On Focusing in 2-Dimensions

On Focusing in 2-Dimensions

Version 1.3
Why Osmic recommends the Confocal Max-Flux™ Optic

Summary

Two-dimensional focusing is an important aspect of many x-ray analytical experiments. It is especially important when small samples are involved or localized features must be probed. Consequently two-dimensional focusing can advantageously be applied in fields as diverse as Protein Crystallography. Microcrystaline studies, High Pressure Single Crystal Diffractometry, SAXS, Micro-Fluorescence etc..

This document describes why we recommend using the Confocal Max-Flux™ (CMF) optic for such applications. After having read this document you should clearly understand why the CMF optic has 4 distinct advantages over other existing schemes.

Ease of Alignment
Monochromatization
Maximizing Flux by proximity
Symmetric beam and focus

Confocal Max-Flux

In order to understand these advantages, we first discuss existing alternative schemes and how the Confocal Max-Flux™ Optic differs from these approaches.

We then describe how each advantage is a direct result of the shortcomings of the alternative systems.

Confocal Max-Flux Optic

Traditional Two-Dimensional Focusing

Introduction

In our view of the world there exists 3 mechanisms for manipulating x-ray above 4 Kev¹.

Diffraction by crystal lattices (Bragg and Laue-diffraction).
Total Reflection at grazing angles.
Bragg Diffraction by Multilayers².

Early forms of one-dimensional focusing were obtained by Crystal Diffraction, in presently well-known geometries, such as Johann and Johansson. However, these techniques have not been routinely employed for two-dimensional focusing in a laboratory environment.

The other two mechanisms have been used extensively in two-dimensional focusing schemes such as

Kirkpatrick-Baez Total Reflection Scheme.
Kirkpatrick-Baez Multilayer Diffraction Scheme (Also called Cross-Coupled)
Confocal Max-Flux™ Scheme

The following text, will describe the first two in general. Then comparisons will be made to the Confocal Max-Flux™ Scheme.

Kirkpatrick-Baez Total Reflection Scheme

Total reflection at grazing angles was first used for two-dimensional focusing in the so-called Kirkpatrick-Baez geometry, where one reflector focuses in one dimension. Focusing in the second dimension is obtained by a second reflector further downstream, as seen in the figure below.

KB - Geometry

Such optics found their way into Protein Crystallography and SAXS under the names of Supper Mirrors, and Franks Mirrors and Yale Mirrors.

Cross-coupled Focusing Multilayer Scheme

By substituting total reflection coatings with appropriate laterally graded multilayers, one can use the Kirkpatrick-Baez scheme for two-dimensional focusing with multilayers. The geometry is thus exactly the same as shown above, except that the angles are a factor of 3 to 4 higher than previously.

These kinds of systems have been available since the mid-90's, but have not become very common since the major instrument manufactures have tried to push the collimating versions. Our opinion is that for most applications where two-dimensional information, the focusing cross-coupled scheme is superior to the collimating cross-coupled scheme. This opinion will be expanded upon in an upcoming Osmic document.

Focusing with the Confocal Max-Flux™ Optic

General Concept

The Confocal Max-Flux™ Optic can essentially be described as a Multilayer-based Kirkpatrick-Baez Scheme with the reflectors positioned side-by-side. Both reflectors are part of the same block but are positioned at a 90º angle to each other.

Confocal Max-Flux Optic

Optical Scheme

The optical scheme of the Confocal Max-Flux™ Optic is similar to that of the Kirkpatrick-Baez scheme in that each reflector focuses the beam in one dimension only. However, the optical scheme is a bit more complicated in that, as shown below, the x-rays can arrive at the focus in two different ways: either by first reflecting from reflector 1 and afterwards from reflector 2 (yellow path) or the other way around (red path). In general, the width of the reflectors will be larger than required and consequently reflection will take place near the intersection of the two reflectors.

Optical Scheme of the Confocal Max-Flux Optic

Advantage #1: Ease of alignment

Ease of alignment is perhaps the most critical improvement of all, not only because it obviously makes initial alignment and fine-tuning less time-consuming, but more importantly because systems that are difficult to align are often not aligned correctly and thus can provide considerably sub-standard performance.

Three very important differences make Confocal Max-Flux™ Optics much easier to align than Total Reflection KB or Cross-coupled optics.

The number of free alignment parameters has been reduced.
The number of coupled alignment parameters has been reduced.
The optimal alignment position is easily verifiable.
The scheme provides a horizontal beam

Reduced number of free parameters

The schematic representation to the left clearly indicates that Confocal Max-Flux™ Optic has 6 independent degrees of freedom: (3) translations and (3) rotations. Since the KB scheme and Cross-coupled scheme both consist of 2 independent optics, each have 12 degrees of freedom, making alignment considerably more difficult. In addition, the total reflection KB scheme typically allows the user to determine the focusing characteristics of the system, making alignment even more difficult. Clearly, this simple consideration shows the alignment of the Confocal Max-Flux™ Optics to be simpler.

The problem of coupled parameters

An additional problem with the KB and the Cross-coupled optical scheme is that the various degrees of freedom are actually coupled when the system is aligned well. For example, when the first optic is slightly rotated the beam exits the first optic at a different angle. Ideally, the second optic should be rotated and translated to intercept the new beam properly, as shown schematically below.

Easily verifiable optimal alignment

An advantage over specifically the Total Reflection Kirkpatrick-Baez scheme is that the Confocal Max-Flux™ uses Bragg-Reflection. Total reflection characteristically has equivalent throughput over a relatively wide angular range, while Bragg-Reflection provides a high throughput only in a very narrow angular range. Optimal alignment of the Confocal Max-Flux™ is verified when a small rotational adjustment produces a definite reduction in overall throughput.

The Confocal Max-Flux™ maintains a horizontal beam

Most sources used in two-dimensional analysis provide a horizontal beam in the laboratory environment. As clearly illustrated in the previous drawing, all the reflectors will change the direction of the beam. For example, for a traditional total reflection Kirkpatrick-Baez system, the first mirror could deflect the beam to the upwards, while the second reflector would deflect the beam to right (see figure below). The resultant beam would propagate upwards and to the right -a direction that is not particularly appropriate to laboratory instrumentation.

This consideration is the same for the Confocal Max-Flux™. However, Osmic has chosen to solve this problem by rotating the optics by 45 degrees around the incident beam. As seen in the figure, this ensures that the total deflection is only in the horizontal plane.

The Confocal max-Flux Optics maintains a horizontal beam

Anecdotal evidence of the importance of simple alignment

Our installation experience, has shown that even though existing Kirkpatrick-Baez systems should be able to deliver 20 - 40% of what the Confocal Max-Flux™ geometry does, in reality we see anywhere from 7 - 40%. A large part of the research groups are simply not able to align the Kirkpatrick-Baez system for optimal performance and consequently some systems are underperforming by a factor of 3.

With the above mentioned advantages in ease of alignment of the Confocal Max-Flux™ scheme this will not happen.

Advantage #2: Monochromatization

The second advantage of the multilayer is that it provides a degree of monochromatization that is appropriate for many laboratory instruments. The monochromatization is an intrinsic property of Bragg-Reflection, but as discussed below the fluorescence lines of the source also play a crucial part.

Intrinsic Multilayer Bandwidth

The theoretical reflectivity of a typical multilayer can be seen in the figure below in red. State-of-the-art multilayers provide roughly the same kind of peak height and width. Peak reflectivity is above 80% and width of the peak (DE/E FWHM) is roughly 400ev or 5%.

Also shown is the reflectivity of a total reflection coating (in this case Nickel) shown in blue. The broad high reflectivity of the total reflection mirror makes it clear that Total Reflection mirrors do not make good monochromators.

Reflectivity of Multilayer and Total Reflection Coating

When combined with a laboratory source

The graph above has been chosen to illustrate what happens when multilayer optics and total reflection optics are used together with laboratory sources. The explanation follows:

Taking a typical Cu laboratory source, whose spectrum consists of Ka1, Ka2, Kb and bremstranhlung we observe the following:

For the multilayer:

The bandwidth of 4% is large enough to let both Ka1 and Ka2 pass through.
The bremsstrahlung radiation within the 4% band around the Ka - energy will also be passed through.
All other energies will be considerably attenuated, specifically one can see that the Kb (symbolized by the black line) will be attenuated by a factor of 100 after one reflection and therefore a factor of 10,000 after the second reflection.

For the total reflection mirror of Ni:

The high reflectivity below the absorption edge of nickel, allows Ka1, Ka2, and ALL lower energy bremstranhlung radiation to pass through.
Even in the best of alignments, roughly 30% of the Kb will pass through the optic, and even after 2 perfect reflections, roughly 10% will remain in the beam.

The difference can clearly be seen in the figure below, which shows the relative spectra for 3 optical schemes³.

no optics (blue).
Kirkpatrick-Baez Total Reflection Optic with a 10 micron Nickel foil for additional Kb reduction (yellow).
Confocal Max-Flux™ Optics (Red)

The lack of low-energy background and Kb - line is clearly seen in the Confocal Max-Flux™ Spectrum.

Spectrum Comparison

Advantage #3: Maximizing Flux

Maximizing flux is naturally a very important part of any optical scheme.

Laboratory sources typically emit x-ray radiation in all directions. The closer the focusing optic can come to the source the more of that radiation can be collected and focused onto the sample or the detector, as seen in the figure below.

The limiting factor in trying to get close to the source is often the housing of the source. However, for the second optic of the Kirkpatrick-Baez or Cross-coupled multilayer scheme, it is the housing PLUS the length of the first optic that is the limiting factor. All other things being equal, the second mirror will intercept less radiation than the first mirror.

Maximizing Flux

This side-by-side approach of the Confocal Max-Flux™ optic solves this problem, and a maximum amount of radiation is intercepted.

In addition, the multilayer has the advantage over total reflection that the angle of incidence are larger and that thus even more radiation can be intercepted even for optic at the same distance from the source (See to right).

Advantage #4: Symmetric beam

The final major advantage of the Confocal Max-Flux™ optical scheme is that it is inherently completely symmetric with respect to the two independent focusing dimensions. Consequently, there is no difference in the convergence and focus size of the beam in the two dimensions. This is not so for the Kirkpatrick-Baez total reflection scheme and the Cross-coupled Multilayer scheme as illustrated in the following discussion.

Focus Size

The size of the image, as in any focusing optical scheme, is essentially determined by the Law of Magnification⁴, which states that

where P is the distance from the optic to the source of size S, and Q is the distance from the optic to the image of size I, as illustrated below:

Focus Size

From this equation it is clear that for two optics located at different distances from the source, the image cannot be the same size. Consequently both the Kirkpatrick-Baez and Cross-coupled multilayer foci will be asymmetric.

Asymmetric Convergence

Clearly, in the general case the convergence from the Kirkpatrick-Baez reflectors and the Cross-coupled reflectors will differ in the two dimensions, since Q will differ for optic 1 and optic 2.

Asymmetry in practice means losing flux or resolution

Although the asymmetry documented above may not seem alarming, the asymmetry actually reveals that the asymmetric systems are not delivering optimum performances.

An argument for this statement may be: Asymmetry essentially means that your system resolution differs in the two dimensions. Very rarely⁵ does your instrumentation take this asymmetry into account. In fact, the majority of instrumentation will most likely work with the worst resolution. Consequently you are not benefiting from the good resolution in the other direction, and might as well have traded the resolution in for higher flux.

Conclusion

We have argued that the Focusing Confocal Max-Flux™ optical scheme has several clear advantages over alternative approaches, such as the Kirkpatrick-Baez Total Reflection Scheme or the Cross-coupled multilayer scheme.

These advantages appear from simple geometrical considerations and specific multilayer characteristics.

These advantages have proven to be so significant that the Confocal Max-Flux™ Optics have become the de-facto standard, especially in Protein Crystallography at the Cu-Ka wavelength, and is beginning to gain acceptance in the fields of Small Angle X-ray Scattering and Microcrystaline diffractometry.

So far only Cu-radiation versions have been available, but with development of multilayers for Cr, Co, and Mo, acceptance in other applications should soon follow.

[1] Others (Fresnel Zoneplates, and Multiple Array Lenses) exists but are only applicable to highly coherent sources such as synchrotrons. (Return to text)

[2] For more information on how Bragg-Diffraction from multilayers work, please refer to the Osmic Homepage and/or our info-page "Multilayer Optics". (Return to text)

[3] The spectra were taken with an energy-dispersive solid state detector. (Return to text)

[4] For the multilayer there is a slight complication in that the angular acceptance of the multilayer is roughly 0.06 degrees. If the angular size of the source is larger than that, S in no longer the actual size of the source but instead P*tan(0.06). (Return to text)

[5] We are unaware of any existing instrumentation that takes this asymmetry into account. (Return to text)