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This spreadsheet illustrates the off-axis performance of a parabolic mirror. It does so by performing a trace of rays arriving at radial zones of the parabola at the angle specified by the user. The rays are distributed in circles, with the spacing between the circles arranged so that the difference between their radii encloses equal areas.
Thus the spaces between the curves will contain equal amounts of light energy. The actual intensity will determined by how much space the energy is spread over.
The spreadsheet is arranged into two worksheet, with the computations performed in the sheet labeled "Calc Sheet", while the graphical results are shown on the sheet labeled "Inputs and Plots".
The "Inputs and Plots" sheet is organized into three zones. First, the data inputs are presented in bold italic font. These inputs are:
1. The distance from the axis at which the image is to be evaluated. Zero is on axis. 2. The focal length of the mirror. 3. The diameter of the mirror. 4. The extent of focus offset desired. For perfect focus this is zero.
All of the above are given in inches.
In addition to the plotting zone, several numerical outputs are shown. These are:
1. The distance of the image from the axis in degrees.
2. The rms diameter of the image, in inches. This represents approximately 75%
of the total energy present in the image.
3. The equivalent diameter of the image in arc seconds.
4. The equivalent resolution in lines per millimeter. Photographic films
commonly achieve 100 lines/mm.
The plotting area shows the results of the ray trace, with the curves displayed representing the imaging of the circles of equal area onto the focal plane. In addition, in the upper right hand corner of the image plot, the Airy disk is shown, to illustrate the size the image would have if the performance of the system were diffraction limited.
In order to see the effects of coma and astigmatism (the primary off-axis aberrations of a parabola), the user should set the focal length to 20 inches and the mirror diameter to 4 inches. This represents a 4" f/5 newtonain telescope. By varying the distance from the center from 0 to, say 0.5 inches in steps of 0.1 inches, the progressive effects of coma and astigmatism can be clearly seen. For small distances from the axis (say 0.1) the primary impact will be coma, which makes the image look like a little wedge, or comet nucleus. As the distance from the axis increases to 0.3 a second effect is seen, producing a small vertical feature near the "head" of the comet. This is astigmatism. As the distance from the center is increased the effect of astigmatism becomes more and more pronounced.
So the reader does not lose interest it might be a good idea to play around with the plots before reading further in this discussion.
The progressive increase in the aberrations is governed by two general trends. First, coma increases linearily with the angle from the axis and varies inversely as the square of the focal ratio. Second, astigmatism increases as the square of the angle from the axis and varies inversely as the forth power of the focal ratio. Thus, for small angle coma dominates, while for larger angles astigmatism is the primary defect in the image.
The "Calc Sheet" does all the work. The top seven rows are basic parameters. The names are self-descriptive. The radius of each of the 10 zones is given in column B, rows 10 through 419.
In order to assure that the radii of the circles are arranged so that they enclose annular rings of equal area the values in column must satisfy:
R1^2 - R2^2 = R2^2 - R3^2, where R1, R2, and R3 are three consecutive (decreasing) radii.
By setting the first value equal to the radius of the mirror and the second value equal to the radius times the square root of 0.90, we will have 10% of the energy contained in the first annulus. The recursion equation does the rest.
Each of the radial zones is in turn broken down into 40 points, with angles evenly spaced around each zone. The angles are given in column C. The radius and the angle are then used to compute the y and z coordinates of the points. These values are given in columns E and F.
Column D contains the x coordinate of the point on the parabola and is computed using the formula:
x = ( y^2 + z^2 )/(4*f) - f
where f is the focal length of the parabola. This puts the focus at the point (0,0,0), the origin.
In order to compute the reflected ray the normal vector N must be computed. This is done by calculating the gradient (using calculus). The three components of the normal vector are:
Nx = 1, Ny = - y/(2*f) Nz = - z/(2*f)
In order to compute the reflected ray this vector should be normalized, as is done in the spreadsheet.
If the reader knows a little calculus the imaging properties of other surfaces can be explored by changing the formulas in the five columns D (the value of x), G (the norm or length of the normal vector), and H, I and J, which are respectively the x, y and z components of the normal vector.
Columns G through J compute the normal vector, N, to the surface at the corresponding point (x,y,z). In turn, Columns L through N compute the incoming ray W. Since there is no optical element in front of the parabola all the incoming rays are the same. The next three columns (P, Q and R) compute the reflected rays. The next column (S) computes the norm of the reflected ray (V). This should be one (and it is). The purpose of computing it at this point is to insure that the computations are correct (at least in this sense).
The relationship between the incoming ray (W), the normal vector (N), and the outgoing ray (V) is:
V = W - 2(W dot N)N,
where 'W dot N' is the dot product of the vectors W and N. Column O computes the dot product, while columns P, Q and R perform the multiplication by two and the subtraction.
The distance T of the focal plane from the point (x,y,z) is now computed in column T. This is computed by the formula:
T = (Focus_offset - x0 )/Vx
where x0 is the x coordinate of the point given in column D, and Vx is the x component of the reflected ray, given in column P.
The next three columns (U, V, W) compute the point at which ray intersects the focal plane. This is done by computing the vector:
Xfocal = V*T + X0
where X0 is the vector (x,y,z) to the point on the mirror, V is the reflected ray, and T is the distance computed above. The value of xfocus (the x-component of the vector Xfocal) should always be the same as Focus_offset. The values of yfocus and zfocus represent the point at which the ray intersects the image plane as seen by the eyepiece or photographic film. This is what we will plot.
The yfocus (column V) and zfocus (column W) values cluster (but not symmetrically!) about the distance specified as the input on the first sheet. As a result the average values of yfocus and zfocus are computed in cells X4 and Y4, respectively. In columns X and Y, rows 10 through 419, the difference of the values in columns V and W and the averages are calculated. These are the values that will be plotted on the plot sheet. Finally, the standard deviations of the values of yfocus and zfocus are computed.
Columns AD through AF compute the coordinates of the Airy disk shown in the graph.
I hope the readers find the discussion useful. Have fun!
Mark Christensen