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Fraunhofer Diffraction and It's Effects on Aperture Masks - A Primer

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When a plane wavefront is constrained to enter a finite aperture, the intersection of the wavefront at all points of the aperture perimeter produce secondary radiators (Huyghens Principle). Where these secondary wavefronts interfere and either reinforce or nullify one another diffraction patterns are produced.

The Fraunhofer diffraction pattern produced by any particular aperture or obstruction placed across that aperture can be modelled using a Fourier Transformation. There are plenty of fast Fourier transformation freewares available that can be used to quickly perform the aperture transformation that produces the Fraunhofer diffraction pattern, and the energy point spread function or PSF ( a quasi-3D histogram of the peaks and troughs of the energy distribution in the diffraction pattern).

The diagrams used in this Forum article were generated using FLT-SE It has its limitations, one being it cannot model circular vector layers, only 256x256 pixel square vector layers. But for the purposes of this Forum article that is all that is required.

Most of the amateur astronomers I speak to at local astronomical society meetings have heard something about diffraction. Many have a vague recollection of ripple tank or laser interference experiments conducted in their High School Science classes. But most do not understand what diffraction is, how a Fraunhofer diffraction pattern is formed, and what sort of diffraction pattern arises from a particular aperture mask.

I shall begin with a simple example, an unobstructed circular mask. All telescopes act as though they have a circular mask in front of the objective. This is because the objective is circular, and defines the shape of the telescope entrance pupil. Most seasoned amateur astronomers know about the telescope exit pupil, fewer seem to be aware that a telescope also has an entrance pupil. So where is the entrance pupil? For most practical purposes it can be considered as lying immediately in front of the objective, and to have a diameter equal to the objective's clear aperture. In reality it either lies at infinity, or a considerable distance in front of the objective. Because star light entering a telescope from an astronomical object is parallel, the distinction in practice does not matter.

A circular mask mimics the circular entrance pupil of a telescope. Diffraction of the plane wavefront as it enters the circular aperture produces the familiar Airy disc.

circular aperture

The image on the left is the aperture mask, the centre image is the Fraunhofer diffraction pattern and on the right the PSF histogram.

A mask with twin circular apertures produces a double diffraction pattern which merges into a single pattern at the focal plane. A phenomenon exploited to aid focusing.

twin-hole mask twin-hole mask fft twin-hole mask fft psf

The classic diffraction experiment uses a single narrow slit. Babinet's Principle states that the diffraction pattern produced by a single slit is the same as that produced by a bar of the same width.

one bar mask 1-bar mask fft 1-bar mask fft psf

Notice that the diffraction spike produced by the bar is perpendicular to the bar and is broken up into a series of regular strips (called orders). The brightest strip in the centre of the spike is the zero order image of the bar, the strips lying immediately either side are 1st order images & so on.

The bar lies across the centre of the entrance pupil. What happens if the bar is tilted diagonally across the entrance pupil?

1-bar-tilt mask 1-bar-tilt fft 1-bar-tilt fft pasf

The diffraction spike still lies perpendicular to the bar (ignore the cobble stone pattern and the horizontal & vertical faint spikes - they are artifacts caused by the FFT program 256x256 pixel square vector layer boundary).

Both the vertical & the diagonal bar cross the optical axis, and their diffraction spikes also cross the optical axis. What happens if the bar lies to one side of the entrance pupil, well off the telescope optical axis? Will translating the bar off-axis also cause the diffraction spike to shift off-axis?

1-bar-tilt-shift mask 1-bar-tilt-shift fft 1-bar-tilt-shift fft psf

Translating the diagonal bar off-axis does not alter its diffraction pattern. Note that it still crosses the optical axis. The bar can be translated anywhere across the entrance pupil and the diffraction pattern will be unaltered. In other words the position of the diffraction pattern has nothing to do with the position of the aperture mask.

What happens when the mask has two equi-spaced bars (the equivalent of twin slits)?

2-bar mask 2-bar mask fft 2-bar mask fft psf

Each diffraction order comprises double images. Note the diffraction spike is no brighter than that produced by a single bar (the FFT program normalizes the peak intensity of the diffraction patterns - in reality the extra bar reduces the amount of light reaching the objective and results in a slightly fainter spike).

Will translating the twin bars alter the nature of the diffraction pattern?

2-bar shift mask 2-bar shift mask fft 2-bar shift mask fft psf

Note that translating the the twin bars does not affect the diffraction pattern which still crosses the optical axis.

As the number of bars in the mask is incrementally increased the separation of the image orders becomes increasing complex.

3-bar mask 3-bar mask fft 3-bar mask fft psf
4-bar mask 4-bar mask fft 4-bar mask fft psf

Energy in the PSF is lost to higher orders and the extra bars block more incoming light producing a fainter diffraction spike.

The brightness of the spike is a function of its width to the that of the entrance pupil (bar width / aperture ratio). A thin 4-vaned spider produces long thin faint spikes.

4-vane spider thin mask 4-vane thin mask fft 4-vane thin mask fft psf

A thick 4-vaned spider produces shorter and brighter spikes.

4-vane thick mask 4-vane thick mask fft 4-vane thick mask fft psf

So what happens if we cover the entrance pupil with a series of equi-spaced bars (a coarse diffraction grating)?

7-bar mask 7-bar mask fft 7-bar mask fft psf

Note the normalized spike brightness looks the same as a 4-bar mask. In reality because more light is blocked the spike will be fainter.

What happens if the entrance pupil is covered by diagonal bars?

9-bar-tilt mask 9-bar-tilt mask fft 9-bar-tilt mask fft psf

The normalized spike brightness looks brighter than the spike produced by a single diagonal bar. In reality, because almost half the light is blocked, it will be noticeably fainter.

Recently there has been a good deal of enthusiastic chatter on Cloudy Nights and Stargazer's Lounge and other astronomy forums about a focusing mask invented by Pavel Bahtinov. Bahtinov masks, as they have come to be known, have been produced Žn mass by keen astrophotographers, and commercial enterprises.

The Bahtinov mask produces three principal diffraction spikes, a vertical spike flanked by two spikes inclined ~20°. The spikes cross on axis at the focal plane. Shift the focus and the vertical spike shifts off-axis.

Lets see how the separate parts of the Bahtinov mask behave. First the horizontal rungs.

bahtinov mask rung side bahtinov mask rung side fft bahtinov mask rung side fft psf

Note the horizontal bars produce a vertical diffraction spike with overlapping orders which is fainter than the diffraction spike produced by a single bar of the same width.

Next the tilted bars.

bahtinov mask tilt side bahtinov mask tilt side fft bahtinov mask tilt side fft psf

And with both sides combined.

bahtinov mask bahtinov mask fft bahtinov mask fft psf

Ignore the reflections of the spikes off the mask boundary, they are artefacts. The combination of the horizontal rungs and the tilted rungs produce three spikes that cross the optical axis at the focal plane.

I have demonstrated that a single bar produces a perpendicular diffraction spike, and that a grid of bars produces a multi-order spike of lesser intensity due to energy loss (blocked light).

The three essential elements of the Bahtinov mask are the horizontal rung and the tilted rungs. What happens if only a single horizontal bar and only a pair of tilted bars make up the mask?

Y-mask Y-mask fft Y-mask fft psf

Note the same diffraction pattern, except much brighter (because less light is blocked). This mask will act in precisely the same way as the Bahtinov mask. Shift the image away from the focal plane and the central spike will translate orthogonally.

I have also demonstrated that it does not matter where in the entrance pupil the bar lies. The same diffraction spike will be produced, crossing the optical axis. So what happens if I use a mask with three non-intersecting bars?

non-intersecting-apex mask non-intersecting-apes mask fft non-intersecting-apex fft psf

Note the same diffraction pattern is produced. What happens if I spread the bars apart?

non-intersecting mask non-intersecting mask fft non-intersecting mask fft psf

Note the diffraction pattern remains unaltered.

And what happens if all three bars cross on axis?

intersecting mask intersecting mask fft intersecting mask fft psf

Again, the same diffraction pattern.

I trust this primer in Fraunhofer diffraction demonstrates how the Bahtinov mask functions, and how a superior mask consisting of a 'Y' shaped bar serves exactly the same purpose, but to better effect.

So save yourself the time and effort in making a Bahtinov mask, or save your money. Instead make a 'Y' shaped mask. Perhaps you could recognize the man who pointed out this economical alternative and christen it the "Lord" mask?

Chris Lord

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