1. Zone plates for gas-jet focusing
Adam Jeff
May 2013

2. 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)

3. 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)

4. 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

5. 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)

6. Zemax simulations

7. 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.

8. 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?

9. 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)

10. 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.

11. 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 …

12. 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

13. 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