Parfocal and Critical Focus Zone

 

Statement of the Problem

The term "parfocal" is generally taken to mean that refocusing is unnecessary among different filters in a particular system. 

The term "critical focus zone" or CFZ is defined based upon the properties of the Airy diffraction disk to represent a region within which there should not be a measurable difference of focus. The CFZ depends upon the wavelength of light and the f-ratio of the optical system. 

When filter manufacturers state that their filters are "parfocal", they mean that the filters are made to a specified thickness tolerance. They can provide filters that will not increase the focus shift that is already in your optical system, but cannot guarantee that  you will not need to refocus among the filters.  Thus, parfocal is a system property.   It is likely that refractors will focus differently at different wavelengths (colors), moreso than pure reflectors. 

Lastly, this article addresses the meaning of CFZ and questions the traditional thinking that there is no need for filter offsets within the CFZ.  Modern focus analysis tools, such as FocusMax, may allow for measureable focus improvements within the CFZ that would indeed call for the use of focus offsets available in many data acquisition programs. So, it raises the question -  Can we measure focus differences in the "zone of no focus difference"?

Equipment

Takahashi FSQ-ED with RoboFocus
Astrodon 92-98 mm adaptor for TAKometer
Astrodon TAKometer (cable not connected in this photo)
Takahashi CAA
Astrodon CAA-APF  (92mm - 2.7" female, 0.4" thick)
Astrodon Spacer27 0.15" long "washer" fitting over 2.7" threads
0.75" Astro-Physics 2.7" extension tube
Astrodon MonsterMOAG off-axis guider with Starlight Loadstar guide camera
Apogee AI-FW50-9R filter wheel with 9 50mm slots
Apogee U8300 with 3326x2504 5.4 mic pixels

The Takahashi FSQ-ED is an improvement over the original FSQ106N refractor, the latter of which had significant focus differences between red, green and blue.

Testing Method

Focus tests were performed on this FSQ-ED equipped with a Technical Innovations ROBO FOCUS attached to the left focus axle shown above and using  FocusMax for focusing.     5 focusing V-curves were used to calibrate the Robo Focus in FocusMax and 6 independent measurements of Astrodon Generation 2 Red, Green and Blue filters were made near the zenith, all within a period of 7 minutes. Each filter was measured 3 times (R,G,B) and then the sequence was repeated to account for thermal changes.   The thickness specification for these filters is 3 mm  +/-25 microns (+/-0.025 mm) and are parfocal on most reflectors.  Parfocal behavior on refractors depend upon how well they are color corrected and the f/ratio.  The critical focus zone (CFZ) decreases with faster optics.

Results

The Robo Focus position value (tick) for each measurement is listed below.            

  Red Green  Blue
  4481 4482 4480
  4481 4482 4480
  4481 4483 4480
  4480 4482 4481
  4481 4482 4481
  4480 4482 4481
Average (ROBO Ticks) 4480.7 4482.2 4480.5
Std. Dev. 0.5 0.4 0.5
Offset ROBO Ticks -1.5 0.0 -1.7
Offset (mic) -40.6 0.0 -45.1
Crit. Focus Zone (mic) 79.3 67.1 54.9

 

The results show that there is a slight color difference at about 3 standard deviations between the green and the other two filters. Please note that Robo Focus and FocusMax report integer values, so the averages would be rounded to 4481, 4482 and 4481 for R, G and B, respectively. 

Each tick was determined to be about 27 microns by measuring (caliper) a large displacement of the focus tube and noting the difference in ticks.  Therefore, the offset between red or blue and green is 40-45 mic. (27 mic if you used the displayed numbers). 

This value is within the critical focus zone for each of the wavelengths, using the standard formula, CFZ = 4.88 * Wavelength(mic) * f/ratio^2.  This CFZ is a theoretical range of best focus accounting for the effects of turbulence, diffraction and other factors that spread the light of the star into a (Airy) disk.  The practical (rather than theoretical) CFZ is 10-30% larger.  The differences in focus for this system are within the CFZ.  The word "system" is used because these variations may include differences in filter thickness. 

This is where it becomes conceptually complicated, because FocusMax is predicting the best focal point (i.e. the center of the CFZ) by projecting a fitted calibration from two defocused images at different focus points OUTSIDE the CFZ.  

The conundrum is that there is a reproducible focus offset between the green filter and the others within the classically defined CFZ.  Thus, all filters are at best focus, i.e. within the CFZ, even though the method appears to have better precision than that. That would make the system parfocal for all filters by definition (within the CFZ).  The question is whether  images taken 1 tick  apart (27 mic.) show any measureable difference zoomed in. 

The filter thicknesses were within 8 micons of each other. However, these thickness variations affect focal shifts by 1/3 their value due to refraction of light, so the 8 micron maximum difference translates to about 3 microns difference in focus (e.g the presence of a 3 mm thick filter adds ~1 mm of backfocus from the telescope).  3 mic. is ~ 1/9 of a tick.  Thus the small color differences are due to the optical design of the  FSQ-ED. 

The following shows a sequence of images with the green filter, starting at 4482 (FocusMax best focus) and decrementing by 1.  A CCDInspector graph of FWHM of these data (using 2.1"/pix)  is also presented.  Note that a return to 4482 from 4476 show excellent repeatability of ~0.05".  The calculated CFZ of 67 mic. for the green filter is ~ 2.5 ticks. 

Image Analysis

There are significant, measureable changes that the system can measure within the CFZ.    Therefore, it could be concluded that assuming any focal setting withn the CFZ would lead to "best" focus is not correct.  FocusMax can predict best focus within the CFZ from outside the CFZ better than that. 

 

Conclusions

  1. Tools like FocusMax allow prediction of best focus within the CFZ.
  2. This calls into question the traditional meaning of CFZ.
  3. Using these tools, it is likely that best focus for refractors will be different for each filter and that filter offsets should be used.
  4. Parfocal filters means that they will not ADD to the existing focus differences in your optical system.
  5. Parfocal is a system property.