An Introduction and Some Basic Optics

This project was especially interesting for me considering I had very little background working with x-ray optics, much less, compound refractive lenses. Having said that, the professor I worked with suggested I start from scratch and do some background reading on a few basic optics principles. In the end, it proved to be an extremely valuable learning tool for me.

My goal was to understand the makings of what a compound refractive lens is, and its advantages and/or disadvantages for x-ray focusing. To start, first I would like to explain what exactly a refractive lens is and how one works. The word "refractive" implies the change in any direction of a ray of light. To understand how the lens works, we should look at it mathematically.

This basically states that the focal length of a lens is determined by the index of refraction (in our case, this would be for an x-ray), the radius of curvature of the front surface of the lens, and the radius of curvature of the back surface of the lens.

So, then, what is the x-ray refractive index? It is represented by the lower case letter n, and is defined as:

where delta is the refractive index decrement (typically 10^-6 for hard x-rays) and beta is the absorption index. The complex refractive index of all materials is very close to 1. Since the real part of the index of refraction (1 - delta) is less than one, the focusing lens must have a concave form. It is also good to note, at this point, that a biconvex lens has a negative focal length whereas a biconcave lens has a positive focal length. "The radius of curvature is taken to be positive if light, after passing through the surface of the lens in question (either the front or back surface), can pass through the center of curvature of that surface. Otherwise, the radius of curvature is negative" [1]. In simple terms, a positive lens is one in which the center is thicker than the edges, and a negative lens would be of the opposite shape and would only produce virtual images.

Compound Lenses

From here, it is easy to see how one can stack lenses up to form a compound lens. Mathematically, you would just add up their focasing power:

If L is << (f1,f2) then you simply add up their inverse focal lengths. The following image taken from [2] better places this in perspective:

Compound Lens Properties

One would think a refractive lens may not be appropriate for focusing x-rays since absorption is so strong and refraction effects are extremely small. Hence the developent of the compound refractive lens. There are two basic types of compound lenses for focusing x-rays. One is the saw-tooth, or alligator, lens which is truly a sherical aberration-free refractive lens for hard x-rays. However, this lens focuses in one-dimension only, and is not optimal for increasing the x-ray intensity. I, however, shall focus on the other type of lens which is useful for focusing hard x-rays. This is referred to as a cylindrical compound refractive lens. These lenses are useful for microdiffraction, microfluorescence, and coherent imaging [3]. Furthermore, these lenses may focus in one or two dimensions, unlike the saw-tooth lens. A compound refractive lens that can focus in two dimensions is technically called a "crossed cylindrical lens" in which case holes would be drilled in the lens in both the vertical and horizontal directions.

To form a good compound refractive lens, one should consider the following:

• low Z materials
• minimun mass absorption
• the lense should be as thin as possible (20 microns)
• x-ray energy in the range of 5-40 keV
• a small radius of curvature (.25 mm)
• shperical aberration

A refractive compound lens for is effective for focusing X-rays in the energy range of 5 to 40 keV. To create one, the use of low Z materials is necessary in order to optimize the transmission and minimize the mass absorption coefficient. Such materials would include aluminum, boron, beryllium, and perhaps even lithium. Aluminum seems, so far, to be the best of these, yielding the best results (i.e. it is easy to manufacture and the drilled holes keep their shape, also the expected focal size is very close to the expected value) [3]. The following is an image of such a lens:

To show the differences between some of the above mentioned materials, I have included the following table:

Table 1. N is number of holes in compound refractive lens at different
photon energies for different low Z materials (f=1 m and R=.250 mm) [3].
======================================================================
Lens Material                  8 keV            20 keV           40 keV
                                        delta | N        delta | N        delta | N
--------------------------------------------------------------
                                           [10^-6]          [10^-6]          [10^-6]     
--------------------------------------------------------------
Be                                 5.260  |  24       0.853  |  147       .213  |  587
B                                   6.960  |  18       1.128  |  111       .282  |  443
Al                                 8.320  |  15       1.348  |   93        .338  |  371
======================================================================

Above, I also have the material lithium mentioned. Though this material, to the best of my knowledge, has only been tested as a viable material for x-rays via a "saw-tooth" refractive lens, it is by far the best candidate for focusing x-rays because its x-ray absorption is the lowest of all solids. However, there are areas of concern such as "surface impurities, which reduce lithium's transparency, and surface imperfections, which reduce the lens' quality" [4]. Also, this type of refractive lens focuses in only one dimension, unlike the cylindrical compound lens (shown above). More research is still needed in this area regarding the use of Lithium as a compound material.

Conclusions

As of right now, the materials which yielded the best results for a cylindrical compound refractive lens have been beryllium and alluminum. A good radius of curvature is about .25 mm, and x-ray energy of 10 keV with approximately N=100 holes is sufficient.

Regardless, the compound refractive lens an efficient, simple, and inexpensive tool for focusing x-rays. "The advantages of the compound lenses can be summarized as follows: they are very robust and small (5-10 cm in length), easy to align and to operate. The focal length and size are easily adjustable by adding or removing individual lenses, and the lenses can withstand a high heatload" [3].

Acknowledgements:

I would sincerely like to thank Professor Chris Jacobsen for his extreme patience and kindness in working with me on this project. He has been of great support. My deepest gratitude to him.

I would also like to thank Dr. John Noe for his patience and support in helping me to create this web site. I had extremely little knowledge of HTML code and could not have created this web page without his assistance. My sincerest thanks.

And finally, I would like to thank Professor Harold Metcalf for giving me the motivation and courage to sign on for a class that I had such little background in. Truly, thank you.

References:

[1] The Lensmaker's Equation

[2] Beamline Research and Development

[3] B. Lengeler and J. Tummler, A. Snigerev, I. Snigireva and C. Raven, Journal of Appl Phys, 84, 11 (1998).

[4] N. R. Pereira, Applied Physics Letters, Dec. 17, 2001, pp. 4085-4087.