The Ronchi Test
by Masud Mansuripur
Published in Optics & Photonics News, pp42-46, July 1997

In this centennial of Vasco Ronchi's birth it seemed appropriate to devote one of these columns to the well-known method of testing optical systems that he developed in the 1920's.^1,2 The essential features of the Ronchi test may be described by reference to Fig. 1.
 

Figure 1. A beam of coherent, quasi-monochromatic light is brought to focus by an optical system that is undergoing tests to determine its aberrations. A diffraction grating, placed perpendicular to the optical axis in the vicinity of focus, breaks up the incident beam into several diffraction orders. The diffracted orders propagate, independently of each other, and are collected by a pupil relay lens, which forms an image of the exit pupil of the object under test at the observation plane.

 
A lens (or more generally, an optical system consisting of a number of lenses and mirrors) is placed in the position of the "object under test." The lens is then illuminated with a beam of light, which, for the purposes of the present article, will be assumed to be coherent and quasi-monochromatic. (These restrictions on the beam may be substantially relaxed in practice.³)

The lens brings the incident beam to a focus, in the vicinity of which a diffraction grating is placed perpendicular to the optical axis. (The optical axis will be denoted the Z-axis throughout this article.) The grating, also referred to as a Ronchi ruling, may be as simple as a low-frequency wire-grid, or as sophisticated as a modern short-pitched, phase/amplitude grating. The position of the grating should be adjustable in the vicinity of focus, so that it may be shifted back and forth along the optical axis. The grating breaks up the incident beam into multiple diffracted orders, which will subsequently propagate along Z and reach the lens labeled "pupil relay" in Fig. 1. (The pupil relay may simply be the lens of the eye, which projects the exit pupil of the object under test onto the retina of the observer. Alternatively, it may be a conventional lens that creates a real image of the exit pupil on a screen or on a CCD camera.) The diffracted orders from the grating will be collected by the relay lens and, within their overlap areas, will create interference fringes characteristic of the aberrations of the optical system under consideration. By analyzing these fringes, one can determine the type and, with some effort, the magnitude of the aberrations present at the exit pupil of the system.

The above description of the Ronchi test relied on its modern interpretation based on our current understanding of physical optics and the theory of diffraction gratings. Historically, however, the gratings used in the early days were quite coarse, and the results obtained with them required no more than a simple geometric optical theory for their interpretation. Typically, one would place the eye at the focus of the lens and hold a grating (e.g., a wire grid) in front of the eye, moving the grating in and out until a clear pattern became visible. At this point the beam would be illuminating several of the wires simultaneously. By looking through the grating and observing the shadows that the wires cast on the exit pupil, one could determine the type of aberration present in the system. The coarseness of the grating, of course, caused several of the diffracted orders (as we understand them today) to overlap each other, thus resulting in reduced contrast and smearing of the patterns near the boundaries. These problems were eventually overcome when finer gratings became available and the diffraction theory of the Ronchi test was better understood.

Choosing an appropriate grating. For best results the pitch of the grating should be chosen such that, as shown in Fig. 2, no more than two diffraction orders will overlap at any given point.
 

 

Figure 2. Diagram showing several diffracted orders in the far field of the grating of Fig. 1. When the grating?s period is chosen properly, each diffracted order (i.e., emergent cone of light) will overlap only with its nearest neighbors. Except for a lateral shift in position, the various orders are identical, carrying the amplitude and phase distribution of the beam as it appears at the exit pupil of the object under test.

 

To determine the appropriate grating period P, one needs to know the wavelength l₀ of the beam used for testing, and the numerical aperture NA of the focused cone of light. (By definition, NA = sin q, where q is the half-angle subtended by the exit pupil of the lens at its focal point. If the lens under test is being used at full aperture, NA will also be equal to 0.5/f-number.) To avoid multiple overlaps among diffracted orders, the angle between adjacent orders must exceed the focused cone?s half-angle. Now, it is well-known in the theory of diffraction gratings that, at normal incidence, sin q_n = nl₀/P where n, an integer, is the order of diffraction, and q_n is the corresponding deviation angle from the surface normal. Therefore, we arrive at the conclusion that P should be less than or equal to l₀/NA. For example, assume that the lens under test has a numerical aperture NA = 0.5. Then, if the grating period is chosen to be 2l₀, each diffracted order will deviate from the zero-order by 30°, making the +1st order just touch the -1st order in the far field.

Figure 3 shows the computed intensity distribution at the observation plane of an aberration-free system in which the relay lens has the same numerical aperture as the lens under test (NA = 0.5).
 
 

Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.

Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l₀/NA, leaving a small gap between +1st and -1st orders.⁴ Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected.

Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.
 

Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.

For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly.

The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively.

Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.⁴ Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l₀. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l₀ in (a) and moving forward to Dz = +25l₀ in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.
 
 

Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.

 

To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.³ It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test.

Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.
 

Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.

 

The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.⁴
 
 
 
 

Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.

 

In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r³ cos f) and spherical aberration (r⁴ ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function.

Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.³ Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.³

 

Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.

 

Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.⁵ We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.⁴
 

 

Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)

 

The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.

 

Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.

 

Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.³ As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements.

A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis.

Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.
 

References

1. V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923).
2. V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923).
3. D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992.
4. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona.
5. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954.
6. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973).

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