EYEPIECE APPARENT & REAL FIELDS of VIEW
The trend towards ever wider eyepiece apparent fields of view appears to be unquestioningly welcomed because it enables wider real fields to be obtained at any particular power. 82° is better than 70°; 100° is better than 82°; 120° is even better. But is it? Do ever wider apparent fields of view always bring extra benefits? The purpose of this article is to caution those whose acquisitive instincts and desire for one-up- manship cause them to overlook certain drawbacks to ever wider apparent fields of view, and to clearly demonstrate there is an optimum apparent field angle, which when exceeded brings little tangible benefit.
VISUAL FIELDThere would seem little point designing and making an eyepiece whose apparent field exceeded the eye's field of vision. Yet anything exceeding roughly 90° does just that. Consider this diagram from NASA SP-3006 Bioastronautics Data Book, edited by Paul Webb MD, for Scientific and Technical Information Division, NASA: Washington D. C., 1964.
The clear area bounded by the dashed line is the eye's peripheral field of vision when the macula lutea is fixated on the optical axis. The shaded area is the peripheral field for averted vision, when the eye is fixated off axis. The axial field of vision is roughly 130° x 120°. Note the up field is cut off at 50° radius.
The eye's central field of peripheral vision (shaded grey) is contained within the 50° circle. This means that anything exceeding 100° cannot be seen by the eye from a fixated position on axis. The only way to access all the field is by shifting the eye off axis.
Furthermore this assumes your pupil is sufficiently dilated to accept the entire oblique exit pupil at the field boundary. This will be so if the exit pupil is smaller than the eye pupil, not so when they are similar. Placement of the eye precisely at the Ramsden disc (assuming there is no spherical aberration of the exit pupil) becomes increasingly critical as the afov widens. In practice saccades means the widest apparent field the eye can accept is about 90°.
Pupillary aperture and dilation are dependent on the ambient illumination, and age. Maximum eye pupil opening reduces with age. Smoking and drinking, drug abuse & depression makes the shrinkage worse. But by the time you reach your 70's its about 5mm - 7mm max, if you're healthy. Of course a smaller eyepiece exit pupil makes a light polluted sky background appear darker, but the eye's sharpest vision happens when the eye pupil is about 2mm. But if you observe from a dark sky site, and you're not interested in trying to detect colour, then a wider exit pupil, approaching your eye's full dilation, is preferable for detecting faint nebulosity. When the exit pupil of the eyepiece gets down to ~1mm, diffraction effects begin to manifest, lowering image contrast.
DISTORTIONThe next thing to consider is distortion. Some wide angle designs exhibit pincushion or positive distortion. Some designs are slightly overcorrected and exhibit negative or barrel distortion. It is not possible to suppress both angular magnification distortion and rectilinear distortion because the former is proportional to the field radius in radians and the latter the tangent of the field radius.
Angular magnification distortion and rectilinear distortion are each classified as geometric distortions. Because it is impossible to make both constant across the field radius, if the designer corrects for one he is left with the other, to equal extent. Angular magnification distortion is sometimes referred to as f-theta distortion, and rectilinear distortion as f-tan-theta distortion, the "f" being the eyepiece focal length. An eyepiece obeying the tangent condition exhibits angular magnification distortion, and vice versa.
When the field radius is small, theta in radians is similar to tan-theta, hence geometric distortion is small, but as the field radius widens, the discrepancy between theta and tan-theta grows, approximately as the cube of the field radius.
The coefficient of distortion E can be easily calculated for a field radius in the absence of astigmatism by taking the ratio of theta and tan-theta and expressing it as a proportion of the latter. I have plotted E as a percentage against the apparent field angle in degrees, between 0° & 140°.
Note how the geometric distortion increases with the apparent field of view. At the apparent fields encountered in classic Orthoscopic eyepieces, even if it isn't controlled, over a 40° field it is only 4%. For a classic Erfle with a 60° apparent field it is 9%, but for a 120° apparent field it is a huge 40%.
Angular magnification distortion is corrected to a large extent in astronomical eyepieces, and rectilinear distortion in binocular & spotting 'scope eyepieces intended for terrestrial use. The point to bear in mind is, that if one is made constant with field radius, the other is left uncorrected and manifests itself to the extent shown in the geometric distortion plot.
I have seen it stated in authoritative text books dealing with eyepiece design, that if angular magnification is held constant, the shape of a round object remains round if it is placed tangential to the edge of field. I have also seen it stated that in the presence of rectilinear distortion, a round shape is distorted into an ellipse at the field edge. Both these statements can be easily shown to be incorrect.
In a 120° afov eyepiece, suppose the real field to be 2°, the solar image, being 1/2° across, will occupy a quarter of the field. The edge of the disc when the image is on-axis, will lie 1/4¼ off axis, 15° apparent, where the rectilinear distortion E = 2.3%. The edge of the disc is stretched outwards 2.3% in all directions, so the Sun still looks round. But shift the Sun's image so the limb touches the field edge, and you have a situation where at the field edge E = 39.5%, at the disc centre E = 21.5% and on the inner limb E = 9.3%. In the radial direction the image is stretched out more from the disc centre to the field edge than from the inner limb to the disc centre, and in the tangential direction more past the disc centre, than either at it or within. The round disc of the Sun is distorted into an oval, with the rounder side outmost.
It is also tacitly presumed that if angular magnification is held constant with field radius. the separation of a close double star will remain the same whatever the placement within the field of view. A moments consideration will demonstrate this cannot be so. If angular magnification is held constant across the field, then rectilinear distortion will go uncorrected, and because it is positive, i.e. pincushion distortion, any object drifting toward the field edge will become stretched outward in a radial direction. Unless the pair are exceedingly proximate, as they approach the field edge, the component furthest from the field centre, will have its position stretched outwards more than the other, and so their apparent separation will increase.
It matters not what design the wide angle eyepiece has, the only trade off in keeping the uncorrected of the two geometric distortions low is astigmatism and field curvature, or zero Petzval sum in the case of an anastigmatic design.
The geometric field of an orthoscopic design is given by:
where is the field stop diameter
& is the eyepiece focal length
In the presence of geometric distortion the apparent field is given by:
where is the geometric field.
I have plotted against for a range 0.30 to 1.85 & 15° to 130°
Note that when the diameter of the field stop equals the eyepiece focal length, when there is zero distortion, the apparent field is 1 radian or 57°.3.
Note also negative or barrel distortion is not a mirror image of positive or pincushion distortion about the zero distortion line, and that even miniscule negative distortion coefficients drastically reduce the apparent field in hyper-wide angle designs.
Now lets take some real cases. The Rodenstock 40mm x 70° Erfle has a 47mm field stop. Therefore e = 1.175. If you see where 1.175 intersects 70° it is slightly above the red zero distortion curve, meaning it has very slight pincushion distortion, barely noticeable.
The Explore Scientific 20mm x 100° has a 63mm internal field stop before the 34mm Erfle II section so e = 1.85. 1.85 intersects 100° below the red zero distortion line meaning it has slight negative or barrel distortion, about 0.2%. (Note the geometric field of the ES 20mm x 100° is 129°; barrel distortion shrinks the field, which implies the Smyth converter is slightly over-compensating in order to reduce the width of the field group in the Erfle II section.)
These wide angle eyepieces perform very well down to about f/7. The exit pupil of the Rodenstock is 6mm and the ES 20mm, 3mm. These are reasonable exit pupils for low power eyepieces, providing a wide real field of view.
Now lets consider the TeleVue Ethos 8mm x 100° - version 3.0. The field stop is 13mm, so e = 1.625, which intersects the red line just below the 100° afov ordinate. The geometric field is 104° meaning there is slight barrel distortion, about 0.2%.
What I'd like you to notice is that very wide low or zero distortion fields make it impractical to make 2-inch series eyepieces that give 4mm or larger exit pupils. The 8mm Ethos will have ~1mm exit pupil on most ED apo's. The snag with exit pupils of the order 1mm and less, is diffraction, which lowers image contrast. The clearest views require exit pupils of 2mm or wider. Your only going to get that with an f/4 'scope.
The Rodenstock Erfle is a well corrected eyepiece, most Erfle types are. I have in my collection a 36mm x 70° 2-2-2 Kaspereit. The field stop, which lies behind the field doublet, is 44mm. Hence Ed = 1.22, and the corresponding geometric field is 72°.5. The calibrated afov is 71°.3 corresponding to 0.4% barrel distortion.
It is frequently stated that hyper-wide apparent fields give the sensation of "floating in space". The only situation in which this can be the case is when the field boundary cannot be seen from the eyepoint. This is going to be the situation for any eyepiece with afov much greater than 90°. The only way the eye can take in a 100° or 120° afov is by moving off axis and looking obliquely to the field edge.
It is also stated that hyper-wide eyepieces offer more visual impact. But this overlooks the other half of the field of view parameter. What also matters is the real field. I use a rule of thumb to calculate what I call the visual impact factor or VIF. Multiply the eyepiece focal length in millimetres by the apparent field of view in degrees. The larger the product, the greater the visual impact. When you use my VIF standard it becomes obvious that a 40mm x 70° eyepiece is going to provide greater visual impact than a 20mm x 100° eyepiece. (2800 vs 2000). The 8mm Ethos VIF is only 800.
MAXIMUM FOCAL LENGTH
The focal length corresponding to the widest feasible field stop for any particular afov is given by:
note it is dependent on the OG focal ratio, f/#. I have plotted against between f/4 and f/20 for a field stop of 48mm (the widest practicable field stop in a 2-inch barrel.
Note as the afov increases the difference in maximum focal length reduces less towards f/4. Of course the eyepiece focal length divided by the OG focal ratio gives the exit pupil. The widest useable exit pupil is that which equals the pupiliary widest dilation, about 7mm on average across adult age range, corresponding to a focal length range 28mm @ f/4 thru' 140mm @ f/20.
Due to mechanical and optical practicalities it isn't possible to produce 120° hyper-wide angle eyepieces in 2-inch series with focal lengths longer than 22mm, and 100°, 26mm. Explore Scientific have a 25mm x 100° hyper-wide about to enter production.
The exit pupils obtained with hyper-wide eyepieces range from 5mm @ f/4 to 1mm @ f/20 for a 20mm focal length. In my opinion Leitz got the balance between afov and focal length right with their 30mm x 88° Weitfeld-Planokular, which had a 70° geometric field and about 8% pincushion distortion. The generous 20mm eye relief meant all the field could be accepted and for mid focal ratio 'scopes (~f/7) the exit pupil is ~4mm.
This clever design did not need a Smyth converter, instead it was an Erfle derivative with a bonded aspherical eye lens shell. Leitz introduced it in 1990.
I also have in my extensive collection a rare 25mm x 90° 3-2 Galoc. Its field stop is 35mm hence Ed = 1.4, & E = 0.02, or 2%. The design is very economical, the edge field performance on my f/7 TEC140ED apo is good, slight astigmatism and no lateral colour. It gives a 3.6mm exit pupil and fov 2° 22' @ x38. It is a very unusual design, having a steeply concave eye lens final surface. Eye relief is a generous 1.0Fe or 25mm. There is moderate astigmatism in the outfield, no lateral colour, and no noticeably rectilinear distortion. The field is dark, contrast is very good and there are no ghosts or flaring. This is my most prized low power wide field eyepiece. It is a remarkable wide field 3-2 orthoscopic design. It's performance is so remarkable for such a simple looking eyepiece that it probably has at least one aspheric surface.
AIR-GLASS SURFACESThe other problem with hyper-wide angle astronomical eyepieces is the number of air-glass surfaces. The ES 20mm x 100° has 9 elements in 5 groups and 12 air-glass surfaces. The TV Ethos 8mm x 100° has 9 elements in 5 groups and 12 air-glass surfaces. The 40mm x 70° Rodenstock Erfle has 5 elements in 3 groups and 6 air-glass surfaces. The Leitz 30mm x 88° Weitfeld-Planokular has 7 elements in 4 groups and 8 air-glass surfaces. Broad band multi-layer anti-reflection coatings produce narrow angle scatter, especially at fast focal ratios where the abxial peripheral field rays are most oblique. Eyepieces with afov greater than 90° require a Smyth converter, which adds to the number of air-glass surfaces. Modern a/r multi-layer coatings can minimize transmission loss but they can't reduce narrow angle scatter, in fact they make it worse. The effects of narrow angle scatter manifest themselves at medium to high power on very bright stars. Narrow angle scatter is the bane of multi-element a/r coated lens systems.
SMYTH-ERFLE DESIGNIf you look at the lens arrangements of the ES20mm x 100° & the TV Ethos 8mm x 100° you will notice that the Smyth converter acts like a built in Barlow lens. Both have 4 elements in 2 groups. Like a Barlow lens a Smyth converter has a negative focal length. The crucial difference is the Smyth converter has a forward negative group followed by a positive field flattener. Its purpose is to diverge the beam and widen the apparent field and balance the distortion of the positive section's hyper-wide field. Now take a close look at the lens arrangement behind the Smyth converter. Both designs have 5 elements in 3 groups in a 2-1-2 arrangement. The sections of both designs are essentially a modified Erfle type II. The trick is to balance the negative distortion of the Erfle II section with the positive distortion of the divergent Smyth converter section.
The field stop of the ES 20mm x 100° lies behind the Smyth converter. This is a preferable arrangement from the standpoint of field stop edge colour correction. Lateral colour correction at the field stop will be only slightly under-corrected (i.e. it will possess a blue fringe), but the eye will be unable to see the field edge without looking obliquely so this on the face of it isn't that big an issue. The TV Ethos 8mm x 100° has its field stop at the conjugate focus of the negative triplet and the field flattener, which leads to overcorrection and a reddish fringe.
Why is the Smyth converter necessary? If the Erfle design is modified to produce apparent fields in excess of 70° noticeable positive distortion and astigmatism is introduced, as well as lateral colour. If the lens surfaces are spherical there is no way of avoiding this. However if one or more of the surfaces are aspherized, it is then feasible to widen the field of the Erfle to ~90°, with perhaps the addition of only one or two elements to control lateral colour.
EXIT PUPIL VIGNETTINGVignetting can be caused by undersized lens groups, but it can also be caused by a mismatch between the eye lens diameter, eye relief and the exit pupil. When the eye lens is undersized for the exit pupil it leads to exit pupil vignetting.
The Rodenstock 40mm x 70° has 25mm eye relief, and a 35mm eye lens. According to the chart the eye lens should be 40mm to avoid vignetting. The ES 20mm x 100° has 14.5mm eye relief and a 30.5mm eye lens. According to the chart the eye lens should be 37mm. The effect of an undersized eyelens is to cause vignetting of the outfield. The Nagler type 5 20mm x 82° has a 17mm eye lens and 12mm eye relief. The eye lens ought to be 25mm diameter. As you can tell this design limitation, trading off eyepiece size for a certain amount of vignetting is not uncommon. Erfle's don't usually suffer from an undersized eye lens because the lens widths are dictated by the field doublet, and there is little to be gained by undersizing the rest of the groups. The aforementioned 36mm x 70° Kaspereit has 11mm eye relief and a 46mm eye lens, wider than the field stop. It needn't have been wider than 20mm, so no vignetting due to an undersized eye lens in this instance.
CONCLUSIONAll the aforementioned reasons are why I prefer wide angle eyepieces based on the original Orthoscopic or Erfle design. I own and use Nagler 82° wide field types 5 & 6, and the ES 20mm x 100°. They're fine eyepieces, but they do not out perform the Rodenstock 40mm x 70° Erfle, when it comes to low power wide real fields. The visual impact of the Rodenstock is obvious to anyone who looks into it! It is also obvious that an eyepiece like the 25mm x 90° Galoc offers far more in terms of economy of elements and air-glass surfaces than any 13+ element 25mm x 100° design using a Smyth converter.
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