The radius of curvature of an optical component is a critial factor for proper performance. As a result, it is critically evaluated and specified during design and maticulously measured during manufacture and testing. There are two primary methods for measureing the radius of curvature of an optical surface; the first is to measure the radius of curvature using an interferometer and a radius slide and the second is the view the interference fringes between a known test plate the test article. In general, the test plates are characterized using an interferometer and a radius slide. This tachtalk will provide an overview of the measurment process involving an interferometer.
The concept of measuring the radius of curvature is, on the surface, straght forward. Two position measurements are taken at two specific locations in front of the interferometer and, ideally, the radius of curvature is the difference between the two position measurements. The difference in position between the two measurements is determined using either a linear encoder or an optically based distance measureing device mounted to a rail system. The optic under test slides along the rail system parallel to the optical axis of the interferometer. The rail system is know as a "radius slide".
The location of the two measurements are know as the "cateye" position and the "confocal" position. The cateye position, shown in figure 1, is the position where the interferometer test beam comes to focus. Placing a mirror at this location that is normal to the optical axis will reflect the test beam such that it appears to be diverging from a point located at the mirror surface. As a result, the interferometer illumination focus and the apparent focus point of the test beam are co-located. The resulting ideal wavefront will perfectly match the reference beam producing a single infinite fringe known as a "Null" fringe. It is the generation of this Null fringe that informs the user that the test acticle is at the proper position for the first measurement.
The confocal position is the position where the test wavefront is normal to the test surface as shown in figure 2. This occurs when the test article is located at a distance equal to the radius of curvature away from the cateye position. At the ideal confocal position the test wavefront is reflected directly back on itself. The return beam from the test article will focus such that it overlaps with the cateye point, resulting in a Null fringe.
In effect the radius of curvature is the distance along the optical axis of the interferometer between two measurements where the fringes are "fluffed" out to a Null state.
While the measurement concept is straight forward, in practice it is not practical to align the test artical at the cateye and confocal positions for a perfectly Nulled fringe. As a result, it is necessary to use the interferometric measurements at both positions to correct the radius of curvature estimate provided by the radius slide. The power term from the cateye and confocal surface measurements is used to compensate for the slight misalignments at the two positions. The correction term is given by
$$\delta R = \frac{\phi\lambda}{1-[1-{(1/(2~F/\#_{limit}))^2}]^{1/2}}$$where $\phi$ is the measured power term in waves, $\lambda$ is the wavelength of the source in the interferometer and $F/\#_{limit}$ is the limiting F/# or R/# of each of the measurements. The limiting F/# and R/# will be discussed in the following section. The same equation is used for corrections and cateye and confocal, however the sign of the correction differs.
The correction required for a given power measurement increases with F/# as shown in figure 3.
Bringing it all together the radius of curvature measurement equations becomes
$$ROC = (P_{con}-P_{cat})-\delta R_{con}+\delta R_{cat}$$The sign of the corrections will flip depending upon whether the interferometric measurements represent and optial path difference or a surface. Commercial interferometers will almost always report surface.
In [3]:
def correction(phi,wavelength,fnum):
return (phi * wavelength) / (1 - sqrt(1 - (1 / (2 * fnum))**2))
figsize(12.5,4)
fnum = linspace(1,10,100)
plot(fnum,correction(1, 0.0006328, fnum), color="#348ABD", lw=3)
grid(b=True, which='major', color='k', linestyle='--')
xlabel('F/#')
ylabel('Correction (mm)')
title('Correction Trend for Fixed\nPower Measurement of 1 wave at 633nm');
The T-sphere F/# defines the cone of light light leaving the T-sphere. It is defined as the effective focal length divided by the exit pupil diameter as shownin figure 1.
$$F/\# = \frac{EFL}{D}$$The surface R/# defines the cone of light incident on the test surface in the confocal configuration. It is defined as the nominal radius of curvature divided by the diameter of the part shown in figure 2.
$$R/\# = \frac{ROC_{nom}}{d}$$At the confocal position the cone of light used for the measurement can be limited either by the T-sphere F/# of the R/# of the surface. If the T-sphere F/# is smaller than the surface R/# then the cone of light from the T-sphere will overfill the test surface and the R/# represents the limiting cone of light for the measurement. In this case the R/# of the surface is used to calculate the ROC correction. On the other hand, if the T-sphere F/# is larger than the surface R/# then the cone of light from the T-sphere will underfill the test surface and the F/# represents the limiting cone of light for the measurement. Figure 3 shows the two cases discussed.
Since there the surface R/# is involved in the cateye measurement the T-sphere F/# is always the limiting cone of light. The T-sphere F/# should be used to caculate the radius of curvature correction.
To determin the sign of the interferometer power measurement in the confocal configuration one can examine the ralative curvature of the test surface and the incident wavefront. The incident wavefront is a projection of the test surface wavefront and the interferometer measures the difference between the test and reference wavefronts. Typically the sign convention of an interferometer is such that a negative relative optical path difference between the test and reference wavefronts is a postitive height in the measurement. In other words, a bump on the surface is a bump in the measurement.
Using figure 4 as a reference, at the Null position both the test and reference wavefronts have the same radius of curvature. Where as when the test article is moved to either side of the Null position the curvature of the wavefront no longer matches the curvature of the test article. At position 1 the rays passing through an off-axis point travel a longer distance to reach the test surface than they would for the reference wavefront resulting in a positive optical path difference or a negative relative height in the measurement. For this case, the edges of the surface measurement would be turned down equating to a negative power coefficient. At position 2 the ralative curvature of the test article and the refrence wavefront are swapped and the power coefficient becomes positive. To summarize, when the measured power at the confocal position is positive, the ROC correction term calcualted from the power is subtracted from the estimate and when the power is negative it is added.
At the cateye position the sign of the measurement can be determined by comnparting the relative position of the imaged focus point positions in figure 5 to those in figure 4. In comparing the position we can see that the relations between the position of the test article and the focus spots is proportional between the confocal and cateye configurations. As a result, at cateye position 1 corresponds to a negative power measurement and position 2 produces a positive power measurement. When the measured power at the cateye position is positive, the ROC correction term is added and when the power is negative it is subtracted.