 mm diameter of ion beam, respectively. The turning point means that the change of relative figuring error (the ratio of actual figuring error to its desired error) is slower below this point, while faster above it. For example, of the “dot” curve in Fig. 29, when the pitches are smaller than 1 mm, their relative figuring errors are smaller than 10 %. When they are larger than 1 mm, their relative figuring errors are increased rapidly. So the result that the 1 mm pitch would be selected as the raster pitch for 5 mm diameter of ion beam is obtained, which makes good trade-off between the relative figuring error and the figuring efficiency. With the above analysis, it is known that a curve of pitch vs. its figuring error can be set up to help in selecting the optimal pitch (Xuhui et al. 2011).