The quick and time-honored answer to the ATM's question, "How good is good enough?" is the Rayleigh Criterion: the max minus min (or "peak minus valley", P-V) wavefront error cannot exceed ¼ wave. However, it has been noted that there are good ¼-wave mirrors and poor ¼-wave mirrors, so work on a mirror is often continued until the wavefront P-V is significantly and perhaps unnecessarily less than ¼ wave. Two criteria, solidly based on wave optics, are introduced here which provide a more precise answer: the Root Mean Square (RMS) error and the Strehl Ratio. It is shown that these criteria can be found by reduction of data from the usual amateur procedures, the Foucault, classic caustic, and the "poor-man's" caustic tests. Their use could reduce the amount of effort required to produce a satisfactory mirror.

These concepts have been incorporated in a DOS test data reduction program, SIXTESTS.EXE:

http://home.earthlink.net/~burrjaw/public/sixtests.zip

or a Windows version:

http://home.earthlink.net/~burrjaw/public/win6.zip

for the popular amateur mirror tests, which can be used for testing parabolic mirrors without understanding the underlying details. The details are presented here for those interested; they can also be applied to more complicated projects such as the classical or Ritchey-Chrétien Cassegrains.

The usual method of reducing the data from amateur tests, e.g., the Foucault test [1, p. 101-104] and the caustic test [2, p. 298], is to find the errors in the readings relative to the readings which would be obtained from a parabola of assumed focal length and then integrate these errors to find the mirror surface errors. It is more general to process the readings to obtain the surface profile directly without reference to a desired surface. In this way the surface errors relative to any desired surface can be determined, and it will be shown that for the parabola, the distance to the point of best focus can be computed rather than assumed. This ensures that the estimated errors are minimized, and that no errors are introduced by assuming a less well focused reference surface.

Fig. 2.1 shows a cross-section of the mirror with vertex at the origin; the x-axis is along the optical axis. (*x*, *y*) is a point on the surface of the mirror; *y* is the zone radius. A small patch of the mirror around the point (*x*, *y*) may be thought of as flat (the tangent plane to the surface at the point) with the

Figure 2.1. Mirror in Tester.

normal vector * n* shown in Fig. 2.1. Light from a
point source at any point along the normal will be reflected back along
the normal, and if cut by a knife-edge (KE) also on the normal, the
patch will just start to "gray".

Two points along the normal are of particular interest: the point (*f*, 0) at the intersection of the normal and the optical axis, with *f* measured in the Foucault test, and the point (*X*, *Y*), such that the distance from the surface point (*x*, *y*) is equal to the local radius of curvature *r* of the surface patch. *X* and *Y* are measured in the caustic tests and the curve traced out by the point (*X*(*y*), *Y*(*y*))
as a function of the zone radius is known as the evolute, or caustic of
the mirror cross-section. The caustic has a "trumpet" shape, two arcs
above and below the the optical axis, the caustic arc below the optical
axis corresponding to the mirror surface zones above the axis.

For either of these tests, if the x-coordinate of the surface at zone radius *y*
is known, the measurements can be processed to give the angle of the
normal and the slope of the surface, which allows one to find the
x-coordinate of the surface for a slightly larger zone radius. This
process is known as numerical integration of the differential equation
which expresses the slope of the surface at each point. The result is
the surface profile *x*(*y*) as a function of the zone radius *y*.
The details of setting up the differential equations are given for six
tests: fixed and moving source; Foucault, caustic, and poor-man's
caustic tests. The last section gives some methods for integrating the
differential equations.

The Foucault test has the source and KE on the optical axis.

The moving-source Foucault test has coincident source and KE,
implemented by a beamsplitter or a common knife edge as used in the
"slitless" tester. The vector (*x*', 1) with *x*' = *dx*/*dy* is parallel to the tangent to the surface at (*x*, *y*), and the vector (*f*-*x*, -*y)* is parallel to the normal, so the dot product

...........(2.1)

Solving for *x*', one gets the differential equation

...........(2.2)

Figure 2.2. Fixed-Source Tests.

Fig. 2.2 shows the mirror in a fixed-source tester. It will be
seen that the fixed-source equations are more complicated. For the
Foucault test with the source and KE on the optical axis, the mirror
patch normal is found by using the reflection law: consider two unit
vectors along
the rays from the mirror surface point to the source and to the KE; the
reflection law implies their sum will be parallel to the normal.
Usually, the source is assumed to be at the mirror's paraxial center of
curvature, but that is not necessary for the following formulation; the
distance from the mirror vertex to the source can be found with a tape
measurement. With the source at (*S*, 0), and the KE at (*F*, 0), the two unit vectors are

with *
* the distance from mirror point to source:
, and similarly for the mirror point to KE distance *
*. Using similar reasoning to that used to obtain Eq. (2.1), the dot product

......(2.3)

Solving for *x*' and some algebra gives the differential equation for the surface using fixed-source Foucault readings:

...........(2.4)

When used in the caustic test, the KE is mounted on an x-y stage so the two coordinates (*X*, *Y*) of the KE can be either set or measured. It should be mentioned that the caustic test, as usually implemented - set *X*, measure *Y* - is inherently less accurate than the Foucault test and the *Y*
(cross-axis) measurements should be made with an extra digit of
accuracy. Since the measurements are easier to make (no shadow
comparisons), averaging several readings at each zone and using
narrower zones is recommended.

The moving-source caustic test (or Gaviola) test has the coincident source and KE mounted on an x-y stage. The position (*X*, *Y*)
of the source/KE is thus on the normal to the surface (Fig. 2.1), and
with both coordinates of the position known as a function of zone
radius *y*, the surface differential equation for the moving-source caustic test is

...........(2.5)

When surveying a parabolic mirror with paraxial radius *R*, the zone's center of curvature is at

...........(2.6)

The accuracy of the Gaviola test can be improved by moving the source/KE until its x-coordinate equals *
*, and then measuring the y-coordinate *Y *at the actual center of curvature, which would be *
*for the errorless parabola (the minus sign on *
*
means the zone's center of curvature is on the opposite side of the
optical axis). These choices are not required or expected using this
differential equation formulation; in particular, the desired surface
does not have to be a parabola.

We will assume that the fixed source is at (*S*, 0) on the
optical axis; for a parabolic mirror, it is usually placed at the
paraxial center of curvature. In that case, as for the moving-source
caustic test, the accuracy can be improved by moving the KE until *X* equals the x-coordinate *
* of the image of the source from the zone being measured [3],

...........(2.7)

and *
*
is the expected y-coordinate of the image. As before, these settings
and expectations are not required for this formulation. In an extreme
case, if the desired surface were an ellipsoid, the source could be
placed at the far focus, and the measurements made in the vicinity of
the intended near focus.

With the same source-mirror point distance *
* as in the fixed-source Foucault test, and
, the surface differential equation for the fixed-source caustic test is

...........(2.8)

In the poor-man's test, either fixed- or moving-source, the along-axis value *X* is set and the zone radius *y*
is found for which this is the x-coordinate of the evolute
(moving-source) or of the image of the source (fixed-source). This zone
is found by the shadow behavior as the knife edge moves across the two
images from opposite sides of the mirror - the zone radius is the one
which darkens first as the knife edge starts to darken the mirror, and
the corresponding zone on the other side of the mirror darkens last
before the knife edge cuts off all light from the mirror.

Fig. 2.1 shows the unit tangent vector

and the surface normal vector
. Corresponding to the surface point
is the evolute point (the *X* value is measured for several zone radii)

with the local radius of curvature
. The first component
, or
, which, introducing the intermediate variable *v*, can be represented by the two first order differential equations

...........(2.9)

After interpolating the moving-source poor-man's observations , this is the pair of differential equations to be integrated using standard numerical techniques.

Deriving the differential equations for the fixed-source poor-man's test is considerably more complicated. The first step is to formulate the off-axis object-image relationship for a small patch of a spherical mirror, Fig. 2.3.

Figure 2.3. Small Spherical Mirror Object-Image

The relationship between the object and image distances is

with *R* the mirror's radius of curvature. I have not seen
this relationship in any reference, although it is closely related to
the paraxial Gaussian thin lens formula. It can be proven (with
difficulty) by either ray-tracing to first order in the mirror zone
radius *y* (methods nearly the same as in Banerjee's paper [3]), or requiring, to second order in *y,* constant object-image path length.

Fig. 2.2 shows the fixed source at distance *S* from the mirror vertex along the optical axis, distances
, and unit vectors
from the surface point to the source and image, respectively; as before, *r* is the local radius of curvature at the surface point. Solving the lemma for the local curvature,

The second derivative of *x* is found by the following lengthy computation. First,

By the law of reflection, the sum of the two unit vectors is perpendicular to the tangent vector

with . We get the two equations

Solve the first equation for , and substitute into the second:

Let
, and solve the resulting quadratic equation for *u:*

The complete algorithm for input to a numerical integrator of a pair of first order differential equations is

...........(2.10)

The popular 4th order Runge-Kutta method for numerical integration
of the surface differential equations has been used successfully for
reducing data from these tests. It was found that 1 mm steps in the
zone radius *y* were needed to match analytical results for some critical test cases.

With 1 mm steps in *y*, the Runge-Kutta integrator requires
derivative values every 0.5 mm. This in turn requires interpolation of
the discrete test readings, *
* at the measured zone radii *
*,
depending on the test. For Foucault and poor-man's caustic data
reduction, quadratic interpolation has been successfully used, but to
avoid false errors due simply to the interpolation, reasonably narrow
zones must be used. If the caustic tests were performed with the *X*-setting
in the vicinity of the local center of curvature (for the parabola,
Eqs. (2.6)and (2.7)), for greatest accuracy quadratic and cubic spline
fits could be made, in the form , with *n*=2, 3,* *to the *X* and *Y*
coordinates, respectively. This matches the form of the expected
position of the center of curvature for the moving-source test, or the
image for the fixed-source test. In the recently proposed Lateral Wire
Test (LWT), the test *X*-settings are usually constant [8]. It
was found that data from either the classical caustic test or the LWT
can be reduced using quadratic interpolation for *X* and *Y*/*y*.
As for the other tests, the zones should be fairly narrow, which is
already required to obtain accurate results from these lateral
measurements.

With the mirror surface profile known, e.g., by reducing the data
from one of the above tests, finding the best-fit parabola is
equivalent to finding the point of best focus, i.e., the focal length
and displacement of the parabola relative to which the deviations
minimize the 2D mean square error. This is derived in Section 5.3. A
parabola through the origin with paraxial radius *R* is given by the equation
, so the curve *
* is a parabola with displacement *A* and focal length 1/4*B*.

Given a mirror, radius *r*, surface profile *x*(*y*),
, the 2D mean square error between the surface and the parabola *
* is

...........(3.1)

The standard process for finding the *A*,* B* which minimize *F* is to set the partial derivatives equal to zero (
stands for
):

Written in matrix form, the two simultaneous equations in the two unknowns *A, B *become

Solving,

(3.2)

The minimum mean square error is, with the values
and *
*,

...........(3.3)

This is numerically difficult to compute. It was found to be
advantageous to integrate the two integrals in Eq. (3.2) and the
integral involving *
* in Eq. (3.3) at the same time the differential equation for *x* itself was integrated. Finally, as mentioned before, the focal length of the best-fit parabola is
, and the surface Root Mean Square error

With the mirror surface *x*(*y*) as a function of zone radius *y*, obtained by methods such as those described in Section 2, and a desired surface *
* (if the desired surface is a parabola, the methods of Section 3 apply, and
), define the surface error

The sign is important here; looking at Fig. 2.1, if then the actual surface is to the right of the desired surface (a mental rotation so the "glass is below" says that point of the surface is high). A plot of the surface error is valuable to see where to concentrate figuring action.

Figure 4.1. 6"f/8.15 Sphere Deviations.

For example, Fig. 4.1 shows the deviations of a 6" spherical mirror surface from the best-fit parabola. The deviation units are nanometers (nm) and the adopted wavelength of light 550 nm.

Figure 4.2. Random Mirror Deviations.

Fig. 4.2 shows the surface deviations of another mirror, a "random" mirror: the surface errors are chosen at random every millimeter. The random number generator was set to generate numbers uniformly distributed ± 1/16 wave = 34.4 nm.

The surface P-V is simply

...........(4.1)

To calculate the surface P-V of a spherical mirror analytically,
we show some of the details omitted in [1, p. 19]. Expand the equation
for a circle with radius of curvature *R* through the origin into a power series:

and subtract the parabola with paraxial radius of curvature *
*,
:

For a spherical mirror with radius *r*, it can be shown that the last term forces the parabola's *
* which minimizes P-V to be slightly smaller than the radius of the sphere:

and the spherical mirror's surface

...........(4.2)

To have a spherical mirror satisfy the Rayleigh Criterion, the surface P-V must be at most 1/8 wave (68.8 nm); for a 6" mirror (*r*=3"), Eq. (4.2) implies *R*
must be at least 97.8" (focal length 48.9", f/8.15), which is the
radius chosen for the sphere whose deviations are shown in Fig. 4.1.

For the random mirror of Fig. 4.2, the actual surface P-V = 64.8 nm, so the wavefront P-V is slightly smaller, 1/4.2 wave, and the mirror also satisfies the Rayleigh Criterion.

The RMS has already been introduced (Section 3) in the definition of the best-fit parabola as the square root of the result of a least-square fit. This is equivalent to the statistical weighted standard deviation of a general deviation which gives the surface

...........(4.3)

with the weighted mean deviation

Finding the RMS of a spherical mirror analytically relative to the best-fit parabola proceeds in much the same fashion as the calculation of its P-V in the last section, except the power series must be carried to 10th order; Mike Peck has carried out the computation, and details are available:

http://zebu.uoregon.edu/~mbartels/weighted-rms/paraboloid.html

The sphere's surface

Specifically, for a spherical mirror satisfying the Rayleigh Criterion (from Eq. (4.2), ), to first order the wavefront

...........(4.4)

This extremely important result can be used as a criterion for evaluating a mirror. To the extent that the RMS can be used to give a more accurate representation of a mirror's diffraction pattern than the P-V, a mirror with wavefront RMS = 1/13.4 wave is more likely to be satisfactory then one with wavefront P-V = ¼ wave.

The computation of the RMS of a random mirror satisfying the
Rayleigh Criterion, such as that shown in Fig. 4.2 is much simpler.
Combining Eq. (4.3) with the fact that the standard deviation of a
random variable uniformly distributed ±*a* is
, the wavefront

...........(4.5)

Derived from this simpler computation, wavefront RMS < 1/14 wave is often quoted as a criterion for a satisfactory mirror. Due to the variations of random numbers, the wavefront RMS of the random mirror of Fig. 4.2 is slightly better at 1/14.0 wave.

For the spherical and the random mirrors, the P-V and RMS criteria are pretty much in agreement. Where they differ are the mirrors with most of the error concentrated in either the center or edge. For example, consider two mirrors with ¼ wave wavefront P-V errors, i.e., both satisfying the Rayleigh criterion, a "high center" mirror with a constant 1/8 wave surface error from the center to 20% zone, and a "high edge" mirror with a high zone from 80% to the edge. The wavefront RMS for the high center mirror is 1/20.4 wave - a good mirror; the high edge mirror has a wavefront RMS of 1/8.3 wave - a bad mirror. This may explain the observation in the Introduction about there being good ¼ wave mirrors and bad ¼ wave mirrors.

This criterion comes directly from the wave theory of optics. In words, the Strehl Ratio is "the ratio of the light intensity at the peak of the diffraction pattern of an aberrated image to that at the peak of an aberration free image." [4, p. 311] It is shown in Section 5.3 that the definition is equivalent to

...........(4.6)

with the wavefront phase error
(rad) of the light coming from the zone with radius *y*.
The integrals in Eq. (4.6) are carried out numerically using Simpson's
Rule in the test data reduction program mentioned in the Introduction.
To my knowledge, the Strehl Ratios of a spherical mirror or a random
mirror satisfying the Rayleigh Criterion have not been computed
analytically. The Strehl Ratios of the mirrors of Figs. 4.1 and 4.2
were computed numerically with the results .800 and .809, respectively.
The criterion SR > .8 has come to be regarded as the best test for a
satisfactory mirror. The Strehl Ratios of the "high center" and "high
edge" mirrors are .918, .533, respectively, agreeing with the RMS
evaluations.

It is shown in Section 5.3 that the Strehl Ratio has the following approximation in terms of the wavefront RMS *W*:

...........(4.7)

This approximation is only good for small ; for values larger than this, the approximation goes negative, while Eq. (4.6) says that . The approximate Strehl Ratios of the two mirrors of Figs. 4.1 and 4.2 are .781, .800, respectively, showing that the approximation generally underestimates the Strehl Ratio (certainly true if the approximation goes negative!). Finally, the approximate Strehl Ratios, followed by the Strehl Ratios in parentheses are .905 (.918), .426 (.533) for the "high center" and "high edge" mirrors (both of which satisfied the Rayleigh Criterion), respectively.

Ref. 7, p. 238, Eq. (13.2) gives another approximation for the Strehl ratio in terms of the wavefront RMS *W*:

...........(4.8)

This approximation is apparently a slight overestimate for the Strehl Ratio, but it does have the advantage of satisfying the same limits as the Strehl Ratio.

The approximation in Eq. (4.8) is valuable in situations where the phase error is difficult to determine, such as Cassegrain systems, in which the RMS of each surface can be determined, and Eq. (4.8) is used to find the approximate system Strehl Ratio.

The details of the optical image formation from a mirror with or without errors are derived from the physical theory of the diffraction produced by a circular aperture. We will consider the case of a plane wave from an on-axis star impinging on a circular aperture the same size as the mirror. If one were to look at the aperture with a telescope a long ways off (so that the wavefronts from the aperture are also plane, the point of light would flash as the telescope was moved away from the "optical axis". This is known as Fraunhofer, or far-field diffraction. The aperture thus produces plane waves moving off at various angles to the optical axis, which we can imagine being focused by a perfect mirror replacing the actual mirror. This aperture/perfect mirror combination will be used to model a mirror with errors by "moving" the errors to the aperture as an aperture transmission function which will affect the plane waves moving away from the aperture and thus the image formed by the replacement perfect mirror.

In order to limit the complications, it is assumed that the mirror errors are a function of zone radius only; symmetry then tells us that the resulting image of a point source will have "zones" too, i.e., rings with brightness a function of the angle from the optical axis only.

The magnitude of the electric vector in the direction , for small angles, actually the half-angle of a cone with axis along the optical axis because of the symmetry of the circular aperture, is [5, p. 361]

...........(5.1)

with
and *
*
a Bessel function of zero order. The integral can be found in terms of
another Bessel function, so the brightness in that direction is

...........(5.2)

The brightness has been normalized to unity along the optical axis (the center of the Airy disk) by the factor in front of the integral in Eq. (5.1). Using a Bessel function table [6, p. 126-210], the brightness, Eq. (5.2), is plotted as magnitude below the peak brightness in Fig. 5.1.

Figure 5.1. Diffraction Pattern Brightness, 6" Parabola.

Fig. 5.1 is a plot of the brightness of the rings in the image. The valleys are black rings - zero brightness, - which actually go down to -infinity on the logarithmic plot. If the peak of the Airy disk has the brightness of a zero magnitude star, the second ring, at magnitude 6, is barely visible; the third and higher rings could not be seen.

It is of interest to derive the famous statement that 84% of the energy lies in the Airy disk. The total energy inside radius *w*' is

...........(5.3)

Therefore, the fraction inside the first dark ring, the smallest *w*' such that
, is
.

Unfortunately, it is necessary to use complex numbers to describe the zone errors (which have been "moved" to the aperture). We will be able to get rid of the complex numbers when a brightness expression is found. The aperture transmission function

Recall that
is the wavefront phase angle caused by mirror zone surface errors (cf. Eq. (4.6));
is the magnitude of the transmission. We will assume that
for every zone, i.e., there are no losses. The effects of apodization could be found by setting
for some zone radius *y*, but that is beyond the scope of this report.

Including the transmission function, the **diffraction integral** [5, p. 350] gives the (complex) magnitude of the electric vector

...........(5.4)

which differs from the errorless diffraction only by the complex exponential which includes the zone phase angle and thus the mirror zone errors. The brightness of the ring at angle is

...........(5.5)

Given the mirror zone errors, the integrals can be computed numerically to give the brightness of the diffraction pattern rings as a function of angle. For example, with the phase errors corresponding to the errors of the 6" spherical mirror satisfying the Rayleigh Criterion (Fig. 4.1), the Fig. 5.2 diffraction pattern is obtained.

Figure 5.2. Diffraction Pattern from 6" Spherical Mirror.

The blue curve is a copy of a parabola's diffraction pattern brightness from Fig. 5.1, and the red curve is the diffraction pattern brightness from the spherical mirror. Several interesting observations can be made about this plot. First, these integrations are performed in the programs mentioned in the Introduction and the plots are available in the form of plot files. Second, this is essentially a preview of a star test using only bench measurements, without all the difficulties of performing an actual star test, mounting the mirror, factoring out atmospheric effects, etc.

Note that the peak of the red curve is slightly below the peak of the blue curve; the plotted difference is small because of the logarithmic chart. According to the definition, the brightness ratio of the peaks is the Strehl Ratio. The width of the Airy disk doesn't change much, but the dark rings of the errorless diffraction pattern have been filled in with the light stolen from the Airy disk, making the rings less distinct and brighter. If the peak is as bright as a zero-magnitude star, the second ring is again the last one really visible at 6th magnitude, but it has a larger radius.

Figure 5.3 shows the effect of a 20% obstruction (1.2" obstruction diameter) on the diffraction patterns of two 6" mirrors: the blue curve is from an obstructed parabola, and the red curve is from the ¼-wave sphere with the same obstruction.

Figure 5.3. 6" Mirrors with 20% Obstructions.

The red curve in Fig. 5.4 is the random mirror's (surface deviations in Fig. 4.2) diffraction pattern. Remember the Strehl Ratio is nearly the same for this mirror as for the spherical mirror, but the "star test" behavior of the random mirror is substantially different. The Airy disk and the first two rings are virtually unchanged, but the third and fourth rings are nearly as bright as the just-visible second ring. The random mirror apparently scatters light stolen from the Airy disk further from the peak.

Figure 5.4. Diffraction Pattern from 6" Random Mirror.

The definition of the Strehl Ratio, Eq. (4.6), is easily derived from Eq. (5.5) by substituting
(to find the on-axis brightness), which implies *w*=0, and noting that
.

We wish to show that the process of minimizing RMS to find the best-fit parabola (Eq. (3.1)) also maximizes the Strehl Ratio for small deviations. Set up the function to be maximized (as before, stands for ):

with
. Taking the partial derivatives, it is easy to show that
, even without making the small angle assumption, which means that the Strehl Ratio is independent of *A*; this is obvious because *A* doesn't affect the relative phase between zones. Setting the partial derivative with respect to *B* to zero,

and using the small angle assumption,

Solving for the optimizing *B*,

which is equal to the *
* found which minimizes the RMS, Eq. (3.2).

To prove the formula (4.7) for the approximate Strehl Ratio for small deviations in terms of the RMS, start with the definition of the Strehl Ratio, Eq. (4.6),

Using the small angle (deviation) assumption (note that it is necessary to go to second order here),

Substituting the expression of the wavefront phase error in terms of the surface deviation after Eq. (4.6),

The expression in brackets is in Eq. (4.3) for the surface RMS *
*, so the approximate

Eq. (4.7) is obtained with the observation that the wavefront RMS is twice the surface RMS.

The EER is the ratio between the total energy inside a given angular radius from a mirror's diffraction pattern to the total energy inside the errorless circular aperture's diffraction pattern, Eq. (5.3). A plot of the EER is available in the plot file generated by the program mentioned in the Introduction. For example, the 6" spherical mirror's EER is the red curve in Fig. 5.5 and the random mirror's EER is the green curve.

Figure 5.5. Spherical and Random Mirrors' EER.

These curves back up the observation in the discussion of Fig. 5.4, that the random mirror tends to scatter light further out than the spherical mirror. However, the EER may overemphasize this, since it does not take into consideration that the random mirror's light is scattered over larger areas, so the brightness of the scattered light may be so dim as to be undetectable. In Fig. 5.4, the random mirror's diffraction pattern brightness only becomes significantly more than the errorless pattern for the fourth ring and beyond.

The amateur mirror tests generally depend on the smoothness of surface slope as evidenced by the interpolation schemes required to integrate the differential equations describing the various tests. We choose a third order linear stochastic process, white noise into a sinusoidal 2nd derivative, to model the surface error. This choice is to allow test accuracy comparisons to include the poor-man's caustic test, which depends on curvature smoothness. Comparisons between different test scenarios, such as the number and placement of zones, can also be studied. The surface error model is

.................(6.1)

with first and second derivative errors
. The surface error wavelength *L* and variance parameter *q* of the white noise *w* will be chosen later from the stated accuracy of the Foucault test on a standard mirror.

The best-fit parabola's parameters , Eq. (3.2), satisfy

(*r* = mirror semi-diameter). The errors
are the final values of the solution to the corresponding error equation

...........(6.2)

From Eq. (3.3), the weighted mean square error (MS, square of the RMS) of a mirror with surface *x*(*y*) is

The fit parameters can be used as coefficients causing only second order errors vs. using the true surface, giving

.......(6.3)

Similar to treatment of the fit parameter errors, the first term in Eq. (6.3) may be regarded as the final value of the solution to the error equation

It is numerically advantageous to combine the coefficients of in Eq. (6.3) with the differential equations (6.2) to obtain the combined error model in the form of a set of linear stochastic differential equations

..........(6.4)

Given a set of linear stochastic differential equations
, with white noise vector* u*, the Lyapunov equation for the covariance matrix

For this problem, choose
, define the vector *
*, then, from Eq. (6.3), the variance of the weighted mean square estimate is
, with

An iteration of a Lyapunov equation solver showed that the variance parameter
of the white noise input to the second derivative error with wavelength *L* = 25 mm gave a fourth root
of 27.4 nm for the Texereau 200 mm f/6 "standard" mirror. For reasons to be detailed later, the choice of *L*=25
mm was chosen in the surface error model for all mirrors. A fairly
large Monte Carlo simulation of a stochastic process with those
parameters showed that .

In its simplest form to apply Kalman filtering, discrete linear observations
are added to the stochastic model, with observation matrix *H* and a vector of normal observation errors * v*.
The general principle to form the observation matrix is to linearize
(take the differentials of) the measurement differential equation and
substitute quantities from the parabolic equation in the differential
coefficients. A Kalman filter implementation requires the covariance
matrix

From Eq. (2.2) for the moving-source Foucault test,

Substituting coefficients from the parabola,

giving the observation matrix for moving-source Foucault

...........(6.5)

In the wire test version of the Foucault test, zone radii measurements are made at longitudinal settings. Parallel to substituting parabola quantities for the coefficients, the wire test observation matrix will be calculated by finding the relationship between zone measurement errors and longitudinal measurement errors for the parabola :

Solving for and using Eq. (6.5) gives the observation matrix for the moving-source wire test

...........(6.6)

Similarly, from Eq. (2.4) for the fixed-source Foucault test, the observation matrix

.......(6.7)

where the paraxial can be used in the coefficients. Using the zone radii vs. longitudinal errors for the parabola, the observation matrix for the fixed-source wire test

..(6.8)

An interesting comparison can be made here between fixed- and
moving-source tests. For the fixed-source Foucault test with the source
at the center of curvature (*S*=*R*),
; substituting *F*=*R*
in Eq. (6.7) shows that the elements in the resulting observation
matrix are twice those in Eq. (6.5). This seconds the known fact that
the KE increments in the fixed-source Foucault test are almost exactly
twice those in the moving-source test, due to the angle of incidence on
a mirror being equal to the angle of reflection. This affords some
accuracy benefit to the fixed-source test. However, making the same
substitution in Eq. (6.8) for the wire test and comparing with Eq.
(6.6) shows that this factor of two does not apply to the fixed- and
moving-source wire tests. For both of these tests, the fixed-source
coefficients could be increased by moving the source closer to the
mirror.

Texereau states, p. 85, that using 4 Couder zones with a
reading accuracy of ±.005" gives "a precision ten times better than the
Rayleigh tolerance." This 10:1 ratio between P-V estimates with and
without those Foucault observations also results when the P-V is
modelled by the standard deviation of the edge surface deviation and
the 2nd derivative error model (*L* = 25 mm, *q* = .039) of Section 6.3 is used.

From Eq. (2.5) for the moving-source (Gaviola) caustic test, the observation matrix

...........(6.9)

and, from Eq. (2.6), the observation matrix for the fixed-source caustic test is

..........(6.10)

Again, there is a two-to-one advantage to the coefficients of the fixed-source caustic test at the center of curvature which can be further increased by moving the source closer to the mirror.

In the poor-man's test, the longitudinal KE setting is fixed and the zone for which this is the longitudinal coordinate of the local center of curvature is measured. As for the wire test which makes zone radii measurements, it is necessary to refer radii measurement errors back to errors in the longitudinal setting. From Eq. (2.6),

Taking the differential of the second equation in (2.9) and substituting the above gives the observation matrix for the moving-source poor-man's test

.........(6.11)

The equations for fixed-source poor-man's test are complicated and constructing its observation matrix requires an extensive computation. The resulting observation matrix for fixed-source poor-man's test is

(6.12)

Substituting *S=X=R* shows that the fixed-source poor-man's
test at the center of curvature, like the wire test, does not have a
two-to-one coefficient advantage.

It is hoped that it has been demonstrated that the Strehl Ratio, and, for small deviations, the RMS, are better descriptions of the quality of a mirror. For example, it was shown that two mirrors with the same P-V value could have much different Strehl Ratios and RMSs, and it is likely that the relative performance of these two mirrors would vary as indicated by the Strehl Ratio.

Although the wave optical theory behind these measures is fairly deep, it must be emphasized that they can be computed as part of the reduction of data from the usual amateur mirror bench tests such as the Foucault and caustic tests.

However, as a topic for further research, two mirrors have been shown with the same Strehl Ratio, but quite different behavior of their respective diffraction patterns and EERs. It is not known whether a search for yet another single-number, stronger criterion should be pursued or this is not of significance for the visual observer and the astrophotographer.

I wish to thank Bratislav Curcic for suggesting the writing of this report as a reference for an article in the ATMJ, Nils Olof Carlin for correcting several erroneous conjectures and misinterpretations and making possible programming of the application of this theory to data reduction of typical amateur mirror tests, and Michael Peck for sharing the very important analytic formula for the RMS of spherical mirror which provided a critical test case for the programs.

[1] J. Texereau, *How to Make a Telescope*, 2nd English Edition, (Willmann-Bell, Richmond, VA, 1984).

[2] J. Strong, *Concepts of Classical Optics*,* *(W. H. Freeman, San Francisco, 1958).

[3] D. P. K. Banerjee *et al*, "Improving the accuracy of the caustic test,", *App. Opt.* **37**(7), 1227-1230 (March, 1998).

[4] W. J. Smith, *Modern Optical Engineering*, (McGraw-Hill, New York, 1966).

[5] M. V. Klein, T. E. Furtak, *Optics*, (John Wiley, New York, 1986, ISBN 0-471-87297-0).

[6] E. Jahnke, F. Emde, *Tables of Functions*, (Dover, New York, 1945).

[7] H. R. Suiter, *Star Testing Astronomical Telescopes*, (Willmann-Bell, Richmond, 1994, ISBN 943396-44-1).

[8] J. Francis, "Improved Test Methods for Elliptical and Spherical TCT Mirrors,", *ATM J.* **14**, 34-39, (ISSN 1074-2697).

-- Jim Burrows

-- mailto:burrjaw@earthlink.net

-- JD2451003 = 1998-07-08

-- 1075 = -09-18

-- 1263 = 1999-03-25 "WRMS" -> "RMS".

-- 1606 = 2000-03-02 caustic interpolation.

-- 2142 = 2001-08-21 error models