190508 Foucault test tolerance origin

You asked about the origin of the tolerance (x=2pR/r), and I didn't know, as it came from page 3 of Tom's 4-page Foucault test description.  The detailed description is also avoided in Harbour's paper.  Harbour says:  The amount of error that is allowed for the location of the focal plane to deviate from its ideal location for any given zone on a mirror has been worked out for us with the science of geometry. For our purposes it is not necessary to elucidate the entire method of determining the allowed error.  Thank you very much Mr Harbour!

My curiosity was piqued, so I did a Google search.  It comes from the Rayleigh criterion and the Airy disk first minima width (small angle sine approximation used, i.e. sin(x) = x).  See: https://www.quora.com/Where-does-this-%CE%B8-1-22-%CE%BB-D-come-from-Whats-its-derivation. 

The variable names used in the quora paper are different than in Tom's derivation, i.e., Tom uses D as the mirror focal length, while the quora reference uses D for the lens (mirror) diameter (I'll call it D' herein to avoid confusion).  They are related by f/, the focal ratio (your mirror is f/5.09). 

So, the quora paper has p=1.22λ/D', and Tom uses 1.22λf/D.  These formulas are equivalent and confusing because of the D's represent different measurements.  Note that if D' is the mirror focal length, and Tom's mirror diameter is D,  then D = D'/f, where f is the f/ ratio.

I know the above is a lot of information.  If you have any questions, I welcome them.

Enjoy,

Bruce

190502 Foucault mirror measurement data reduction spreadsheet

Foucault measurement data reduction spreadsheet

bkm	Thu, May 2, 1:45 PM (7 days ago)

to Tim, Thomas, me, Mike

TimC, TomT,

We were rushed a the end of Tuesday's workshop, and we didn't have much time to explore the Foucault data reduction spreadsheet I created for you.

In the short time we had, you picked up very quickly on what Excel can do, and asked some great questions.  The purpose of this email is to explain spreadsheet operation in more detail so you can peruse it at your leisure.

The attached pdf file has three pages of Excel screen captures for the Foucault data reduction.  The first page is the spreadsheet as normally presented, i.e., numbers in the cells.  The second page shows the actual cell contents (numbers, formulas, labels, etc.), and the third page is the resulting plot showing the average measurement results for the dummy data I created in cells M17 through Q21.

You can follow along what the Excel spreadsheet does from the Foucault hand calculations page also appended.  The hand calculations simply progress from left to right, i.e., the number calculated in a column is used in the next column to the right.  In the hand calculations, the formulas used are shown at the top of each column.

Every Excel cell has an address consisting of a letter (for the column), and a number (for a row).  The column letters and row numbers are shown in the borders at the top and left side.  An Excel cell can contain any of several things: a label, a number, an equation, or be blank.  For example, cell B17 contains the number 0.9, and cell C17 contains an equation that references cell B17.  More on this later.

Labels are simply typed in and 'enter' depressed.  Labels provide spreadsheet instructions, guidance, or information, such as in data entry, or data results output.

In a cell containing a number, the number is just typed in and 'enter' depressed.

A cell containing an equation always starts with an equal sign"=".  If you wanted to calculate 3 + 4, you would type in =3+4 (no spaces), then enter.  The number 7 would be displayed in that cell.  If that cell is highlighted with the cursor, the cell contents will be shown at the top left of the spreadsheet as "=3+4".

Excel provides algebraic and trigonometric functions.  For example, the relative percent stick position is the square root of the percent Zone area; hence, the numbers displayed in column C are the square root of the numbers in column B, i.e., 0.95 shown in cell C17 is the square root of the number (0.9) shown in cell B17.  "SQRT" is the mnemonic Excel uses for square root, thus cell C17 contains "=SQRT(B17)".  The argument is always enclosed in parenthesis and no spaces.  You can have multiple nested parentheses for complicated expressions.  The actual cell contents (numbers, equations, or labels) can be seen on the second page of the attached pdf file.

Excel aids in copying cell contents by automatically indexing cell callouts.  For example, When I created the spreadsheet, I typed into cell C17 "=SQRT(B17)".  I then highlighted that cell and selected 'copy' (ctrl-C, or copy from the edit menu).  I then pasted this expression into cells C18 through C21 (ctrl-V, or paste from the edit menu).  Excel automatically indexed the SQRT argument (cell reference) to B18 through B21.

Dollar signs indicate cell references that are not supposed to be automatically indexed, such as constants that are calculated and stored in other cells, such as f/ ratio, as calculated in cell G5, or p as calculated in cell G6, or numbers that are input, such as the mirror diameter in cell C5, or the radius of curvature in cell  C6.

The radius of curvature is used in the calculation of tolerance as shown in column E.  In the second pdf page (with the actual equations), C5 is referenced through dollar signs, i.e., $C$5, which tells Excel to not automatically index this reference.  You can limit automatic referencing to just cells or columns, i.e., $C5 would not allow automatic referencing on columns, but would allow automatic referencing on rows.  Likewise for the converse: C$5.

A cell can have any of 65,000 different background colors, sixteen of which are standard Windows colors, such as turquoise.  I have colored in turquoise the backgrounds of data entry and data output cells to make them stand out.  The background color has no effect on any operations.  A cell can have one of many outlines in the same or different color (or none).  I have outlined the data entry and output cells in dark blue.

Text can also be many colors.  I made the pin location output numbers dark blue, otherwise, the text is black.  These pin location numbers are what you can easily calculate as I detailed in the hand calculation page previously appended, and again appended here for convenience.

I may be approaching information overload, or maybe have exceeded it.  If you want to explore Excel and the Foucault data reduction spreadsheet, I'll bring my laptop to the next workshop.  Let me know.


Bruce