We all know the importance of achieving the best focus achievable before starting an imaging session. This can be a long and tedious process if attempted without the proper tools. Such a tool is a Focus Mask. Several focus masks have been designed. Some have been barely useful others are nearly a must for serious astro-photographers. The amazing thing about these masks is that they rely on a simple process, using the diffraction spikes generated from a mask to achieve a good focus. Most of the masks designs have been done using trial and errors or limited understanding of the patterns generated by such masks. A scientific approach in designing a mask can result in best possible focus by eliminating the flaws in the mask geometry and trying to optimize the mask/software design used in reporting how far we are from critical focus.

The Bahtinov mask was designed as a visual tool, to achieve better focus quality by visually centering a diffraction spike with respect to the crossing of 2 other spikes. It works pretty well compared to other types of mask, but it was never intended to be used with a computer assisted program. Although you can achieve a better focus using a computer to evaluate the spike location, the design of the Bahtinov mask limits the accuracy achievable. The GoldFocus mask was designed specifically for computer assisted focusing.

The following analysis is my own independent analysis and is supplemented by information provided by Jeff Winter designer of the GoldFocus mask . He has taken the time to analyze thoroughly how the diffraction spikes generated by focus masks behave and based on this analysis he has the developed a more optimum design. He has agreed to provide to me detailed information on the design of is GoldFocus mask in order to help me and other people, like me that are skeptical about new products advertising how much better they are when some of them cannot fulfill their claims. Some of the information that was given to me to complete this analysis is proprietary and therefore cannot be shown here but the simplified information supplied here is derived from it and should be sufficient to understand why the GoldFocus mask is superior to the Bahtinov mask.

The Bahtinov mask is not an optimum design. The displacement of the spikes is not same for all the spikes. This phenomenon is caused by the geometry of the mask, the spikes move at different rates depending on their location. Although not exactly the same, you can visualize this effect on your own by looking at a severely out of focus image using a Bahtinov mask, as show below. This image was taken on a Newtonian telescope, which clearly shows the central obstruction, and a homemade Bahtinov mask which is not perfectly centered. If you are far enough from optimum focus, you will see an image of the Mask, as you close in on the focus the image of the mask shrinks and the slots move in closer as shown in the following image. The length of the green lines shows the horizontal motion and the point where the green light intersects the yellow lines shows the vertical motion of the lines. Those distances are clearly different depending on which slot you consider.

The diffraction spikes behave in a similar way. This shows that the motion is not the same depending of the various locations of the slots. Radially the distance to travel is the same but horizontally the distance varies as a function of how far from the center of the pattern the slots are. The slots that are further away from the center move faster to reach the center. In the case of the diffraction spikes, the resulting pattern is the sum of all the diffraction patterns generated by all the slots and this causes the spikes to get distorted and also to become thicker so that their locations will not be same as the ideal spike. This results in an error in the spike position which translates into not reflecting exactly how far from ideal focus you really are and with less accuracy than possible if the design is optimized to reduce this effect.

The following image was produced by the freely available software Maskulator available here. It shows that the central line of the left image, generated with a Bahtinov mask if composed of a multitude of little moon crescents. Look at the ones in yellow they can clearly be seen. These crescents change the rate at which the central line moves. You can also see that the picture on the right generated with the GoldFocus mask is nearly free from this defect, although it is not possible to completely eliminate this effect it is so much smaller that it can no longer be seen in the image. Note that this image is for a very narrow band of right light (< 10 nm), when using the full spectrum of light these crescent will merge together and make it impossible to see them. It just shifts the spike and makes it appear to be thicker. It appears like noise when attempting to measure the spike location

The following image generated for a Bahtinov mask in white light and zoomed in clearly shows that the center spike is much wider than the other two. The bright area of the spikes is delimited by the red lines and the edges of the spikes are delimited in green. It can be seen that the central spike is about 2 times wider than the other spikes, and the dimmer portion of the spike is also over 2 times wider. Quantifying the Signal to noise ratio based on this is a difficult task, but we definitely say that the results of this effect is a decrease of signal to noise ratio of the central spike, making it more difficult to measure its accurate location with respect to the other spikes.

Jeff has provided me with the math that quantifies exactly this effect and although I am not at liberty to specify the lost of sensitivity of the Bahtinov mask, I can tell you that it contributes to the overall improvement in accuracy of the Gold Focus mask. The geometry of the GodlFocus mask also increases the Signal to noise ratio significantly (8 times better than the Bahtinov mask) improving the capability to make accurate measurement of the spikes, resulting in more stable measurements (less fluctuations on reported focus error).

The improved geometry in the design of the GoldFocus mask causes the overall geometry of the spikes to move 4.8 times faster compared to the Bahtinov design. You can see part of this in the first image above and even clearer in the image below. Look at the intersection of the 2 nearly horizontal spikes, in the right image (GoldFocus mask), it is located much more to the right then the central spike in the Bahtinov mask (left image). Jeff also discovered that his initial approximation of 4 was very conservative as is turned out to be closer to 4.8. Although not a large increase, the mask is even better than the initial value quoted.

In the image below the spikes location has been highlighted by drawing a black line going through the spikes, allowing us to see clearly the intersection point of the spikes otherwise masked by the bright central core of the star. The following zoomed image shows the increased sensitivity of the GoldFocus mask. Let's start by using the intersection of spikes A and C as a reference for our measurements. Using the green scale which has tick marks every 2 pixels, and if you zoom in the picture you can clearly see that the vertical spike (spike B) moves about 7 pixels to the right while the intersection point of the two near horizontal spikes (spikes D and E) moves about 18 pixels to the right. This is a bit misleading since the vertical spike (A) is moving right while the other two (A and C) are moving to the left. The net result is that the center spike (B) as moved half of the distance (3.5 pixels). The real displacement becomes 3.5 for the central spike (B). For spikes D and E, we must take into account that the 18 pixel value includes the motion of the spikes A and C so we must subtract its motion from the motion of the horizontal spike: 18-3.5 = 14.5, the net displacement is 14.5 vs. 3.5 or about 4.1. Considering that with this picture we can only measure the spike displacement with a resolution of 1 pixel, it matches pretty much the calculated value above.

Note that the mask entered into Maskulator for the GoldFocus mask is only an approximation of the GoldFocus mask. It has been generated based on the picture of the mask and its geometry may differ slightly from the real thing. As a result the measurements done above may deviate from theoretical calculated values. So we can only get close to those values and not expect a perfect match between measurements and calculated values. Considering this fact the values come pretty close.

Software of the GoldFocus mask

So now we can see that we have a better mask design, but what about the software used to report focus error. Jeff has supplied me with a copy of the user's manual and based on the information in it I can tell you that using his software requires a little more preparation then the Bahtinov mask.

• The mask needs to be aligned toward the North for an equatorially mounted telescope or up for an alt-azimuth mount. This allows the software to report the direction of focus in/our properly and allows the software to remember the in and out direction the next time you use the software.
• The software needs to be told where your acquisition software is going to store its image files. It will be remembered next time you start the software, but since it is a good idea to use a separate directory for your focusing you should change it every time, otherwise the software could delete the files you are imaging if you forget to close the software once focusing is complete.
• The software must learn the way your acquisition software is storing its files, this occurs automatically and only takes a few seconds.
• For optimum performance or for advanced users, some tweaking may be required, such as changing the annulus radius, width and number of images to stack. If your CCD pixels are not square, you may also need to change the slot angle, unless your acquisition software is re-sampling the data to produce square pixels in its output files. Unless your CCD pixels width to height ratio exceeds 1.25 you should not have to worry about it since the software can cope with this automatically

Jeff's software is able to read in fits files allowing him to get the full 16 bits of resolution that your imager generates. This gives a finer resolution in the pixel data as opposed to doing a screen capture which is limited to 8 bits per pixels. This results in more accurate calculations. Jeff is not attempting to measure each spike individually but analyzes the whole pattern. This allows all the pixels to be used for the analysis, as opposed to using only one third of the pixels for each spike. He is using the data from all 5 spikes at once. This way he gets a better averaging of errors so he can get a more stable reading of the focus error. Another advantage of this is that doing one analysis instead of three should require less time to perform. Yes, I know that the analysis can get more complicated and take more time, but the approach which I cannot discuss here due to its proprietary nature, attempts to do it in a way that is very fast and efficient, sort of like solving simultaneous equations.

Helps imaging with multiple filters

If you are using filters to do your imaging, you have known the pain of having to refocus after each filter change, and not having a bright enough star to do the focusing, resulting in slewing the scope to a nearby bright star and then back to the object you want to take a picture of, only to find out that the image is slightly displaced so either you have to reposition the scope or expect to have to crop the image where the image for each filter do not overlap. Well there is a better way using Jeff's software. A onetime filter calibration can be done like this.

1. Select a long exposure time and carefully focus using the first filter or combination of filter (ex: O3 + IR filter). This step should be done using long exposure times and stacking 15 images to provide the best focus possible.
2. Change filter to a clear filter (or IR only) and reduce the exposure a little to compensate for the difference in the filters, the software will report a focus error.
3. Record this focus error as your filter calibration value.
4. Change to another filter you want to use and repeat step 1-3 above for each filter you want to calibrate.
5. Once you have the calibration value for each filter, when you want to use one of your calibrated filters, you switch to the clear filter and adjust focus to get the calibration value of the filter instead of 0 focus error, this results in brighter diffraction spikes so that you can focus on dimmer stars, reducing the possibility of not having a bright enough star to use.
6. You now have optimum focus for your filter and can start capture data

Conclusion

We now have for the GoldFocus mask relative to the Bahtinov mask:

• Increase in sensitivity of 4.8 times.
• Increase in Signal to noise ratio (8 times).
• Analysis Software that uses all data (5 spikes) at once to compute focus error instead of dividing data into separate spikes yielding 5 times more data to average (better averaging = more stable measurement)

Jeff has obviously been studying and analyzing the diffraction patterns caused by the geometry of the Bahtinov mask and has identified the problems to be addressed to produce a much better design. The GoldFocus mask is such a design. This is by no way the perfect mask, there could still be some improvements that could be made to the mask to make it even better but at this point, the improvements might not improve the final results due to limitations of focuser accuracy. The quote Jeff: The resolution of the GoldFocus mask focus measurement is slightly better than the best commercially available electric stepper motor focusers so there is not much reason for making it better.

Additional information

Focuser motion estimation

Here is some very interesting information about the conversion of the pixel errors reported by the software. It is possible to convert this value into focuser displacement. This provides additional data to support the fact that the GoldMask has plenty of sensitivity to achieve critical focus

The critical focus zone has been highlighted to visualize the target to be reached during focusing. It can clearly be seen that the critical focus zone extends to over 4 pixels, so getting critical focus means that if the GoldMask software reports anything below 4 pixels so are within the critical focus zone and any improvements you make will provide the sharpest focus achievable. Note that the CFZ covers the range of maximum and minimum focuser travel so to be in good focus you must me within 1/2 of this value, indicated by the red line. Even so the GoldFocus mask is within 2 pixels. That's plenty of resolution to report the focuser correction to apply. Note that the CFZ value shown here applies to Halpha spectral line frequency, and should be recalculated for whatever actual wavelength you intend to image.

For other Focal ratios the CFZ decreases as a function of the square of the ratio of the F ratio of the telescope (if the F ration goes from 4 to 3 the CFZ will decrease in a ratio of 3/4 squared (9/16) roughly half of the value applicable at F4.

The focuser displacement depends on the size of the CCD pixels and on the Focal ratio. The approximate formula to relate pixel error to focuser displacement is

Focuser Travel (microns) = PixelError x CCDPixSize (microns)* FRatio /2.5

The above formula is only valid for a small value of Pixel Error. If the Pixel Error is too large the defocusing of the spikes causes the relationship to become non-linear.

The critical focus value used for the above graph was based on the value reported by the Maskulator software, Jeff Winter has documented how to calculate the critical focus zone here and also provides additional information on an improved critical focus zone for astro-photographers on his web site at on this page. The calculations based on his formulas give a lower value for the CFZ. This data can be used to see that even with this reduction in critical focus zone the GoldFocus mask has enough resolution to reach well into the critical zone.

Real world testing

This section will be completed once I have received and had time to test run and analyze the focus achieved using the GoldMask to confirm that all that theory matches reality. I plan to use a digital caliper to measure focuser displacement and produce a graph showing focuser displacement vs. Focus error reported by the software. The setup that will be used is an Orion StarBlast imager version (114mm aperture at F4) with a DSI Pro2 imager:

To be completed...