- Step 2: Replace the magnitude of the amplitude reflectance at each wavelength with √{square root over (R_k)} whenever measuring a freshly completed silica film with an intensity reflectance greater than 9% or a low-index film with an intensity reflectance greater than the limiting value of the high-index material.

While the intensity measurement provides useful information (most of the time) about the magnitude of reflectance, it unfortunately provides no information about the phase angle in the complex plane, φ. Much of the remainder of the present description relates to how to obtain these phase angles in at least some circumstances.

Monitor Curves Can Give Non-Redundant Calculations of Phase

In most cases, the magnitude of r_kat the base of a niobia layer can be obtained from the measured reflectance of the film stack terminating in a fresh silica layer. The phase of the amplitude reflectance is more difficult to ascertain, but there are two general approaches. The first is to consider what values of phase are consistent with the final value of reflectance after the next layer is added. To use this information, the optical thickness of the next layer must be known. This is sometimes a redundant calculation since the estimation of optical thickness is usually based on an understanding of the initial reflectance. This is, in fact, a weakness of the usual matrix modeling approach—the calculation is somewhat redundant.

Without additional information, redundant calculation would normally be the only option. However, monitor curves are usually recorded during deposition, and those curves contain all the information necessary to compute the phase, φ_k, without the need for redundant calculations. This involves the use of reflectance maxima in the monitor curves.

- Step 3. Consider the modeled monitor curve for each wavelength of a niobia (high-index) layer. Discard any monitor curves without maxima in their reflectance during the niobia layer deposition. If none meet this criterion, deposit the layer using a pure model approach.

Based on Equation 14 above, the maxima and minima of a monitor curve can be shown to depend solely on |r′_k|, the magnitude of the buried interface's reflectance, and not at all on its phase. The maximum and minimum reflectance during the monitor curve are given by Equation 21:

\begin{matrix} R_{\max} = {(\frac{\langle r_{k}^{'} \rangle + \langle r_{2} \rangle}{1 + \langle r_{2} \rangle \langle r_{k}^{'} \rangle})}^{2} R_{\min} = {(\frac{\langle r_{k}^{'} \rangle - \langle r_{2} \rangle}{1 - \langle r_{2} \rangle \langle r_{k}^{'} \rangle})}^{2} . & 21 \end{matrix}

Therefore, R_maxor R_mincan be used in the monitor curve to convey the magnitude of the buried reflectance. This magnitude can be related to the reflectance as follows.

\begin{matrix} \langle r_{k}^{'} \rangle = \frac{\sqrt{R_{\max}} - \langle r_{2} \rangle}{1 - \langle r_{2} \rangle \sqrt{R_{\max}}} or \langle r_{k}^{'} \rangle = \frac{\sqrt{R_{\min}} + \langle r_{2} \rangle}{1 + \langle r_{2} \rangle \sqrt{R_{\min}}}; \frac{\langle r_{2} \rangle - \sqrt{R_{\min}}}{1 - \langle r_{2} \rangle \sqrt{R_{\min}}} . & 22 \end{matrix}

Thus, from a monitor curve covering at least a quarter wave at the monitor curve wavelength, |r′_k| can be determined.

A caveat to using these equations is as follows. The R_minexpression has two failings. First, there are two possible solutions for |r′_k| based on R_mindepending on whether |r′_k| is less than or greater than |r₂|. If it is less than |r₂|, the right-hand solution is appropriate. If it is greater than |r₂|, the left-hand solution is appropriate. The R_maxexpression also has two solutions in principle but can be discarded because it provides nonphysical results. The second problem with the equation from R_minis the issue of experimental error. The error expected in the estimation of |r′_k| is related to the error in measurement of R_minand R_maxby Equation 23:

\begin{matrix} ɛ_{\langle r_{k}^{'} \rangle} = \frac{(1 - \langle r_{2}^{2} \rangle)}{2 \sqrt{R_{\min}} {(1 + \langle r_{2} \rangle \sqrt{R_{\min}})}^{2}} ɛ_{R_{\min}} \approx \frac{ɛ_{R_{\min}}}{2 \sqrt{R_{\min}}} ɛ_{\langle r_{k}^{'} \rangle} = \frac{(1 - \langle r_{2}^{2} \rangle)}{2 \sqrt{R_{\max}} {(1 - \langle r_{2} \rangle \sqrt{R_{\max}})}^{2}} ɛ_{R_{\max}} \approx \frac{ɛ_{R_{\max}}}{2 \sqrt{R_{\max}}} . & 23 \end{matrix}

In other words, the error goes up as the key reflectance diminishes. Since the minimum is, by definition, smaller than the maximum, the error expected in estimating |r′_k| goes up accordingly. Thus, for both reasons, the calculation of magnitude from a maximum reflectance (i.e., a minimum in the transmission monitor curve) is preferred.
Choosing the Best Monitor Wavelength

The phase angle φk at the monitor wavelength can be determined from |r′_k| and the Rk as:

\begin{matrix} \cos (\pm ϕ_{k}) = \frac{{\langle r_{k}^{'} \rangle}^{2} (1 + R_{k} {\langle r_{2} \rangle}^{2}) - {\langle r_{2} \rangle}^{2} - R_{k}}{2 \langle r_{2} \rangle \sqrt{R_{k}} (1 - {\langle r_{k}^{'} \rangle}^{2})} . & 25 \end{matrix}

This, of course, provides 2 solutions. Once the phase angle φk from Equation 25 is determined, r′_kcan be computed usingEquation 20 and a monitor curve can be generated if desired for comparison with the actual to help determine which solution is better. Afterwards, a value for phase at the monitor wavelength should have been obtained that is as correct as possible. It depends, of course, on accurately measuring the maximum reflectance value and Rk. Thus, not all wavelengths are created equal as potential monitor wavelengths. A full-spectrum monitor (acquiring many monitor wavelengths) is the best solution, but if only a single wavelength is available, then there is a systematic approach to choosing the best.

Equation 25 depends, ultimately, on only two measurements: the measurement of the initial reflectance and the measurement of the maximum reflectance. For those wavelengths that exhibit a maximum reflectance during deposition, these can be evaluated quantitatively as possible monitor wavelengths.

It can be shown that the anticipated standard deviation of the phase calculation can be written as:

\begin{matrix} σ_{ϕ_{k}} = \sqrt{A σ_{R_{k}}^{2} + B σ_{R_{\max}}^{2}} A = \frac{{((1 + r_{2}^{2}) R_{k} + 2 r_{2} (1 + R_{k}) \sqrt{R_{\max}} + (1 + r_{2}^{2}) R_{\max})}^{2}}{4 R_{k}^{} (\begin{matrix} 4 r_{2}^{2} {R_{k} (1 - R_{\max})}^{2} - \\ {(\begin{matrix} 2 r_{2} \sqrt{R_{\max}} + (1 + r_{2}^{2}) R_{\max} - \\ R_{k} (1 + r_{2}^{2} + 2 r_{2} \sqrt{R_{\max}}) \end{matrix})}^{2} \end{matrix})} B = \frac{{(1 - R_{k})}^{2} {(r_{2} + (1 + r_{2}^{2}) \sqrt{R_{\max}} + r_{2} R_{\max})}^{2}}{{R_{\max} (1 - R_{\max})}^{2} (\begin{matrix} 4 r_{2}^{2} {R_{k} (1 - R_{\max})}^{2} - \\ {(\begin{matrix} 2 r_{2} \sqrt{R_{\max}} + (1 + r_{2}^{2}) R_{\max} - \\ R_{k} (1 + r_{2}^{2} + 2 r_{2} \sqrt{R_{\max}}) \end{matrix})}^{2} \end{matrix})} . & 27 \end{matrix}

This equation is helpful selecting the best monitor wavelength for the purpose of determining the phase of the amplitude reflectivity at the monitor wavelength.

FIG. 4 shows a representation of the regions of permissible values of Rk and Rmax, with a color code for the lowest values of σ_φk for silica (assuming r2=−0.2 and the standard deviation of the reflectance measurements is 0.003). The lowest value possible under these conditions is 0.0262 radians (1.5 degrees). The lower axis, Rmax, represents possible values of Rmax, while the left axis, Rk, gives possible values of Rk. Note that large regions of reflectance are not possible—there are many combinations of Rk and Rmax that cannot coexist. On the boundaries of those disallowed regions, the error in estimating the phase angle, φk, becomes infinite.

The same plot for niobia, assuming r₂=−0.4, is given inFIG. 5. The minimum value of error here under the same conditions is 0.00875 radians (0.5 degrees)—a much better phase calculation.

It is possible to develop a histogram of the number of combinations of allowed Rmax and Rk that provide a specific level of error in φk. This is accomplished first by considering the range of possible Rmax values when depositing a layer: it cannot, asFIGS. 4 and 5 illustrate, be less than the reflectance of the thin film material being deposited itself (e.g., Rmax for niobia inFIG. 5 cannot be less than −0.4²=0.2 as shown in a leftmost portion ofFIG. 5 in dark yellow). Possible values of Rk can be evaluated for each value of Rmax, and the values of phase precision those values provide can be determined using Equation 27. This is accomplished by dividing the phase thickness of the top layer into increments and computing the reflectance at each increment (this is done because the reflectances at the turning points are more likely than those in between the turning point values). In the end, a histogram of the resulting precision values can be formed and a determination made as to how likely each will appear for a given film material. This has been done for silica (assuming r₂=−0.2, and reflectance standard deviations of 0.003) and for niobia (assuming r₂=−0.4) as shown respectively inFIGS. 6 and 7.

For silica, 30% of all observed combinations will have a phase error given by Equation 24 that is less than 2.4 degrees. No values less than about 1.5 degrees error in φk is possible for silica under these conditions. For niobia, the same fraction will have φk errors less than 0.9 degrees, as illustrated in theFIG. 7, also derived from a full numeric simulation.

Thus, the calculated phase error of possible monitor wavelengths will tend to be considerably better for niobia films than for silica. If a limit of 0.9 degrees phase error is placed on the monitors before this calculation is performed, only niobia will give possible monitor wavelengths, and 30% of all wavelengths (overall) will meet this criterion. On some layers, it is possible that no wavelengths will meet this criterion, while on others many may do so.

- Step 4. For the remaining possible monitor wavelengths in a niobia layer deposition, determine the anticipated standard deviation in φk. Discard any with cy greater than 0.9 degrees (0.016 radians). If none remain, proceed with a pure model deposition.

For low-index layers, bootstrapping is not recommended. For layers with larger magnitudes of r₂(such as niobia), the precision of the bootstrap is almost always better, but there is no guarantee that a specific layer will include a set of R_kand R_maxvalues anticipated to provide excellent precision in calculating the phase φk. If no wavelength with an anticipated reflectance maximum meets this criterion, modeling alone should be relied upon to deposit the layer until a suitable bootstrap layer is reached.

An ingenious characteristic about this calculation is that it provides φk—and thus also φ′_k—that is consistent with the monitor curve and is independent of δ, the phase thickness of the film. Once a valid solution for rk is determined, the valid solution for r′_kcan be obtained. Thus, the final transmission value at the monitor curve wavelength can be used to determine what value of δ is most accurate for the monitor wavelength.

If a wavelength meets the criterion specified above, then consideration should be given as to whether the anticipated end of the layer will have a reflectance, Rf, suitable for estimating δ, the phase thickness of the layer with some precision. This calculation is performed with Equation 26, where δ depends on Rf and rk′.

\begin{matrix} δ = \frac{ϕ_{k}^{'}}{2} \pm \frac{1}{2} \cos^{- 1} (\frac{R_{f} + R_{f} {\langle r_{2} \rangle}^{2} {\langle r_{k}^{'} \rangle}^{2} - {\langle r_{2} \rangle}^{2} - {\langle r_{k}^{'} \rangle}^{2}}{2 \langle r_{2} \rangle \langle r_{k}^{'} \rangle (1 - R_{f})}) . & 26 \end{matrix}

This equation provides fairly unique solutions for δ that can be used to correct the reflectance at all wavelengths. The possible solutions for δ that are obtained can be tested against the observed monitor curve to determine which is correct.

For a given monitor curve with a given expectation of Rmax, and with a known value of r2, the sensitivity of δ to errors in Rf can be determined according to Equation 28:

\begin{matrix} σ_{δ} = \langle \frac{(1 - \langle r_{2}^{2} \rangle) (1 - R_{\max})}{2 (R_{f} - 1) \sqrt{A}} \rangle σ_{R_{f}} A = (R_{\max} - R_{f}) (\begin{matrix} 2 \langle r_{2}^{2} \rangle (R_{\max} R_{f} + R_{f} - R_{\max} - 2) - \\ (1 + \langle r_{2}^{4} \rangle) (R_{\max} - R_{f}) + \\ 4 (\langle r_{2} \rangle + \langle r_{2}^{3} \rangle) (1 - R_{f}) \sqrt{R_{\max}} \end{matrix}) . & 28 \end{matrix}

If the type of numerical analysis above using Equation 28 is repeated, a plot as shown inFIG. 8 is obtained (shown as the logarithm of error vs. Rmax and δ because otherwise the scale would be difficult to see). This plot is made over the entire range of possible values of Rmax between r₂²and 1 (on the receding axis) and δ angle between 0 and 2π (the front axis).

The data inFIG. 8 can also be rendered as a histogram as shown inFIG. 9. The histogram inFIG. 9 implies that the error in phase thickness is usually satisfactory compared to the error in φk. At a cutoff of 0.9 degrees error in δ, about 51% of the remaining monitor wavelengths for niobia, for instance, should be usable. Thus, given a good monitor wavelength for determining the phase angle φk, there is a good chance of having one that also provides a good precision in δ.

To conserve time, and when modeling is running fairly well, it is reasonable to only select monitor wavelengths for niobia (in a silica/niobia stack), and only when they meet these two criteria (standard deviation of φk<0.9 degrees, and standard deviation of δ<0.9 degrees at the end of the layer). When no monitor wavelengths meet these criteria, it is reasonable to proceed with a pure model matrix approach to depositing the next layer.

- Step 5. Compute the expected error in δ for the remaining wavelengths at the target thickness of the niobia layer. If no wavelengths have an error less than 0.9 degrees, proceed with a full model deposition of the layer. If some do meet this criterion, select the lowest error in this category.
  Reflectance of a New Film and Determination of the Old Phase at Wavelengths Other than the Monitor Wavelength.

Why bother with determining the φk and δ from a monitor curve as precisely as possible? First, if a monitor wavelength for bootstrapping has been selected successfully, all connection to the previous dependence on the matrix calculation for the monitor wavelength may be avoided. For all other wavelengths, there is at least the opportunity to determine the magnitude of rk, but only the original modeled estimate of phase. The question arises: how to “repair” the phases of all other wavelengths? Since in a single-channel monitor there are no monitor curves at those wavelengths, the phase at each wavelength cannot be directly obtained. (If there was spectrograph recording all wavelengths all of the time, such as with an FTIR system, the steps provided below would not be needed). However, to reach this point in bootstrapping, there must be a good value for δ for the monitor wavelength, plus good values of Rmax and Rk. With this, at least some of the other measurements can be “fixed” to accord with measurements already taken. This will restrict errors to those of a single layer at the worst for those wavelengths where the following calculation is possible.

From δ for the monitor wavelength, the physical thickness of the layer consistent with the modeled refractive index of the film material can be estimated. This physical thickness and the modeled refractive index of the film can be used to estimate δ for all other wavelengths.

- Step 6. Compute the value of δ for all wavelengths based on the value calculated for the monitor wavelength.

Returning toEquation 16 and replacing rk′ with the definition inEquation 20, the following is obtained:

\begin{matrix} r_{f} = \frac{r_{2} (1 - r_{2} r_{k}) + (r_{k} - r_{2}) (\cos (2 δ) - ⅈ \sin (2 δ))}{1 - r_{2} r_{k} + r_{k} (r_{k} - r_{2}) (\cos (2 δ) - ⅈ \sin (2 δ))} . & 29 \end{matrix}

In Equation 29, the exponential has been replaced with a trigonometric expression using Euler's relation. If r_kis replaced with a+i b, a conventional expression for r_fcan be obtained. The complex conjugate of r_fcan be formed and the product taken of the two. This provides the intensity reflectance of the top interface in terms of values from the new film, plus a and b. The following can then replace a and b:
a=cos (φ_k)√{square root over (R_k)}
b=sin (φ_k)√{square root over (R_k)} (30.)
where use is made of the intensity reflectance measured in vacuum for interface k to represent the magnitude of the amplitude reflectance vector at the interface k in vacuum.

The resulting expression can be simplified as Equation 31:

\begin{matrix} R_{f} = \frac{2 r_{2}^{2} + R_{k} (1 + r_{2}^{4}) + 2 r_{2} Q}{1 + r_{2}^{4} + 2 r_{2}^{2} R_{k} + 2 r_{2} Q} Q = r_{2}^{2} R_{k}^{} \cos (2 δ + ϕ_{k}) + R_{k}^{} \cos (2 δ - ϕ_{k}) - r_{2} (1 + R_{k}) \cos (2 δ) - (1 + r_{2}^{2}) R_{k}^{} \cos (ϕ_{k}) . & 31 \end{matrix}

In this expression, everything is known EXCEPT the phase angle φ at interface k, allowing it to be determined from the measurement of intensity reflectance at the subsequent interface.

Dispensing with numerical solution methods, this expression can be solved for the Cosine of the angle:

\begin{matrix} \cos (ϕ_{k}) = (\frac{A (1 + r_{2}^{2}) \sin (δ) \pm B \cos (δ)}{C}) A = R_{f} + r_{2}^{4} (R_{f} - R_{k}) - R_{k} + 2 r_{2}^{2} (\begin{matrix} (1 - R_{f}) (1 + R_{k}) \cos (2 δ) - \\ (1 - R_{f} R_{k}) \end{matrix}) B = \sqrt{D (1 + r_{2}^{12}) + F (r_{2}^{2} + r_{2}^{10}) + G (r_{2}^{4} + r_{8}^{2}) + H r_{6}^{2}} C = \sin (δ) (4 r_{2} (1 - R_{f}) R_{k}^{1 / 2} (2 r_{2}^{2} \cos (2 δ) - 1 - r_{2}^{4})) D = - {(R_{f} - R_{k})}^{2} F = 2 (\begin{matrix} R_{k} (2 + R_{k}) + R_{f}^{2} (1 + 2 R_{k}) + 2 R_{f} (1 - 5 R_{k} + R_{k}^{2}) - \\ 2 (1 - R_{f}) (1 - R_{k}) (R_{f} + R_{k}) \cos (2 δ) \end{matrix}) G = - 6 - 4 R_{f} - 5 R_{f}^{2} - 4 R_{k} + 38 R_{f} R_{k} - 4 R_{f}^{2} R_{k} - 5 R_{k}^{2} - 4 R_{f} R_{k}^{2} - 6 R_{f}^{2} R_{k}^{2} + 8 (1 - R_{f}^{2}) (1 - R_{k}^{2}) \cos (2 δ) - 2 {(1 - R_{f})}^{2} {(1 - R_{k})}^{2} \cos (4 δ) H = 4 (\begin{matrix} 3 + 2 R_{f}^{2} - 10 R_{f} R_{k} + 2 R_{k}^{2} + 3 R_{f}^{2} R_{k}^{2} - \\ 2 (1 - R_{f}) (1 - R_{k}) (2 + R_{f} + R_{k} + 2 R_{f} R_{k}) \cos (2 δ) + \\ {(1 - R_{f})}^{2} {(1 - R_{k})}^{2} \cos (4 δ) \end{matrix}) & 32 \end{matrix}

In principle, Equation 32 can be used to solve for the phase angle. By doing so, the deposition system process control is effectively “bootstrapped” by detaching the system completely from everything that came before the last layer. Equation 32 provides four (4) solutions for the phase angle; two come from the +/− portion of the calculation; two more from the fact that cosine is an even function, so positive and negative angles both work equally well. However, only two of these solutions are consistent with the measured value of R_f. Thus, solutions should be checked via Equation 31 for validity. Only two solutions should be left after this process is complete.

- Step 7. Compute the two possible values of phase angle for each wavelength other than the monitor wavelength.
  Estimating Error for Non-Monitor Wavelengths

The phase angle is dependent on three reflectivity measurements (R_max, R_fand R_k), those being scrambled together in Equation 32. While this works well for a hypothetical system with no noise, a real spectrometer exhibits errors in measurement of the intensity transmittance.

Based on work already done, the analysis of Equation 32 is fairly straightforward for extending phase information to other wavelengths. The following expression can be constructed from it:

\begin{matrix} ɛ_{ϕ_{k}} = \frac{- 1}{\sin (ϕ_{k})} ((\frac{\partial \cos (ϕ_{k})}{\partial R_{k}}) ɛ_{R_{k}} + (\frac{\partial \cos (ϕ_{k})}{\partial R_{f}}) ɛ_{R_{f}} + (\frac{\partial \cos (ϕ_{k})}{\partial δ}) ɛ_{δ}) σ_{ϕ_{k}} \approx \frac{σ_{R}}{\langle \sin (ϕ_{k}) \rangle} {({(\frac{\partial \cos (ϕ_{k})}{\partial R_{k}})}^{2} + {(\frac{\partial \cos (ϕ_{k})}{\partial R_{f}})}^{2})}^{1 / 2} . & 33 \end{matrix}

In Equation 33 error in phase thickness has been omitted from the calculation since it is of minor concern at this stage. The phase thickness was settled previously for the sake of argument; therefore, making that approximation, computing the sine of the angle and the sensitivity of the cosine functions to errors in R_kand R_fremain. There is a subtle issue to be considered at this point. What angle should be focused on—the modeled angle or the computed angle using Equations 31/32? In principle, both values are available.

Thus, the conservative answer to the foregoing question is “both”. Assume, for example, that the model has a complex reflectance rk at a given wavelength from which the phase angle φk and the predicted value of Rk can be obtained. Using a best estimate of the phase thickness at the wavelength, Rf can be computed using Equation 31. Next, compute the phase angle using Equation 32 after varying Rk and Rf each by a small amount—e.g., by 10⁻⁵, 1/100^thof a percent transmission. Since the modeled phase is known exactly, it will be trivial to identify which results are the ones closest to the modeled phase; therefore, the sensitivities are computed as:

\begin{matrix} \frac{\partial \cos (ϕ_{k})}{\partial R_{k}} \approx \frac{\cos (ϕ_{k} (R_{k, m}, R_{f, m}, δ)) - \cos (ϕ_{k} (R_{k, m} - 10^{- 5}, R_{f, m}, δ))}{10^{- 5}} & 34 \\ \frac{\partial \cos (ϕ_{k})}{\partial R_{f}} \approx \frac{\cos (ϕ_{k} (R_{k, m}, R_{f, m}, δ)) - \cos (ϕ_{k} (R_{k, m}, R_{f, m} - 10^{- 5}, δ))}{10^{- 5}} . \end{matrix}

From the model, the sine of the phase angle that appears in Equation 33 is trivial to obtain.

Thus, the standard deviation of the phase calculation can be estimated from the model. If the model were absolutely trustworthy, this would be sufficient; however, this is not the case.

- Step 8. Using information extracted from the model for r_kat each wavelength and the computed best value of δ, compute the estimated standard deviation of phase at all wavelengths except the monitor.

Since the model is not completely trustworthy, the process is repeated using values taken from the calculated value of phase closest to the model value of phase. In other words, for the two possible values of φk, the one closest numerically to the model phase is selected. Now, keying on that value, compute the sensitivities according to Equation 34 using the measured values of R_k, R_fand the estimate of δ for the wavelength being tested.

- Step 9. Using the computed phase closest to the model phase for r_kat each wavelength, the measured R_fand R_kvalues and the computed best value of δ, compute the estimated standard deviation of phase at all wavelengths for which the magnitude of r_kwas estimated (Step 2) other than the monitor.

Now, compute the estimated error in phase as follows:
σ_φk(estimated)=(σ_φk²(model)+σ_φk²(calculated))^1/2 35.

What are the probable limits to this calculation? A bit of experimentation suggests that the value of standard deviation in phase angle could be between about that of the monitor measurement (on the low end) to nearly infinite. A sufficient approach is to replace the modeled values with calculated values only if a rather conservative cutoff is obtained. If the error estimates are less than about 1.3 degrees from Equation 35 (square root of two times 0.9), and if the reflectance was greater than 9 percent at the start of the layer (Step 2), then the model and calculated values are averaged together to obtain a new estimate. If either the phase error OR the reflectance criteria are not met, the estimate is returned alone to the model.

- Step 10. If the phase error estimate is less than 1.3 degrees AND the criterion ofStep 2 is met, average the calculated and modeled reflectance and phase values to obtain a new value that will be used in all future modeling at that wavelength.

A further refinement of this approach is to fit the phase values obtained above to a Kramers-Kronig model of the phase to fill in values of the phase that have not been determined previously.

While preferred embodiments of the invention have been shown and described, those of ordinary skill in the art will recognize that changes and modifications may be made to the foregoing examples without departing from the scope and spirit of the invention. Furthermore, those of ordinary skill in the art will appreciate that the foregoing description is by way of example only, and is not intended to limit the invention so further described in such appended claims. It is intended to claim all such changes and modifications as fall within the scope of the appended claims and their equivalents.