Focal-plane, optical testing methods like the Foucault, wire, and Ronchi tests are useful in optical shops [1
1. V. Ronchi, “40 Years of History of Grating Interferometer,” Appl. Opt 3,437–451 (1964). [CrossRef]
]. Among these three test methods, the Ronchi test is widely used and is powerful given the use of a square periodic grating which makes it possible to increase the illumination and most importantly to quantify the measurements. Although optical interferometry has become a standard in optical testing, geometrical tests are still important in the optics shop. The fabrication of large aspheric surfaces requires assessing large figure errors during the grinding process. Geometrical tests which have a large dynamic range permit the optician to assess and reduce surface figure error to a level where optical interferometry becomes practical. Although the Ronchi test is an excellent test it has, however, some limitations when it is used in combination with a Spatial Light Modulator (SLM) to perform phase-shifting and obtain a quantitative measurement. To increase the measurement precision, the spatial frequency of the Ronchi grating is usually increased at the expense of grating diffraction which causes multi-beam interference and makes the fringe observation difficult or
impractical. Thus, the Ronchi test measurement precision can be ultimately limited by the effects of grating diffraction. The technique of phase-shifting has been applied to the Ronchi test. For example, Yatagai made a phase-shifting Ronchi test with a moving Ronchi grating and quantified the measurements with high precision [2
2. T. Yatagai, “Fringe Scanning Ronchi Test for Aspherical Surfaces,” Appl. Opt 23,3676–3679 (1984). [CrossRef] [PubMed]
]. Castro-Ramos designed an automatic phase shifting Ronchi tester with a square Ronchi ruling and measured x and y components of wavefront derivative simultaneously [3
3. C.-R. Jorge and S. Jose, Automatic phase-shifting Ronchi tester with a square Ronchi ruling, O. Wolfgang and N. Erik, eds. (SPIE, 2004), pp.199–210.
]. Hibino proved that the dynamic range of the Ronchi test is larger than the dynamic range obtained in Fizeau interferometry [4
4. K. Hibino, D. I. Farrant, B. K. Ward, and B. F. Oreb, “Dynamic range of Ronchi test with a phase-shifted sinusoidal grating,” Appl. Opt 36,6178–6189 (1997). [CrossRef]
]. Phase-shifting Ronchi testing has been shown to be feasible for improving the Ronchi test given the ability to quantify the test results without using a high spatial frequency grating. In the early days of phase-shifting Ronchi testing, a binary grating was used in combination with a synchronous phase detection scheme to measure the phase of the fringes. However, the measurement precision was limited by both the number of grating moving steps within one grating pitch and the random errors of each stepping distance. To increase the measurement precision more measurement steps are required which slow down the measurement process and introduce further errors. The use of a sinusoidal intensity grating makes the measurement process efficient since the phase measurement can be completed in just four steps or less [5
5. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt 13,2693–2703 (1974). [CrossRef] [PubMed]
]. In practice, it is difficult to make a high quality sinusoidal intensity grating that has a sinusoidal transmission ratio independent of the incident angle of tested beam. This is a critical factor in phase-shifting measurements especially when a fast converging beam is measured.
With the development of high pixel density, spatial light modulators (SLM’s), it is attractive and feasible to create a high quality, sinusoidal grating that is computer controlled to perform phase-shifting Ronchi testing. The pixel based time sequential or liquid crystal (LC) polarization gray-scaling operation of a modern SLM ensures the sinusoidal grating transmission’s ratio to be independent of the incident angle of the light. Thus by using a SLM in phase-shifting Ronchi testing the difficulty of manufacturing a high quality sinusoidal grating is overcome and the need to physically step the grating is eliminated. In addition, gratings of variable frequency can be generated at a snap through controlling the SLM and the test sensitivity can be adjusted dynamically. Gonzalez successfully utilized a liquid crystal (LC) type SLM to generate a Ronchi grating and performed phase-shifting [6
6. M. M. Gonzalez and N. A. Ochoa, “The Ronchi test with an LCD grating,” Opt. Commun 191,203–207 (2001). [CrossRef]
]. We tried to use a digital mirror device (DMD) SLM to perform phase-shifting Ronchi testing. However, the diffraction effects induced from the inter-pixel gap caused multi-beam interference effects and the fringe analysis became impractical. The inter-pixel dark gap, which is a common feature in most types of SLM’s, has a reflection ratio much less than the active pixel area and this makes the SLM to act as a diffraction grating. In addition, as the ratio of cell thickness to pixel width increases in a LC type SLM, the acceptance angle of the modulated beam decreases. If the incoming beam cone angle is larger than the maximum acceptance angle of the LC cell, then part of the modulated light becomes reflected at the gap of the LC cell and introduces undesired interference fringes at the observation plane. Some post image processing can be used to remove such additional artifacts at the expense of phase measuring precision [6
6. M. M. Gonzalez and N. A. Ochoa, “The Ronchi test with an LCD grating,” Opt. Commun 191,203–207 (2001). [CrossRef]
]. Thus, due to these problems it is difficult to use a SLM to generate a grating to obtain high precision, phase-shifting measurements in a Ronchi test.
The grating-slit test that we discuss in this paper does not have the difficulties mentioned above when using a SLM in the Ronchi test. In contrast to the Ronchi test which uses the SLM in the image space, the grating-slit test uses the SLM in the object space. Since the SLM is used to generate multiple incoherent illuminating light sources, both the SLM interference effects induced in a Ronchi test are eliminated. As a consequence, the grating-slit test is more suitable to be used in combination with a SLM than the Ronchi test. To our knowledge, the concept or the advantages of the grating-slit test have not been previously documented. In this paper we discuss a novel geometrical theory of the grating-slit test and the concept of spatial bucket integration with respect to the slit width to obtain the fringe contrast. This geometrical derivation shows that grating-slit test has the same sensitivity as Ronchi test and the dependence of the geometrical fringe contrast with respect to grating pitch and slit width. In a future paper we will discuss the physical theory of the grating-slit test. We believe that phase-shifting, grating-slit testing can become a key test tool for assessing large figure errors during the fine grinding process of a large aspheric surface where optical interferometry or profilometry are impractical or where infrared interferometry is expensive.
2. Transverse ray theory
With reference to Fig. 1
, we assume a simplified optical system with a wavefront W
at the exit pupil that has a paraxial radius r; the transverse ray aberration at the observation plane is TA
Fig. 1. The relation between the wavefront and the transverse ray aberration at an observation plane.
The wavefront aberration, W, is usually very small when compared with r and therefore we can neglect W and simplify the equation to,
This is a well-known and simple equation that relates the Y-component of the transverse ray aberration and the derivative of the wavefront. The quantity TA
Y is the Y-component of the ray interception position in the observation plane. The transverse ray aberration at the image plane TA
Y is the quantity that is measured in focal plane tests like the Ronchi test.
3. Geometrical optics model
Fig. 2. 3-D view of the grating-slit test method.
Consider a grating in object space that is illuminated by spatially incoherent light. The grating is imaged by the optics under test as shown in Fig 2
. The width of the single narrow slit is d and the slit length is parallel to the Y-axis. The light intensity seen at the exit pupil is bright only when light passes through the slit. For small fields of view, the wavefront function W
can be approximated as independent of the field position P
) and the resultant geometrical point spread function of the aberrated wavefront is linearly shifted over the small field. This property of linear system invariance over a small field is the basic and important assumption in the grating-slit test theory.
Consider an illuminating point source P
) on the sinusoidal grating and assume that the optical magnification ratio of the optics under test is m
. The sinusoidal grating is parallel to the Y-direction and it is illuminated uniformly in all angular directions. Light passes through the slit with width d only when the ray interception position FPx at the slit plane satisfies Eq. (3
is the magnification of the test configuration and TAx
is the transverse ray aberration in x direction. With the assumption that the optics under test is a linear invariant system within a small field, the transverse ray aberration TAx
is a function independent of the field
). By inserting Eq. (2
) into Eq. (3
) we obtain Eq. (4
For simplicity, we can assume that the slit width d is infinitely small. Therefore, only when the ray interception position in the x-direction is zero, light is set to pass through the slit. In mathematical terms this requires,
Now if the grating used has a pitch p
, the peak of fringe intensity occurs only when the incoming ray from the object position Xa
satisfies Eq. (5
is an integer number. After combining Eq. (5
) and Eq. (6
) we obtain a similar result to the obtained for the Ronchi test fringe [8
8. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop TestingD. Malacara, ed. (Wiley-Interscience, New York, 1992), p. 321.
By analogy, when the sinusoidal grating is rotated by 90 degrees and is parallel to the X-direction, the transverse ray aberration in Y direction can be measured.
where, M is an integer number.
In the grating-slit test, a point at the grating plane produces an image or spot that represents a geometrical point spread function. This geometrical point spread function is linearly shifted by the lens imaging and the intensity is determined by the position on the object sinusoidal grating. The slit of width d in turn samples and modulates each of the shifted point spread function. Both the point spread function and the slit are spatially coincident. The air slit can be considered to scan linearly the point spread function from different object locations. The lateral position of the spots is modulated into the fringes at the observation plane. The observed fringes of the grating-slit test look just like the observed fringes in the traditional Ronchi test, or so called “Ronchigram”. However, the modulation of the fringes is indeed from the modulation of sinusoidal intensity grating in the object space. This is in contrast to the traditional Ronchi test where the entire Ronchigram is created from a single point source. Although the traditional Ronchi test can be used with an extended light source, each point in the extended source contributes the same whole Ronchigram and is linearly superimposed in intensity with the Ronchigram contributed by other points.
4. Spatial bucket-integration in the geometrical model
Consider the grating slit test setup as shown in Fig. 1
, the slit extends infinitely in Y-direction and modulates the tested beam as a function of the X coordinate of the transverse ray aberration as expressed by Eq. (3
). Therefore, we can simplify the grating as a one dimensional illuminating source with an intensity modulation direction perpendicular to the slit. The irradiance of the illuminating sinusoidal grating in the object space is a sinusoidal function that varies with the x coordinate and is assumed to be independent of the radiation angle. Therefore, we can write the irradiance L from the grating as,
Where, p is the period of the illuminated sinusoidal grating.
In the previous discussion of Eq. (7
) the slit width d was assumed to be infinitely small. This assumption results in a one-to-one mapping between the observed pupil position and the transverse ray aberration function as it is in the Ronchi test. With the small solid angle approximation, the intensity contribution from each point inside the modulation slit can be expressed as,
where KdTAx is the infinitely small solid angle extending from the grating to the observation plane. Since the slit has a finite width d, we should consider the intensity contribution from each point inside the slit. Similar to the concept of temporal bucket integration seen in phase-shifting interferometry, the observed Ronchigram is the integration of the transmitted rays with lateral transverse ray aberration TAx ranging from -d/2 to d/2. We can write the pupil intensity map I (Xp , Yp) as the linearly superimposed intensities from different transverse ray aberration contributions, this is,
The integral in Eq. (11
) can be considered as the concept of spatial bucket integration. This is the bucket integration of the light intensity coming from the finite slit of width d. We can rewrite Eq. (11
and the fringe contrast ratio V is recognized to be,
) gives the fringe contrast as a function of the slit width and the imaged grating pitch. Like in the Ronchi test, to increase the sensitivity of the grating slit test, the spatial frequency of the imaged grating pitch mp
must be increased. But to keep a useful fringe contrast ratio, the slit width must be reduced as predicted by Eq. (13
). If the slit is not perfectly aligned parallel with the grating, the fringe contrast ratio will be less than its theoretical value. Thus, this property can be used to align the slit with the sinusoidal grating.
In the geometrical modeling of the grating-slit test, the diffraction nature of light has been ignored. The diffraction effects from the single slit deteriorate the fringe contrast ratio and are not accounted in the geometrical optics model. As a result, the geometrical fringe contrast ratio modeling would have a significant error when the slit width is in the order of the wavelength of the diffracted light. The problem of slit diffraction can be avoided by increasing the beam defocus amount such that the transverse ray aberration at the slit plane would be large and a wider slit could be used. The increased defocus is equivalent to the wavefront tilt used in shearing interferometry. Thus, by introducing defocus the diffraction effects from the single slit can be negligible.
We present the grating-slit test which uses an incoherent illuminated sinusoidal intensity grating and a slit that modulates the tested beam. A novel geometrical theory is developed which shows the equivalence to the Ronchi test in terms of functionality and that also provides the fringe contrast. The test can be used in combination with a spatial light modulator to produce phase-shifting and obtain quantified measurements. The geometrical model shows that the fringe contrast ratio depends on the imaged grating pitch and the slit width. The intensity integration through the width of the slit can be considered as the spatial bucket integration similar to the concept of temporal bucket integration in phase-shifting interferometry. The grating-slit test does not suffer from several diffraction effects that limit the Ronchi test when used in conjunction with a SLM to produce phase-shifting and obtaining useful quantitative measurements. The main advantage of the grating-slit test over the Ronchi test is not in the accuracy or dynamic range. Instead, the advantage is the ability to do phase shifting measurement in combination with a spatial light modulator. This ability can make the grating –slit test a much more useful testing device than the Ronchi test in terms of quantified measurements.