Alvarez varifocal X-ray lens

Abstract

Visible light optical elements such as lenses and mirrors have counterparts for X-rays. In the visible regime, a variable focusing power can be achieved by an Alvarez lens which consists of a pair of inline planar refractors with a cubic thickness profile. When the two refractors are laterally displaced in opposite directions, the parabolic component of the wavefront is changed resulting in a longitudinal displacement of the focus. This paper reports an implementation of this concept for X-rays using two planar microfabricated refractive elements. The Alvarez X-ray lens can vary the focal distance of an elliptical X-ray mirror or a planar compound X-ray lens over several millimetres. The study presents the first demonstration of an Alvarez X-ray lens which adaptively corrects defocus and astigmatism aberrations of X-ray optics. In addition, the Alvarez X-ray lens eliminates coma aberration in an elliptical mirror, to the lowest order, when combining the lens with an adjustment of the pitch angle of the mirror.

Fig. 1. Schematic experimental setup for AXL characterisation with the focusing mirror (VKB).

For zero translation between the two structures of AXL (a), the outgoing wavefront of VKB remains unaffected. For opposing translations, a positive or negative parabolic perturbation of the wavefront results (b, c). Inset (d) Profile of one of the two fabricated AXL structures viewed from the side. Each of the three AXL refractors (AXL1, AXL2, AXL3) can be moved into the X-ray beam by a vertical translation.

Fig. 2. AXL varifocal calibration curve for VKB and CRL.

Measured change in focus position of (a) VKB mirror and (b) CRL as shearing varied in AXL1, AXL2 and AXL3 is illustrated. The solid line represents the calculated values obtained using Eq. (8).

Fig. 3. VKB defocus coefficient with and without AXL.

The dots show the measured change in defocus coefficients of the outgoing wavefield of VKB with Alvarez X-ray lenses for different knife-edge positions ±16 mm around VKB’s focal length. The role of the AXL is apparent in keeping defocus term of the elliptical mirror constant which would otherwise vary linearly along the z direction.

Fig. 4. Focusing by an elliptical mirror with pitch angle θ.

Rays from the source at $F_1$ are reflected from the mirror and focused to the point $F_2$, giving rise to a virtual source $F_1^{\prime }$. As shown, a change to the mirror pitch angle results in an oblique translation of the virtual source and a change to the AXL shearing gives rise to a longitudinal translation of the virtual source along the optical axis.

Fig. 5. Coma free VKB focus variation with VKB pitch angle offset (δθ) and AXL shearing (Δ).

The open squares show δf obtained by pitch angle change only and other dots show δf variation with the cubic term reduced to zero by the three AXLs—AXL1, AXL2 and AXL3.

Fig. 6. VKB beam caustics.

The beam Intensity distribution after numerical propagation of the measured complex field to the intended focal positions of (a) VKB at 235 mm (b) VKB with aligned AXL2 at 235 mm, and when defocus and coma aberrations are minimised (c) and (d) VKB with AXL2 at 235 mm $~\mp$2 mm, after shearing of AXL structures $△$ = $\mp$15 µm and $△$ = $\pm$15 µm, and VKB pitch angle offset $\delta \theta$ = $\mp$72 µrad, respectively.

Fig. 7. Focused beam profiles of the VKB shifted focal planes.

The solid line shows the intensity profile at the VKB focal plane (235 mm from the mirror centre) with the correct incident angle for the mirror (3 mrad) and with the AXL shearing at zero. The dashed line shows the focus profile of VKB focal plane displaced by 4.0 mm after a shearing of AXL1 ($△$ = $\pm$39.7 µm). The solid line with a square dot shows the focus profile of 4.4 mm displaced VKB focal plane by adjusting the mirror pitch offset angle ($\delta \theta =$127.8 µrad) and then adjusting the shearing ($△$ = $\pm$45.4 µm) to remove the coma aberration.

Fig. 8. CRL defocus coefficient with and without AXL.

The dots show the measured change in defocus coefficient of the CRL wavefront in the energy range ±1.2 keV around 15 keV monochromatic beam and calculated coefficients without AXL (solid line with circle). The location of the focus place was found approximately constant (variation of −40 to 90 µm) for 13.8 keV to 16.2 keV, which otherwise shifted in the same energy range from −64 mm to 69 mm.