Nonnull test is often adopted for aspheric testing. But due to its violation of null condition, the testing rays will follow different paths from the reference and aberrations from the interferometer will not cancel out, leading to widely difference between the obtained surface figure and that of the real, which is called the Retrace-error accordingly. In this paper, retrace error of nonnull aspheric testing is analyzed in detail with conclusions that retrace error has much to do with the aperture, F number and surface shape error of the aspheric under test. Correcting methods are proposed according to the manner of the retrace errors. Both computer simulation and experimental results show that the proposed methods can correct the retrace error effectively. The analysis and proposed correction methods bring much to the application of nonnull aspheric testing.
©2009 Optical Society of America
Aspherics can provide more degrees of freedom for aberration control, yielding higher performance while reducing system weight, size and complexity. But due to their arbitrary forms, aspheric testing has always been an extreme challenge for optical researchers. Null tests which use null optics, Dall or Offner compensator [1,2], CGH (Computer-generated Holograms)  for instance, enable very precise measurements. But every unique aspheric needs a corresponding null optic, that has much high requirement on design, fabrication and adjustment. Therefore, they are suitable for projects with demands in testing precision.
However, in many cases all we are interested in is a quick but rough conformity validation to test new aspheric fabrications, so the nonnull test will definitely be a better choice. Those configurations, which can test large wavefront distortion [1,4,5], can be adopted to perform nonnull test, for instance, long wavelength interferometry , two wavelength interferometry , radial shearing interferometry [8,9], and etc.. The nonnull tests can liberate aspheric testing from the different and intricate instruments, while performing a general test easily and quickly.
But some problems must be taken into consideration when we use nonnull configurations for aspheric testing. In a null test, rays reflected from the test part follow nearly the same path through the optical system as the reference rays and effectively make errors from the interferometer negligible. But when the null condition is violated, rays from different pupil regions will follow different optical paths through the system, varying with apertures and dependent on the test part, and lead to considerable testing error that cannot be negligible. The wavefront measured on the detector will contain errors due to the test part which are indistinguishable from errors due to the retrace errors.
Geary et. al  carried out a case study on retrace error in testing a lens with large spherical aberration by standard spherical interferometer. Gardner et. al  used the RBT(Random ball test) technique in calibrating a figure measuring interferometer. Blümel et al  proposed a method to test an asphere in a spherical test setup by an interferometer, but the method is applicable only in rotational symmetrical aspherics with small deviations from a sphere or flat. Murphy et. al  applied aberration theory to interpret and predict imaging errors for aspheric surface testing. But for a real system, ray tracing is easier, faster and more practical.
In this paper, we use the simple ray tracing method to analyze the retrace error in nonnull aspheric testing. Based on the analysis of retrace error, practical methods are proposed to correct the errors in nonnull aspheric testing. Computer simulation shows the methods work well when the figure error is small, despite large deviation between the theoretic value of the aspheric and the reference wavefront. An experiment is carried out and the testing result reaches close agreement with the commercial instrument.
2. Retrace error analysis of nonnull aspheric testing
Figure 1 gives the scheme of the testing arm of a nonnull aspheric testing system. Since the analysis of retrace-error is more accessible for rotational symmetrical aspheric, we just focus on these aspheres.
Asp is the aspheric under test, whose function is z = f(x),Sp is the reference sphere, L 1 , L 2 are the aplanat and imaging lens, respectively. For convenience, we presume they are ideal lenses, with focal lengths f 1 and f 2 , and a distance d between them. The focus point of L 1 superposes on the center of the reference sphere at C 0 , and C is the intersection of the ray reflected from the aspheric with the optical axis. F 2 is the focus point of the imaging lens L 2 , and the detect plane η is ∆L away from the focal plane of L 2. P is the intersection of the testing ray with the detect plane η . The marginal ray of the plane wave intersects with L 1 at point I , and intersects with the aspheric at point A[x,f (x)] after focused by L 1. Because of the deviation between the aspheric and the reference sphere, longitude aberration will be yielded. The ray off the aspheric will intersect with optical axis at point C which is different from C 0, and pass through L 1, then L 2, and finally intersect with the detect plane at point P.
As shown in Fig. 1, the ray IA intersects with reference sphere at point S , and consequently the deviation between the aspheric and the reference sphere is
where, R Sp is the radius of the reference sphere.
According to geometric calculation and imaging theory, the testing rays are traced  from point I (h from optical axis on the aplanat) to point P on the detect plane η , and the OPL (Optical Path Length) from point I to point P on the detect plane can be easily obtained.
And the OPD (Optical Path Difference) between OPLh and the OPL of the ray along the optical axis OPL0 is
which is the wavefront W recorded by the detector.
Spherical or planar testing methods are always adopted to test aspherics which are close to a sphere or flat in nonnull testing. In sphere or planar testing, the wavefront detected by the wavephase sensor is interpreted to be twice the deviation between the test aspheric and the reference sphere, that is
where W is the wavefront detected by the wavephase sensor, and N 0 is the deviation between the aspheric under test and the reference sphere. We can call Eq. (3) the Coherenttesting-principle (CTP for short). But it is a pity that the CTP is not always appropriate in nonnull aspheric testing . Figure 2 shows the testing rays reflected from the aspheric cannot follow their original paths due to the large deviation between the aspheric under test and the reference sphere. In Fig. 2, AS is the deviation between the aspheric and the reference sphere, which is N 0 in Eq. (3). It is plain to see that the wavefront W, which would be recorded by the detector, is not the simple twice of the deviation N 0.
The retrace errors greatly decrease the accuracy of the nonnull aspheric testing, and simply adopting the CTP would result in great difference between the measured phase and that of the real. The retrace error can be categorized into two forms, one is the Retrace path error and the other is the Retrace element error. As the testing rays could no longer follow the same path as the reference after reflected by the aspheric in nonnull test, there would be testing errors, which can be called the Retrace path error. In addition, there do be some errors in fabrication and adjusting of every optical element. When the testing and the reference rays go though different optical paths, they would experience different element errors, which would bring on Retrace element errors. In the remainder of this paper, we consider that the fabrication and adjusting of all optical elements are excellent and the retrace element error can be neglectable. For the retrace path error, according to the form, can also be categorized into two forms, the retrace phase error(RPE for short) and the retrace coordinate error (RCE for short).
2.1 Retrace phase error
Make Err the absolute testing error when the CTP is adopted in nonnull aspheric testing, then
and the relative error can be expressed as
Figure 3 shows the testing error when the CTP is adopted to test an f/2 paraboloid with the aspheric placed in the so-called vertex matched position. “Vertex matched position” is a situation in which the aspheric has been positioned that the focal point of the aplanat and the center of curvature of the vertex of the aspheric are coincident. Figure 3(a) shows half of the PV (Peak-to-Valley) value of the wavefront recorded by the detector (W/2) and the deviation between the aspheric under test and the reference sphere (N 0). Figures 3(b) and 3(c) are the absolute and the relative testing error respectively when CTP is adopted in nonnull aspheric testing. As is shown in Figs. 3(b) and 3(c), for paraboloids with identical F number, the testing error rapidly increases as the aperture of the aspheric under test grows.
Figure 4 shows the testing error when the CTP is adopted to test a 550mm paraboloid with the aspheric placed in the vertex matched position. Figure 4(a) shows half of the PV value of the wavefront detected by the detector and the deviation between the aspheric under test and the reference sphere. Figures 4(b) and 4(c) are the absolute and the relative testing error respectively. From Fig. 4, it is also obvious that for paraboloids with identical aperture, the testing error increases as the F number of the aspheric under test decreases.
As Fig. 3 and Fig. 4 illustrate, the testing error grows up as the deviation between the aspheric and the reference sphere increases. Even if the aspheric under test is perfect, which means there is no figure error, the recorded wavefront by the detector will still contain a phase with some form, corresponding to the aspheric under test. Because the induced testing error is along the optical axis, we name it retrace phase error. If the figure error of the aspheric is relatively the same as the retrace phase error, the detected wavefront will not show the actual figure error of the asphric when the CTP is adopted, and only when the aperture and relative aperture are both very small can the CTP be used in nonnull aspheric testing.
2.2 Retrace coordinate error
A wavefront preserves its shape as it travels only if it is flat or spherical. And if the wavefront is aspherical or has big aberrations, it will continuously change its shape as it travels though the optical system, as is shown in Fig. 5. Simply adopting the CTP would result in another error besides retrace phase error, called retrace coordinate error.
According to the system in Fig. 1, the coordinate of point P on the detected plane can be obtained using geometric calculation when the aspheric is perfect. Figure 6 shows the normalized coordinates of the testing rays at the aspheric surface and the detect plane, respectively (The aspheric under test is an f/2, 160mm parabloid, positioned in the vertex match position and the detector is at the focal plane of the imaging lens). Figure 6(a) is the regularized coordinates of the testing rays at the aspheric surface, sampled at equally spaced zones and Fig. 6(b) is the corresponding normalized coordinates of the testing rays at the detect plane( The same marker means the same zone rays). Comparing Fig. 6(b) with Fig. 6(a), the equally spaced zone rays from the aspheric reach the detect plane at different spaced zone. That is mainly due to the transverse-ray aberration on the detected plane. For a unique asphere, the imaging lens can be designed to project the test part to the detected plane without distortion . But every test asphere may require an imaging system and we can not perform a general test then. In a nonnull system for general aspheric testing, the detector may not be placed on the plane conjugate to the test part and the transverse-ray aberration will be yielded. For this reason, the coordinates of the testing rays at the detect plane have been encountered a radial change, and we call this error retrace coordinate error (RCE for short). If the test optics has a small local error at the 0.3 zone, due to the RCE, it will be present at the 0.1 zone in the testing result. Furthermore, lots of simulations have proved that, the change effect also alters with the aspheric under test and the optical system. Consequently the testing result in nonnull test can hardly give the real figure of the surface under test.
As is illustrated above, due to the retrace phase error and retrace coordinate error, the detected wavefront will deviate to a great extent from the actual figure error. The CTP can only be adopted when the aperture and relative aperture of the aspheric under test are both very small, otherwise the testing data will be doubtful. When the aperture or relative aperture is relatively large, which means the deviation between the aspheric under test and the reference wave is not small, the recorded wavefront by the detector does contain the information of the figure error of the aspheric and also the retrace errors. Measurements should be taken to distinguish the figure of the aspheric from the retrace errors.
2.3 Retrace error for aspherics with figure error
Notice that, the above analysis is carried out in the condition that the aspheric is perfect, and in this subsection, we will focus on the retrace errors for aspherics with figure error. Due to the existence of figure error, the retrace error will be more complicated and unpredictable. Then the simple geometric calculation and imaging theory mentioned above can not be adopted, so we should resort to ray retracing software.
Figure 7 show the simulation results of nonnull testing for an f/1.8 , 150mm parabloid, which has a figure error and is fixed in the vertex-sphere situation. Figure 7(a) is the deviation between the theoretic value of the parabloid and its vertex sphere (N A-V). Figure 7(b) shows the simulated figure error of the aspheric (WAsp). Figure 7(c) is half of the wavefront recorded by the detector (WCCD/2). Figure 7(d) is the figure error (WTest) reconstructed from the wavefront detected by the detector following the CTP. Comparing Fig. 7(d) with Fig. 7(b), the reconstructed figure error deviates greatly from the actual figure error. And Table 1 is the numerical results of the simulations.
We can find from the above that due to the combined effect of the RPE, RCE and the figure error of the aspheric, the testing results deviate much from the real figure error when the CTP is adopted in nonnull aspheric testing. The nonnull aspheric testing is widely used to direct production in industry. However, the nonnull testing would not give convincing guidance if the aspheric under test differs greatly from the reference wave.
3. Retrace error corrections for nonnull aspheric testing
It is essential to find a method to correct the retrace error in nonnull aspheric testing when the deviation between the aspheric and the reference is relatively large. According to the above analysis, the wavefront on the detector W Test contains not only the surface figure of the test aspheric, but also the retrace phase error, retrace coordinate error and the error induced by the nominal figure of the aspheric. So the wavefront on the detector W Test can be expressed as
where, Z is the retrace phase error, R is the retrace coordinate error, E is the error induced by the surface figure of the aspheric and ∑ is the surface figure of the test aspheric. Notice that, the “⊕” in Eq. (6) denotes all the variables are not simply added up. In fact, the tested wavefront on the detector is the coactions of the RPE, RCE and the figure of the aspheric.
The retrace phase error is the dominating error in nonnull aspheric testing and would increase as the deviation from sphere grows. Due to the deviation between the aspheric and the reference, there would still be a wavefront with a relative PV value on the detector plane even though the aspheric under test is ideal. The ideal wavefront Wideal on the detector plane could be obtained simply by ray tracing. Removing Wideal from the tested wavefront by the detector, the retrace phase error Z would be reduced to deviations from nominal behavior.
Due to the asphericity of the surface being tested, distortion would be yielded on the detected plane in nonnull general aspheric testing system. From Fig. 6 we can find that, there is a transform between the pole coordinate in Fig. 6(a) and that in Fig. 6(b). The retrace coordinate error could be removed by a reverse transform between the two coordinates. Figure 8 shows a reverse transform between the pole coordinates in Fig. 6(a) and that in Fig. 6(b).
Because the nominal figure of the aspheric can not be determined in advance, the error induced by the figure error of the aspheric can not be removed directly. When the aspheric surface is well fabricated, the figure induced error could be negligible. In fact, the same problem occurs in both spherical testing and planar testing. When the surface under test has figure error, the testing rays can not follow the exact path of the origin and testing error would occur. But most of the time, if the surface is well fabricated, the figure induced error can be negligible.
Figure 9 shows the computer simulation of error correction. The aspheric under test and its surface figure are the same as that in Section 2.3. Figure 9(a) is the phase error resulting from the deviation between the nominal aspheric and the reference. Because the RPE remains the same when the nominal aspheric surface, reference wavefront and the optical system are determined, if the surface figure of the aspheric is the same magnitude with the retrace phase error or smaller, we can hardly resolve the real figure of the aspheric from the testing result. Figure 9(b) is the result after eliminating retrace phase error from the wavefront in Fig. 7(c) and we can find that when the RPE is removed from the testing data, the resulting figure already has the characteristic of the figure error of the aspheric, but with a distortion in the radial direction. Figure 9(c) is the result with both RPE and RCE removed from the phase in Fig. 9(a), and obviously it is very similar with the surface figure of the aspheric. As is discussed above, because the surface figure of the aspheric under test can not be determined in advance, we can only presume the error induced by surface figure is negligible.
4. Experimental validations for retrace error correction in nonnull testing
In order to validate the methods proposed above for retrace-error correction, a control experiment has been carried out in a home-built Twymann-Green interferometer (which is referred to as Nonnull interferometer in the following). The diagrammatic layout of the experiment is shown in Fig. 10.
The surface to be tested is a long radius sphere with 1933.2mm in radius and 18mm in diameter. Because the surface is very close to planar, it can be tested in a planar-testing setup for nonnull testing. All lenses in the system have been characterized and modeled in ray tracing software, along with the distances between the elements. Ray tracing the system and obtain the RPE and the RCE plot, correct the measured data using the methods proposed above and compare it with the testing result by ZYGO GPI interferometer. Figure 11(a) is the result by ZYGO GPI interferometer and Fig. 11(b) the nonnull testing result after retrace-error correction. Table 2 is the numerical results of the experiment, where WNN is the result obtained in nonnull interferometer without error correction, WPCNN the phase with RPE corrected from WNN, WPCNN the phase with RPE and RPE both corrected from WNN, and WZYGO surface figure obtained in ZYGO interferometer. We can find that combined with the retrace error correcting method, the nonnull interferometer shows satisfying performance with high accuracy.
5. Error considerations and system optimization
The deviation between the aspheric under test and the reference wavefront is usually very large in nonnull aspheric testing. We can hardly obtain null fringe even if the surface under test is perfect. The reference wavefront here can be any wavefront which makes the wavefront off the aspheric resolved by the detector. Aplanats can be adopted to generate reference sphere for mild aspherics, but for deep ones, the most important thing is to find the appropriate reference wave which can reduce the wavefront slope on the detector [17, 18].
Although amounts of calculation has been done in the process of error correction and fringe demodulation, a calculation error of λ/400 can be obtained. Thus, modeling error of the system and retrace error from the figure of test part remain the main error sources of the correcting method leaves.
The accuracy of the retrace error correcting method largely depends on the modeling accuracy of the testing system. Given the condition that the precision of the spherometer is better than 0.01%, the adjusting error smaller than 2 μm, and the locating precision of the transfer 0.5μm, computer simulation shows the accuracy of the correcting method proposed in this paper can be better than PVλ/15 . Notice that, the correcting method proposed here is more suitable for home-built testing system than commercial interferometer, because the designs of the commercial system are usually proprietary. And for a home-built interferometer, as long as the optical elements been modeled precisely, the only thing left in every testing is to determine the distance between the aspheric and the interferometer. Actually, most of the aspheric testing methods should model the testing system, such as sub-Nyquist interferometer  for aspheric testing and aspheric null test with CGH. The CGH method is usually employed in high precision aspheric testing while the nonnull testing is not. In addition, the system modeling error in null test with CGH is called design error and tolerated, and so is the method proposed here.
The retrace error increases as the deviation between the aspheric and the reference wavefront grows, and the retrace error correction becomes more necessary. The only precondition for the method is that, the fringe pattern produced by the distorted wavefront is within the resolution of the detector, and then the wavephase in the interferogram can be obtained.
The method proposed here can be extended to non-rotational symmetrical aspheric testing. But it should be noticed that, as the retrace coordinate error would vary across individual radial direction, the transform function would vary accordingly.
The retrace error of nonnull aspheric testing is analyzed based on the ray tracing method. Retrace error occurs along and vertical to the optical axis due to the large deviation between the aspheric and the reference. Computer simulation is carried out for the aspheric with certain figure error and shows that the tested result would deviate much with the real figure because of the retrace error. Effective correcting methods are proposed according to the performances of different retrace errors. Computer simulation and experiment result both show the method can correct the retrace error effectively and efficiently. The analysis and proposed correction methods bring much the improvement of the accuracy and the application of nonnull aspheric testing.
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