Lens designs for digital imaging focal planes
( June 2007 )
The pixel pitch is 9µm, giving a Nyquist limiting spatial frequency of ~56cycles/mm. This is relatively poor compared with that available from fine-grain film and is a major limit to the theoretical image quality. By microscanning the image by 4.5µm in the x and y directions, it is possible to recover most of this sampling loss. Microscanning can be achieved by decentring a lens, or by tilting a 'thick' glass plate in the focal space. Decentring a lens is difficult to achieve without introducing unintended tilts and decentrations. The tilting plate is less error prone, but requires an additional element. SBIG expect that they will be able to supply such a device in the early summer of 2007.
The useful area of
the array is 4008 x 2672 pixels, i.e. 36.072mm wide and 24.048mm
high. The diagonal dimension is 43.3mm. It must also
be anticipated that there will be a further reduction in pixel
size, possibly to 6 µm. This would increase the resolving power substantially,
and possibly eliminate the need for microscanning.
Equal spectral weightings were applied to the previous designs for the 35mm SLR and 80mm cameras. No compensations were applied for the human eye photopic spectral response or for photographic films. These responses generally lie in the range 0.45 to 0.68 micrometres. CCDs have a significantly wider response, extending from 0.35 to 1.1 micrometres. In this electronic imaging application, the use of the visual photopic curve to weight the spectral content so as to represent the direct viewing performance is unjustified.
It is interesting to examine the behaviour of nominally apochromatic lenses over such a wide waveband. It is already known however that the chromatic dispersions of optical glasses do not permit diffraction quality imagery in such a wide waveband at useable focal ratios. Consequently, the designs originally developed for the 35mm cameras have been re-examined and modified to optimise their use with CCD cameras. The spectral weightings for the KAI11002 were obtained from the complete datasheet available at
In addition to a cover-glass over the CCD, the carousel of interference filters on 1.5mm discs will be included in the optical path. Allowance will be made for the thickness of the tip-tilt plate used for microscanning and also possibly image stabilisation when it becomes available. The presence of these components requires a substantial back focal distance.
The stray light performances are also important. Bright objects such as the Moon in the nominal field of view can create veiling glare and spurious images. Street lights etc, which are outside the nominal field of view can become visible by multiple reflections at the lens surfaces and inside the lens barrel.
1. Preliminary specification.
Several suppliers of lenses for astronomy offer remarkably large aperture diameters. In general, however, as might be expected, the focal ratio increases with aperture, so as to maintain the image quality consistent with the likely pixel size. For a given aperture diameter, the brightness of the image of a star varies inversely with the square of the speed of the lens. Brighter images shorten the time required to achieve adequate signal to noise ratio from the camera. But as the speed increases, the linear size of an extended object on the focal plane falls. There is thus a trade-off between pixel size, and focal length.
The majority of suppliers indicate that their lenses are apochromatic ( corrected for paraxial back focal distance BFD at three wavelengths). One even claims super-achromatic, suggesting correction at four wavelengths. Note however that designing for such a criterion can be deceptive – there are many theoretical solutions which are super-achromatic but have poor imaging performance because of uncorrected sphero-chromatism. Graphs showing the variation of BFD should be treated cautiously – the best criterion for defining performance is the polychromatic Strehl intensity, or diffraction ‘impulse response’, as a function of field angle. Other useful measures are the ‘ensquared energy’ and the Huyghens theory ‘modulation transfer function’, all calculated with spectral weightings appropriate for the image sensor at a range of field points. There is always an assumption in such simple measures, namely that the spectral weighting is due only to that of the FPA. A more accurate measure would be to include the spectrum of the source.
The aim in this present study was to determine
(a) the fastest lens which could reasonably achieve a focal length of 800mm. The pixel pitch of 9µ then allows an image sampling interval of 11.25µrad or arc sec. A number of suppliers offer such lenses, but do not provide full performance data in the above terms. Consequently, before trying to outdo these suppliers, it is sensible to determine their performances in objective terms. A number of lenses were designed using the brief data such as ‘apochromatic air spaced triplet’ and ‘super-apochromatic, air spaced four elements’. For this writer, the aim is to compete with the dedicated catadioptric spectrographs, such as the Takahashi 300mm F/2.8, i.e 840mm focal length.
(b) the fastest lens which could reasonably be achieved with an aperture diameter of 100mm. The aim here was to create a new design which might reasonably be built by an accomplished amateur, or even by a volume manufacturing specialist.
2. Designs for 800mm focal length.
A typical lens offered for sale is 100mm aperture at a focal ratio F/8, i.e 800mm focal length, but without performance claims. It is interesting to see what performances are possible.
2.1 Air-spaced triplets
This term probably refers to the design in which a central positive ‘crown’ element is enclosed by a pair of negative ‘flint’ elements. By careful selection of glass types, it is possible to correct primary spherical aberration ( halo) and coma and reduce the secondary chromatic aberration and also the spherochromatism ( chromatic halo). It is unfortunate that such a simple lens cannot be corrected for field curvature. However it is possible to correct it for astigmatism on a curved image surface. The field curvature is tolerable with eyepiece viewing, because the eye accommodates as it moves through the field. This lens type is generally limited to relatively narrow fields of view for a plane image surface, making it unsuitable for electronic or film cameras. For these sensors, the field curvature must be corrected by additional elements.
However, it is interesting to assess the type of performances which might be expected from such lenses. This following is an example F/8 design. It uses a modern low index, low dispersion glass, SCHOTT Lithotec-CaF2, and traditional crown K5, and is optimised for the visible band. Although is it not classically apochromatic, the secondary spectrum is virtually zero, and less than that of an ‘apochromat’. It has only four curvatures, so could be made by a skilled amateur.
This following image shows the spectral variation of BFD with the 0.7 zonal ray.
But its field is relatively narrow. This following diagram shows that the field is only well corrected over a field of 1deg, compared with the notional full field of 3deg.
After re-optimisation, this particular choice of glasses can better corrected over the range 0.40µm to 0.90µm
2.1.1 Air-spaced four element design
In this arrangement, the rear negative element is split into an equivalent doublet whose glasses are selected to minimise the residual chromatism. With appropriate glasses, it is possible to extend the useful waveband from 0.4µm to 1.0 µm.
There is a consequent gain in the axial MTF, but the field curvature remains uncorrected.
An additional negative group placed in the airspace to the focal plane allows a substantial increase in useful field. This following image shows an example, for the air-spaced triplet. The field flattener was allowed to be no closer that 70mm to the focal plane. The focal length increases to 820mm.
We see therefore that close-spaced three and four element lenses have only moderate performance in electronic imaging on large focal planes, as a result of field curvature. But adding a field flattener can have a markededly beneficial effect. The element count for such a system is five or six.
3. Petzval Lenses
If one is prepared to make only four separate elements, and eliminate the field flattener option, it may be more sensible to consider the Petzval lens concept of two separated achromatic doublets. Each of the groups is corrected for axial colour, but with significant amounts of halo and coma. By partitioning these aberrations between the two groups, it is possible to overcorrect the astigmatism and thus reduce, but not fully correct, the effects of field curvature. But by well separating the rear group into strong positive and negative elements it is possible to flatten the sagittal and and tangential fields, thus achieving a true flat field. By selecting the glass types it is also possible to create a lens which although only ‘achromatic’ rather than ‘apochromatic’ has a secondary spectrum which is negligible over the visible band. For applications in which only viewing in the visible band is required this is perfectly adequate, and greatly superior to lenses using only closely spaced elements. This following image shows a 100mm aperture F/8 lens. The total length is 780mm. The back focal distance is 220mm.
Extending the waveband to the desired range shows that the lens now becomes apochromatic, and still possesses useful MTF.
3.1 Design for a 166mm F/5
The concept may be extended to such a specification, by splitting the front doublet into a pair of doublets. This design is likely to be a serious challenge for the manufacturer, but offers truly outstanding performances. Furthermore, the negative rear element can now provide a useful ‘telephoto’ effect, in which the overall length is substantially less than the focal length, at 750mm.
But the polychromatic MTF is affected slightly by weak chromatic halo.
3.2 A self-flattening Petzval design for a 120mm F/5
With a shorter focal length it is possible to rebalance the design for a wider waveband.
3.3. A self-flattening Petzval design for a 100mm F/4
the rear split Petzval concept can be pushed to even
shorter, faster types. This is one for a modest specification, and only 6 curvature values.
The following two images show the performances in the visible
The following two images show the performances over the full waveband
4. Cooke Triplet designs
The very highest performances in the Petzval designs required six elements. The element count is reduced to three in the Cooke Triplet lens type. Three powerful elements are substantially spaced relative to their separate powers, in the sequence + - +. By a suitable choice of glasses, it is possible to correct for the primary aberrations of halo, coma and astigmatism, field curvature and primary and secondary longitudinal and lateral chromatism and distortion.
This following diagram shows a layout of a 100mm aperture F/8 ‘apochromat’. It was optimised to maximise the performance over the spectral range 0.44 to 0.66µm, uniformly weighted.
This following diagram is the traditional BFD curve demonstrating that the lens is apochromatic within the Rayleigh limit.
But in fact there is a substantial loss in polychromatic Strehl intensity, resulting in a noticeable loss in MTF over the critical frequency range, as shown in the following diagram. This is caused by uncorrected chromatic halo at the extremes of the spectral band.
By weakly aspherising ( figuring ) one or two of the surfaces, it is possible to reduce the effects of spherochromatism, but the gains are only modest.
The gains from figuring are most marked when the aperture is increased. It is then possible to operate the lens at F/6 while still achieving near Rayleigh performance. There is some useful increase in MTF at the critical frequencies, but the greatest gain is in the speed of the lens. Overall, however, it is probably not a good trade – the size and weight of the elements increases rapidly. This is typical of the Cooke Triplet, where each element has substantial power.
Furthermore, because of the considerable powers of each element, the Cooke triplet is notoriously sensitive to manufacturing tolerances.
5. The Tessar
The second evolution from the Petzval lens is the Rudolph Tessar. In this configuration the front element is split. The following two diagrams show the F/8 design.
There is substantial gain over the triplet, but the gain must be weighed against the extra element.
These analyses suggest that for many applications, particularly fast and wide spectral band, the self-flattening Petzval type performs to remarkably high level. It is suitable for a wide range of focal lengths and focal ratios.
© Don Barron 2007