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This page presents our understanding of the interferometric data regarding the lenses in the IMSS collection as obtained by investigators at the Italian National Institute of Applied Optics (INOA) in 1992 . From Reference 2 (below), it appears that at that time, due to the delicate condition of the telescopes, only the broken objective lens was removed from its holder. The other lenses were evidently tested in their mountings, so that a wavefront could be obtained only within the area permitted by the optical stops. It also appears from the pattern of cracks visible in the photographs, that although the broken lens was removed from its ceremonial mounting, it was tested with a stop of similar diameter. In the present diagrams we represent the full diameter of each lens by a blue circle, and the diameter of the optical stop that seems to have been used by a red circle. We are assuming that the interferograms published in the INOA research papers represent the wavefront visible within the red circle. To aid in visualizing the difference in dimensions between the various lenses, all the images on this page have been adjusted to a uniform scale of 3.75 pixels per millimeter. Depending on the size and settings of your monitor, the images may appear a little larger or smaller than life-size.
Based on photographs posted on a webpage by Jim and Rhoda Morris it appears that the leather covered 21X telescope at the IMSS was completely disassembled during a recent renovation. Small photographs of both telescopes, completely disassembled, may also be seen in the 2002 doctoral thesis of Sven Dupré, and (in part) in his 2003 article "Galileo's Telescope and Celestial Light" [Journal for the History of Astronomy 34, 369-399]. We do not know if the lenses were re-tested over their full apertures at that time. The full diameters of the objectives of the 14X and 21X telescopes were tested by Ronchi in 1923, and his results, which suggest that those two lenses did indeed need to be stopped down to achieve satisfactory performance, are described in an separate section below.
An interferometer tests the performance of a lens at a single wavelength, in this case that of a Helium-Neon laser (632.8 nanometers). The interferograms shown here were all obtained in what is called a double-pass mode. In such a mode, each fringe represents one-half wavelength of light. The basic interpretation of interferometric fringes is simple: an ideal system will produce a set of straight and uniformly spaced fringes. The interpretation of deviations from this ideal is more complicated. In principle, one superimposes over the observed pattern the closest fitting system of straight, uniformly spaced lines and measures at each point on the surface how much the actual fringe position differs from the ideal standard. Although this can be done manually, it is very tedious. A more accurate interpretation can be made by means of a computer. Dave Rowe's excellent freeware FringeXP fringe analysis program provides one such means. Such software transforms the fringe pattern into a re-creation of the optical wavefront, sometimes called a phase map. From this map, the optical properties of the system can be predicted in great detail.
It should perhaps be noted that under the conditions of the tests performed at the INOA, a perfect plano-convex lens with an absolutely precise spherical surface would not produce precisely straight fringes. This is not a defect of the test, since its purpose is not to see if the lens has a spherical surface, but rather to see if light transmitted through the lens would come to a perfect point. The expected slight waviness of the fringes is an inherent defect of the simple plano-convex design known as spherical aberration. It was in fact known by Kepler, from as early as 1611 using a crude approximation to Snell's law, that the ideal surface for bending parallel light to a point would be closer to a hyperbola. However, the present lenses have a very high ratio of focal length to aperture, and for such ratios, the performance of a spherical surface differs from that of the ideal aspherical shape by an almost imperceptible amount.
To compare lenses, the results of the interferometric analysis are usually described using such standard summary statistics as the Peak-to-Valley (PV = maximum minus minimum) or Root Mean Square (RMS = a kind of average over the surface) wavefront distortion, or the Strehl ratio (an estimate of the fraction of incoming light that will fall with the radius of the ideal diffraction spot of a perfect system). Because the INOA research papers do not give these statistics in any clear quantitative way, we have attempted to re-analyze the interferograms shown here using the FringeXP program. Unfortunately this could not be applied to the large broken objective, both because FringeXP can only handle circular apertures, and because the phase relationship between the different broken segments is not known to us. Click on any of the other interferograms to see the FringeXP report on which the wavefront statistics are based (note: FringeXP was designed primarily of testing spherical mirrors. The reports include entries for part diameter and radius of curvature. These entries are ignored when comparing the fringe pattern to an ideal plane wavefront, as is done here. Also, since the fringe order cannot be deduced from a single interferogram, the deduced wavefront may be inverted with respect to the true one. Again, this does not affect the statistics presented here.)
A lens system producing a wavefront differing from spherical by less than about 1/4-wave peak-to-valley, 0.07 waves RMS, or giving a Strehl ratio greater than about 0.8 is generally considered to be diffraction limited. Further improvements in wavefront will not sensibly improve the quality of the point image. The bottom line is that the INOA researchers felt that, over the apertures tested, the three objectives shown below, particularly the large broken one, were close enough to this standard that they would produce images indistinguishable from those produced by a good modern singlet lens of the same design. In fact, in Reference 3, Molesini says that, if corrected for the difference in angles between the broken parts, "there is no doubt that the quality of the wave front [transmitted by the broken lens] is of the order of λ/8 for a diameter of 38 mm," that is, 1/8-wave peak-to-valley.
The wavefront distortions given here are at the test wavelength of 633 nanometers, which is towards the red end of the visible spectrum. At shorter visible wavelengths the distortions will be greater, both because the wavelength being used as the standard of measurement is shorter and because the refractive power of the glass is larger. We have not attempted to make this correction. Full optical specification of the three lenses shown below are given on a separate page, where they are identified by the codes GAL 1A, etc. as used in the figure captions.
| Objective (GAL 1A) | Objective Alone | Objective with Eyepiece GAL1B | |
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| Image source: | Ref. 1, Fig. a | Ref. 2, Fig. 6e | |
| Wavefront (633 nm): | 0.54 PV / 0.069 RMS | 0.50 PV / 0.068 RMS | |
| Strehl ratio: | 0.83 | 0.83 |
The objective lens of this telescope (51 mm full diameter stopped to 26 mm) was tested both by itself and in combination with the eyepiece lens. For the former test, the beam of converging light leaving the lens is reflected back into the interferometer by a spherical mirror. For the latter test, the beam of nearly parallel light leaving the telescope is reflected back into the interferometer with a plane mirror. Interferograms of the eyepiece by itself (not shown here) indicate its surfaces are much poorer than those of the objectives. However, as pointed out by the INOA researchers in Reference 2, only a very small portion of it is used at any one time: at the eyepiece, the diameter of light from a point source at infinity is only about 1.8 mm. Over this very small diameter, the distortion of the wavefront is slight and its effect on the overall performance of the telescope is, as the researchers say, "negligible." See Figures 9, 10 and 11 of Reference 2 for a detailed phase map of the wavefront from this telescope and its interpretation in terms of aberrations and image quality.
Curiously, Abetti and Hale reported the resolution of this telescope to be poorer than that of the 20X telescope (see next example), even though the wavefront appears better. Baxandall (Replica Ref. 1, below) attributed this to "an inferior eyepiece." Since the INOA researchers demonstrated that the contribution, even of the poorest eyepiece, to the overall performance is negligible, Baxandall's conclusion is hard to accept. We find it possible that an illusion of lower resolution was created by the extremely small size (at 14X magnification) of the tiny astronomical features Abetti and Hale were trying to evaluate.
Note: in addition to the references listed below, the 1992 INOA interferograms of Galileo's lenses were published in a number of other locations, yet, as indicated above, we have been unable to find any clear statement of the overall wavefront quality in terms of Peak-to-Valley or RMS deviations. Figure 9 in Reference 2 does show a computer generated plot of the Wave Aberration of the complete 14X telescope, corresponding to the interferogram shown at right above, from which it appears that the peak-to-valley distortion is approximately 0.2 waves. This is roughly half the value reported by FringeXP. The source of this discrepancy is puzzling to us. As a test, we examined a 1994 paper in which the same INOA team, together with Mara Miniati of the IMSS, gave a similar report on a 17th century telescope showing a double-pass interferogram of the objective (Nuncius 9, 677-682; Figure 10), for which Peak-to-Valley and RMS values were given. In that case, the results from FringeXP (0.48 waves peak-to-valley, 0.10 waves RMS) are in good agreement with those given in the INOA paper (0.44 waves peak-to-valley, 0.09 waves RMS). This only adds to the mystery of why the wavefront errors we report here appear to be about twice those shown in Figure 9 of Reference 2. Our results assume that the spacing from one dark fringe to the next corresponds to one-half wave of error in the wavefront, which is normal for a double-pass test of a lens.
| Objective (GAL 2A) | Objective Alone | |
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| Image source: | Ref. 2, Fig. 7c | |
| Wavefront (633 nm): | 0.75 PV / 0.10 RMS | |
| Strehl ratio: | 0.66 |
The objective lens of this telescope (37 mm full diameter stopped to 16 mm) was tested only by itself. This is because the eyepiece that originally went with this telescope has been lost. The designation of this telescope as one of 21 power assumes it is being used with the substitute eyepiece. According to the article by Baxandall, mentioned in the section at the bottom of this page, the draw tubes of this telescope allow the distance between the objective and eyepiece to be adjusted by as much as perhaps 200 mm. It seems, then, that the original eyepiece might have had a considerably different focal length, giving a substantially different power.
Although the fringes displayed in the interferogram are reasonably straight, they are much more widely spaced at the top of the figure than at the bottom. This indicates that this objective suffers from a significant amount of astigmatism.
| Objective with Stop | Objective (GAL 3) | Interferogram 1 | Interferogram 2 |
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| Image source: IMSS | Ref. 2, Fig. 3 | Ref. 3, Fig. 2 | Ref. 2, Fig. 8 |
The left-most figure, adapted from that on the IMSS website, shows the large objective in its decorative holder. This is believed to be the objective lens of the telescope through which Galileo first saw three moons of Jupiter on January 7, 1610. No other particulars of that telescope are known, but from Galileo's description in Sidereus Nuncius it would appear that it was of near 30 power. The IMSS image of the objective has been flipped, rotated and rescaled to match the INOA photograph of the lens. We do not know if this is the view from the plano or from the convex side, nor if the bronze stop corresponds in any way to the one actually used by Galileo on his telescope. The IMSS image is shown here to give an additional view of the pattern of cracks, some of which are dimly visible in the interferograms. The second figure shows the complete lens (58 mm diameter) as it was photographed after being removed from its decorative holder during the 1992 testing. Although the whole lens was available for testing, judging for the pattern of cracks it appears that the published interferograms show only the area inside a 38 mm diameter opening (equivalent to that of the decorative holder) as indicated by the red circle.
Since 1923, all investigators have expressed the opinion that this objective is better than the other two. The fringes over most of the area shown are indeed remarkably straight. However, in the region pointed to by the red arrow, there appears to be a sudden large deviation in the surface profile. The existence of such defective zones might account for Galileo's practice, described in one of his letters, of masking some of his objectives with oval (i.e., not necessarily round or centered) stops. The presence of such zones could have been discovered experimentally by carefully watching a fixed terrestrial target while various parts of the objective were alternately covered and uncovered.
Interferometry References :
V. Greco, G. Molesini, and F. Quercioli, "Optical tests of Galileo's lenses," Nature (London) 358, 101 (1992).
V. Greco, G. Molesini, and F. Quercioli, "Telescopes of Galileo," Applied Optics 32, 6219-6226 (1993).
G. Molesini, "The Telescopes of Seventeenth Century Italy," Optics & Photonics News (June 2003) 35-39.
(translated from March 2002 Giornale di Astronomia)
The original images from Reference 1 are © Nature Publishing Group; while those from References 2 and 3 are © the Optical Society of America.
Note: the modern plano-convex singlet lens used to produce the photographs shown on this website has never been tested interferometrically. Although it appears to produce good images, we have no idea whether an interferometer would show it to be better or worse than the 17th century lenses shown above.
The Galilean artifacts at the IMSS were examined by the famous optical engineer Vasco Ronchi in 1923 (Ref. 1, below). In an almost exact reversal of the events of 1992, Ronchi seems to have had no hesitation about removing the lenses from the 21X and 14X telescopes, yet he decided to keep the large broken lens in its decorative holder to keep the pieces together. As a result, we have Ronchi's interpretation of the 21X and 14X objectives over their full diameters. Since Ronchi's results seem to be so little known, and since, outside Italy, the magazine in which they were published is available in only the largest university libraries, we have taken the liberty to reproduce them here. The images shown here are adapted from a photocopy of the original article obtained through inter-library loan. Ronchi took care to show all his images at the same scale. We have adjusted them to match the scale used above and added a red circle to indicate the clear aperture through which the images were formed in the working telescopes, according to the diameters of the aperture stops as given by Ronchi.
Ronchi's test consists of imaging a grating (a transparent plate with a fine series of rulings) back on itself, which, if the image is distorted, will create a pattern of fringes through a kind of moire effect. The Ronchi test is much more qualitative than the modern interferometric test, but it reveals areas in which the lens figure deviates from its ideal shape. The exact pattern observed depends on how the grating is placed and how the system is focused. Most of Ronchi's 1923 ronchigrams (photographs of the Ronchi fringe pattern) were obtained using linear gratings, but one of the photographs of the first objective is with a circular reticle having 8 lines per millimeter.
| Ronchigrams of Objective of Long 14X Telescope (GAL 1A) - Ref. 1, Fig. 2 |
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The above lens (called GAL 1A on the present website) clearly has some problems outside the 26 mm diameter circle tested by the INOA in 1992. According to the text of Ronchi's 1923 article, light from the zone outside the red circle focuses 27 mm closer to the lens than light from the central zone. Such an effect is similar to what one expects from a "perfect" lens due to spherical aberration; however the distortion here is much more severe. It seems clear that this lens benefits from being stopped down.
| Ronchigrams of Objective of Short 21X Telescope (GAL 2A) - Ref. 1, Fig. 3 |
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This lens (GAL 2A) shows the characteristic pattern of astigmatism, extending the distortion already detected within the 16 mm red circle by the INOA interferograms. This lens also benefits from being stopped down. According to Ronchi, light from the zone outside the red circle focuses 22 mm away from that from the central zone.
| Ronchigrams of Large Broken Objective (GAL 3) - Ref. 1, Fig. 4 |
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The large broken lens (GAL 3) appears to have almost perfectly straight fringes over the 38 mm diameter visible through the decorative stop. Of the three lenses, it clearly has the straightest fringes over the largest area. Unfortunately, Ronchi did not test the zone outside the red circle (the full diameter of the lens is reportedly 58 mm), and the INOA did not publish any interferograms of it either. Therefore, it is difficult to assess the degree to which stopping the lens down to 38 mm improves its performance.
The Foucault test is performed by putting a knife-edge at the focus of the lens while it is imaging a point source. In an ideal system, every part of the lens will contribute equally to the point image, and the light will be extinguished uniformly as the knife cuts through it.
| Foucault Tests of the Three Objectives - Ref. 1, Fig. 5 |
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The results are shown in the same order as before: GAL 1A, GAL 2A and GAL 3, from left to right. The light appears to be extinguished most uniformly for the large broken lens (right), but there is no explanation of why the image shown is less than the expected 38 mm in diameter (represented by the red circle).
As a final test of the lens aberrations, Ronchi imaged an monochromatic artificial star (a point source of light at a wavelength of 460 nm) with each of the three objectives. The first two objectives were tested both with and without their diaphragms (aperture stops); while the broken lens was tested first with its current 38 mm clear aperture, and then stopped down to 20 mm. He does report testing the complete telescopes. He also fails to report the scale of the photos. In theory the scale can be recovered from the lens aperture and step size, said to be 30 mm; but, as indicated below, it is really impossible to tell from these photos how small the image at best focus might actually be. It is extremely difficult, and usually impossible, to guess the size of the smallest spot based on photos on either side of it. To know that one has found the best focus, one needs to see at least three steps with essentially identical spot sizes, and there is no example of that in the current series of photos.
| Star Test of GAL 1A - Ref. 1, Fig. 6: Ia/Ib |
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The upper row shows the star images for GAL 1A operated at its full 51 mm aperture; while in the lower low it was stopped to 26 mm. Despite the aberrations, the star image in the center of the top row is smaller than any in the second row; however, it appears that the focus step was too coarse, and Ronchi may well have missed the best possible position in the bottom row.
| Star Test of GAL 2A - Ref. 1, Fig. 6: IIa/IIb |
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The upper row shows the star images for GAL 2A operated at its full 37 mm aperture; while in the lower low it was stopped to 16 mm. The best spot in the second row is perhaps a little sharper than the best in the upper row, but again, the focus step is so coarse it is hard to tell. There seems to be some hint of the astigmatism from which this lens suffers in the upper row where the images on the left appear to be elongated horizontally, while those to the right are elongated vertically.
| Star Test of GAL 3 - Ref. 1, Fig. 6: IIIa/IIIb |
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As indicated above, the upper row shows the star images for the broken lens, GAL 3, operated over the full 38 mm aperture of the ceremonial frame; while in the lower low it was stopped down to the central 20 mm. It is rather remarkable that this lens forms images as well as it does, considering that it is glued together from its broken pieces, the main crack being clearly visible as a jagged dark line running through the out-of-focus images.
Because the apertures are essentially the same, the upper row here should be comparable to the upper row in the star test of GAL 2A. Once again, the focus step is too coarse to compare results, as it appears that the smallest dot from GAL 3 would probably be found between the third and fourth steps from the right in the upper row. It seems unlikely that GAL 3 really produces a smaller spot at focus when stopped down, as the final row would seem to suggest.
All in all, in this day of laser light sources, one would have to say that Ronchi's star test results are rather disappointing and inconclusive compared to a modern test in which the diffraction rings would be clearly visible in the photographs near focus and the asymmetry between images inside and outside of focus made clear. This probably reflects a limitation of the technology available to Ronchi in 1923.
Ronchi Reference :
V. Ronchi, "Sopra i cannocchiali di Galileo," L'Universo 4, 791-804 (1923).
The original images from Reference 1 from which the above were adapted are © Geographic and Military Institute of Florence.
In 1924, David Baxandall published a detailed description of two carefully constructed Cipriani replicas of the IMSS telescopes, which had recently been placed on exhibit at the Science Museum in London (Ref. 1, below). In his paper, Baxandall presents this curious diagram (reproduced to the same scale as all other diagrams on this page) of the construction of the objective holder for the long 14 power telescope. The diagram is curious because it shows a wooden stop, mentioned in the text of the article, which he found inserted at the inner edge of the objective draw tube.
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| This drawing is adapted from Figure 2 of Ref. 1 (below, © Institute of Optics) |
In conformity with the IMSS description of the original telescope, the lens is held in place by a metal wire exerting pressure against the paper (cardboard) aperture stop, which in turn presses against the lens. The paper stop, which is only dimly visible in this copy, has a 26 mm clear aperture; only very slightly smaller than the inner diameter of the tube, indicated as being about 29 mm. The wooden stop, not mentioned in the IMSS description, appears to have a clear aperture of about 16 mm, and to be placed 95 mm behind the objective. Such a stop, if present, would not only act as a baffle, but would have significantly restricted the available aperture (as seen for the eyepiece end), thereby degrading both brightness and resolution.
We have been unable to find any other reference to this wooden stop. It is NOT indicated in the equally detailed cutaway drawing of another Cipriani replica, also made in 1923, as shown in the 1939 article by Petit (Ref. 2, below). We are also unaware of any mention of it in any other description of the 14X telescope. It is unclear whether the original telescope once contained such a stop, perhaps lost during the 1923 disassembly and testing; or if someone added it, for an unknown purpose, to the London replica before Baxandall had a chance to examine it.
In recent times, replica builders Jim and Rhoda Morris have carefully examined a Cipriani replica of the leather covered 21X telescope in the collection of the Adler Planetarium in Chicago, and compared it to the original at the IMSS. Although outwardly correct, they found what seem to be significant differences in the inner contruction, particularly in the length of the drawtubes. So it is possible that the wooden stop Baxandall found inserted into the drawtube of the 14X telescope may have been added for some reason other than historic accuracy.
Replica References :
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The images on this page have been adapted from the original sources by
Tom Pope and Jim Mosher
Last modified: August 28, 2006