Zone plates for gas-jet focusing Adam Jeff May 2013

Particles or Waves? Davison & Germer 1927 Particles are found to produce interference fringes as if they were waves DeBroglie Wavelength λ=h/p It works even if only 1 particle passes at a time Has been tested with molecules up to Buckminsterfullerene (C60)

Decoherence Typically with wave-particle duality only one behaviour can be observed at one time If we place a (non-destructive) detector at one slit, we ‘force’ the particle to choose one slit. The interference pattern then disappears! This will affect the gas jet – if the atoms interact anywhere between the source and the focus they will cease to act as waves – Quantum Decoherence So we have to keep the pressure low to avoid collisions between atoms Various papers suggest 10-6 mbar as an upper limit – still OK for our case C. Jonsson, “Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten“, Zeitschrift fur Physik 161, 4 (1961)

What is a Fresnel Zone Plate? The path difference between each successive light ring is equal to 1 wavelength (at the focal point) constructive interference. Each zone is equal in area Focal spot size is roughly the width of the narrowest (outer) zone Compared to traditional lens: no spherical aberration, large chromatic aberration Two main types: Transmission – alternate zones are blocked – 50% of light lost Phase – alternate zones have π phase shift Both can be binary or smooth

Matter-wave Fresnel Zone Plate DeBroglie wavelength ≈ 0.05 nm for room temperature Helium Focal length of zone plate Resolution ≈ width of smallest zone 𝑓= 2 𝑟 𝑁 ∆ 𝑟 𝑁 λ radius of outer zone width of outer zone Smallest zone 100 nm T. Reisinger, S. Eder, M.M. Greve, H.I. Smith, B. Holst, “Free-standing silicon nitride zoneplates for neutral-helium microscopy”, Microelectronic Engineering 87 (2010)

Zemax simulations

Figures of Merit I calculate the following figures for each design in order to compare them: Peak Intensity in the focal spot. Ultimately, signal strength will depend on this. Transmitted power. A measure of how ‘open’ the plate is. The higher the better. FWHM of focal spot. As small as possible for resolution. However, all the designs produce spots which are plenty small enough. In reality this will be dominated by chromatic effects (not investigated yet). % encircled in 10 μm or 100 μm. Maybe the most important – shows what fraction is spread out into higher order and ‘zeroth order’ diffraction.

Photon Sieves Underlying geometry is the same as the zone plate The ‘clear’ zones are replaced by a series of holes Lower transmission but less higher-order diffraction Easier to manufacture?

Antiholes If the hole is a bit larger than the underlying zone, it can still have an overall focusing effect. In fact the optimum may be found around d/w =1.35. Above a certain size the hole covers more of the neighbouring zones than its own zone, so it has a negative focusing effect. So it’s positive if centered on a dark zone: an antihole. L. Kipp et al, “Sharper images by focusing soft X-rays with photon sieves”, Nature 414 (2001)

Equal-Hole Sieve All the holes are the same size, for easy manufacture. Hole center is switched between ‘light’ and ‘dark’ zones depending on focusing contribution.

Apodised Photon Sieve We can try to make up for the lack of an infinitely large zone plate by smoothing the transmission towards zero at the edges Also tried: Composite Sieve Fractal Sieve Fibonacci Sieve …

Results Design Description Peak irradiance Transmitted power FWHM of focal spot % in 0.01mm % in 0.1mm micrometers Sieve, size 1, first zone 1, 10 rings 75 5.60E-04 0.35 59 97 Sieve, size 1.35, first zone 1, 10 rings 77.6 7.50E-04 50 Sieve, size 1.5, first zone 1, 10 rings 71 8.40E-04 48 98 Random Angle Sieve, size 1, first zone 1, 10 rings 51 4.70E-04 56 Random Angle Sieve, size1.35, first zone 1, 10 rings 49 6.00E-04 Random Angle Sieve, size 1.5, first zone 1, 10 rings 43 6.80E-04 Sieve, size 3.5 holes, first zone 3, 6 rings 8.7 1.10E-03 0.3 30 99 Sieve, size 3.5 holes, first zone 2, 6 rings 9.4 1.20E-03 27 Sieve, size 3.5 holes, first zone 1, 5 rings 6.9 9.80E-04 0.4 6.6 Apodised Sieve, size 1, first zone 1, Gaussian 0.8,15,0 83 5.90E-04 57 Apodised Sieve, size 1, first zone 1, Gaussian 0.8,15,8 195 9.20E-04 64 96 Apodised Sieve, size 1, first zone 2, Gaussian 0.8,15,8 180 8.80E-04 Apodised Sieve, size 1, first zone 1, Gaussian 0.8,8,0 22 2.90E-04 0.55 47 Equal holes sieve, 1 micron, 16 rings 61 2.20E-03 Equal holes sieve, 2 micron, 10 rings 10 2.50E-03 Equal Holes sieve, 5 micron, 6 rings 1.3 5 Zone Plate, first zone 0, 6 rings 144 0.5 Zone Plate, first zone 0.5, 6 rings 9.10E-04 62 Zone Plate, first zone 1, 6 rings Zone Plate, first zone 1.5, 6 rings 53 Zone Plate, first zone 2, 6 rings

Linear Zone Plate Linear Zone Plate is equivalent to a cylindrical lens Focuses in one plane only Makes a line at the focal plane Equivalent to a screen if the focal length is large