When you run the Ronchi test, often there is found to have to do with the figures of the shadow lines that may be different from those that are usually presented to explain the most common defects in optical processing. Here you need a general method that can explain the behavior of any shadow line, and that is what will be explained in this post. Surely this test is the most simple and intuitive and can make the idea at a glance of the quality of the entire shaped mirror, But it is also true that until that is used to verify the spherical shape can be of great help, as it also assessed by eye straight lines it is not so difficult, while if you have to evaluate a dish or another may be getting help, but more than anything else as a support to other tests. So it will be explained how to evaluate the errors highlighted by the distortion of the lines, however, referring to a sphere. To do this the method exposed as will be explained in Malacara 1965 and that you can find on page 335-336 del libro “Optical shop testing” .
Before proceeding further, It takes a few reminders the geometrical meaning of the derivative: The derivative is nothing more than the value of the calculated slope of the tangent at a given point of a curve or better to say a function f(x).
The thus derived can be either positive or negative, positive when the tangent line is growing (then the profile of the function f(x) He is going up), negative when the tangent line is decreasing (therefore also the profile of the function f(x) It will be descending).
By analogy one may think that the value of the tangent as the slope is in a climb (positive if you go up, negative if it comes down, equal to zero if it is in plan or if you are on the top of a mountain or on the bottom of a valley). This helps us because we are going to calculate the value of the derivative of the error function existing between the analyzed surface and perfect sphere and we go back to the value of the function f(x), which for us represents the error or the difference between the analyzed surface and the surface of a perfect sphere.
The method consists in evaluating the test lines relative to a vertical reference line (which represents a perfect sphere) and observe when this line is to the right or left of the vertical line. When the observed line is to the right of the vertical means that the derivative of the error is positive, while if it is left in the derivative is negative. If it is exactly above the vertical derivative is zero. The more we deviate from the vertical and greatest line will be the value that the derivative at that point. N:B: This is true whether we are analyzing the right half of the mirror, because if we look at the left half is reversed what I said. The derivative is positive and negative to the left to the right of the reference line.
In order to further clarify the relationship between the derivative of the error function and error function, They will be presented and described two examples:
Nell 'image is represented in black the error function (or the deviation from a sphere), in blue while the derivative function (derivative of the error function). Nell 'example above we can see how at a certain point the black curve deviates from zero (point a) and begin to grow to a maximum (point C), and then remain constant. The derivative is zero instead function up to "A" then begins to grow until it reaches a maximum (point B) and then ridecresce down to zero at point C. You can think of interpreting even so, while climbing the slope of the road begins to rise gradually until it reaches the maximum slope about halfway from the top and then the slope becomes more gentle way up to the summit, where the slope that is practically zero.
In the second the principle is the same, However, after the point "C" the error falls and returns to zero at the point E ?? " (from then on we still have a perfect sphere). The derivative at that point becomes negative (because the error function is falling), assumes the maximum value (negative) to about halfway down the road and then returns to zero because we say that the "planes" descent towards the end, and then the slope is gradually decreasing. This step is critical because it will derive the function in blue and then we go back to the black (mathematically this step is resolved by integrating the derivative function, but if the concepts are clear above you you can do it all in mind)…
Returning to the method suggested by Malacara let's see a practical set that perhaps best explains the many theoretical examples ... Suppose that the Ronchi lines so present: what does it mean? I put a single line, us just that and only the dial on the top right:
The method says to compare the test line with a reference ... As a reference line take a straight line that represents a sphere, and then calculate when the surface analyzed deviates from a sphere ... Nell 'test example we can see how up to the radius "Ra" the mirror is already spherical, while the problems arise from the "Ra" radius onwards. Since we have already a part of spherical mirror see how it differs from the rest of the surface of that sphere, here is that the vertical reference line (the green) I put it in such a way that it coincides with the vertical section in blue. Often people think that by "Rb" radius on the surface profile is going down, and I say this because in all the tutorials that describe the most common mistakes, in the edge description retorted, there is shown a figure with lines toward the edge tend to fall and this generally leads to a bit of misunderstandings. As reported in tutorial or guide is absolutely correct, as before that the lines deform you have perfect sphere with perfectly straight lines, which in this example however does not happen (or rather it happens in part), so we have to look at the figure globally, not just locally. As we will now see the edge starts to be ribattersi “Rb”, I give “Rc” on…
Up to the point "a" of the Ronchi line is exactly above the vertical reference line, Therefore, here there is a perfect sphere, and then the error by it is equal to zero. Then moving toward the edge of the line is always farther and farther away from the reference vertical and rightward, This means that the derivative of the error will be positive and growing to the point "b". From the point "b" to "c" of the Ronchi line is right again, but decreases its distance from the vertical, This means that the derivative is still positive, but it decreases in value until it reaches zero at "c". the fact that in this stretch the derivative is still positive, although decreases its value means that the error profile is still growing although less steeply, and not that you are retorting ! From the point "c" to the edge, instead, It is that the error decreases because here the Ronchi line is passed to the left of the vertical, and then here the derivative becomes negative and consequently profile drops. Always in the picture, in black, It has been represented as it presents the error profile with respect to the sphere.
Gthem from elements mentioned above are sufficient to have a general understanding of the problem and explain well whatever the form of the lines that you will see during the test, but for a more rigorous treatment, required if someone is planning to be elaborated a small software to automate the process or for a deeper understanding of the method Malacara, Set out below are the photos of the two pages from the book "OPTICAL SHOP TESTING" from which were taken the information to write this post.
