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.


There 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.
pupil dilation vs age


The 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°.
geometric distortion

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 1/4╝ 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 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°

distortion coeficient vs afov
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.

Rodenstock 40mm Erfle
Explore Scientific 20mm
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.)
20mm Nagler - ES comparison

The late 1880's saw the invention by George Eastman of nitro-cellulose roll-film and the Kodak box camera which heralded the advent of popular photography. George Eastman Kodak Exposure times were fixed, the box camera being supplied pre-loaded. Exposures were made outdoors in good light, and the camera returned to Kodak. "You press the button, we do the rest".
KODAK No.1 Box camera_1888 KODAK No.1 box camera 1888


Kodak introduced their folding pocket camera in 1898. It too had a fixed shutter speed. When their No.1 Autographic Junior folding camera was introduced in 1915, because it had a leaf shutter with variable speeds, exposure calculators came into general usage although Burroughs-Wellcome had incorporated cardboard dial calculators into their Photographic Diary from 1900.



The No.1 Autographic KODAK, Junior 1815 advertisement
No.1a AUTOGRAPHIC KODAK JR. folding camera

Throughout the first half of the C20th exposure calculators were the vogue. .

Direct means of measuring ambient light comprised Actinometers, manufactured by Green & Fčidge (1884); Watkins (1890); Wynne (1893) & Bee (1902), and Extinction meters, manufactured by Decoudon (1897); Heyde (1904); James; Minofot; Dremophot (1926 & 1931). Photoelectric photographic exposure meters became portable by the early 1930's pioneered by Weston, GEC, AVO, & L.M.T.

Actinometers, Extinction, & Photoelectric light meters, are all types of light measuring devices. There purpose is to provide a qualifiable or quantifiable measure of luminous intensity of actinic radiation.

Slide rule type exposure calculators rely on the photographer to estimate the scene brightness. Calculators were provided with lighting subject factor tables; usually determined for a specific and narrow range of latitudes, and film speed factors, either in recognised rating systems or a rating system defined by the calculator's manufacturer.

Because exposure calculators were very much cheaper than Actinometers, Extinction & Photoelectric light meters, they entered into common usage in the early C20th and remained so until the 1960's..

It is possible by comparing exposure indices and emulsion speed rating systems employed in these calculators to make comparisons between the sensitivities of photographic films and earlier dry and wet plates. This I have done and the results have been compiled into an Appleworks spreadsheet, and a page view 1Mb pdf plate_film_sensor_ratings.

Here are a few examples from my collection chosen as representative of the type of exposure calculators in use from about 1930 thru' 1960.




The Amateur Photographer magazine Exposure Calculator c1950

Cappelli - Milano Fotometro c 1930

Cappelli - Milano Fotometro c 1930


Ensign Posometer "dial" calculator c1930


Concentric Indicator Laboratories Fotogram c1949


Nebro exposure calculator manufactured by Neville Brown & Co. Ltd. c1949


Johnson's Standard exposure "dial" calculator c1955 (intro 1945)


Kodak exposure calculator Model II - undated, c1960


How do exposure calculators work?

To determine exposure time three factors need to be known:
i) the relative aperture of the lens (the focal ratio or f/#)
ii) the amount of light either shining on the subject or reflected off it.
iii) the sensitivity of the photographic emulsion or sensor.

i) The relative aperture of the lens is the simplest factor to work out. The relative aperture is the ratio between the lens focal length and diameter (or aperture), i.e. focal ratio = focal length divided by aperture, or f/# = F/D. Photographic focal ratios are defined in a specific way, called f-stops. If we wish to double the amount of light falling on the film or sensor, we need to double the area of the lens. The area of a circular lens is given by one quarter of the square of its diameter multiplied by ╣ (pi ┼22/7). Because the area is a function of the square of the diameter, if we double the lens diameter we quadruple the lens area. To double the lens area we must increase the diameter by the square root of 2 or &radic 2 &asymp 1.414. So if we start our f-stop scale at f/1, it proceeds as an arithmetic sequence:

1; &radic 2; 2; 2 &radic 2; 4; 4 &radic 2; 8; 8 &radic 2; 16; 16 &radic 2; 32; 32 &radic 2; 64; 64 &radic 2; 128..........&c

if we wish to intercalate half f-stops it becomes necessary to increase the arithmetic sequence not by a multiple of &radic2 but by a factor that increases lens area by &radic 2, which in terms of lens diameter equates to the square root of the square root of 2 or &radic &radic 2 or the 4th root of 2 i.e. 4 &radic 2 &asymp 1.2, and the arithmetic sequence proceeds thus:

1; 4 &radic 2; &radic 2; &radic 2 4 &radic 2......2; 2 4 &radic 2; 2 &radic 2; 2 &radic 2 4 &radic 2.....4; 4 4 &radic 2; 4 &radic 2; 4 &radic 2 4 &radic 2; 8........&c

or in common photographic parlance:

f/1; f/1.2; f/1.4; f/1.7; f/2; f/2.4; f/2.8; f/3.4; f/4; f/4.8; f/5.6; f/6.7; f/8.........&c

NB f/#'s are reciprocals

ii) The amount of light shining on the subject is termed the "Illuminance" and in photographic terms is measured in either SI lux units or Imperial "Foot Candles". The amount of light reflected off the subject is termed the "Luminance" and in photographic terms is measured in SI units "Candelas per square metre" or Imperial "Foot-Lambert" (Lambert - named after Johann Heinrich Lambert 1728-1777, German mathematician, phycisist and astronomer Johann Heinrich Lambert) or "Candles per square foot". All the exposure calculators in my collection employ light values based on Candles/sq.ft.. Many early photoelectric photographic exposure meters provided direct light value readings in Candles/sq.ft. & the exposure value was calculated using a circular slide rule built into the face of the meter.

Illuminance Table
Because the specific photometric terms "Illuminance" & "Luminance" are so easily mixed up, I do not think them helpful, and neither did most other photographers at the time. Tables of exposure values or Ev (nowadays EV) indices were compiled to which the photographer could refer based on Ev=0 corresponding to an exposure of 1 second at f/1 using an ISO100 rated film. This table comes from the "Fred Parker Photography" website Fred Parker Photography

Exposure Value Chart

EV chart

iii) The sensitivity of the photographic emulsion or the equivalent sensor sensitivity is nowadays expressed as an ISO value. The ISO scale is based on the old ASA scale and is arithmetic, a doubling of the ISO number corresponds to a doubling in the emulsion or sensor sensitivity. (A CCD or CMOS sensor has a base sensitivity, increasing the ISO rating does not in reality increase the sensor's sensitivity, it equates to an increase in the readout A/D amplifier gain setting).

So how is the calculation made?

The exposure duration is inversely proportional to the area of the lens, emulsion speed and subject brightness. If we call the exposure duration "t" for time in seconds; the lens area "A", the emulsion or sensor speed "S" and the subject brightness "B"; we have:

A x S x B = 1/t

"A" can be expressed in terms of the f/#, but because the f/# is based on the square root of the lens area, if we substitute f/# for "A" we must square it, and because f/# is a reciprocal: A = 1/(f/#)2


t = (f/#)2/S x B

We know what units "S" is expressed in - ISO speed numbers, but what about "B", what units is it expressed in? The brightness value "B" is determined from the exposure value (Ev). B = (2^Ev)/ &Pi

An article based on this method, applied to astrophotography exposure time calculation was published by Richard Levy in Sky & Telescope's "Observer's Page" July 1962.

(When Sky & Telescope was a subscription only magazine dedicated to the serious amateur astronomer and not today's dumbed down off-the-shelf hobby mag. It is a disgrace that none of S&T has been digitized and archived, and that neither NASA's ADS service or Google or Biblioteque have been given permission to digitize the archive. So I reproduce the pages here.) S&TJUL1962_p30 S&TJUL1962_p31

Is there a simpler way?

Yes, its called the APEX or "Additive Photographic EXposure" system APEX developed in 1960. Although, in general, it did not catch on with photographers, it did prove a boon to astrophotographers.

We can rewrite the algebraic equation in terms of base 2 logarithms thus:

from: A x S x B = 1/t

is obtained: lg.A + lg.S + lg.B = -lg.t

The APEX system defined pairs of these terms in terms of exposure value Ev from which
the APEX equation follows: Ev = Av + Tv = Bv + Sv
& putting:

Av = 2lg.f/#
Tv = -lg.t
Bv = lg.B.&Pi (B rationalised into foot-lamberts)
Sv = lg(ISO/&Pi)
is obtained:
Ev = 2lg.f/#+ -lg.t = lg.B.&Pi + lg(ISO/&Pi)

This may not at first glance appear simpler, but it is, because using the APEX system you can compile a simple additive table or draw nomograms nomograms or construct slide rules slide rules. For example the LORD ASTROPHOTOGRAPHY EXPOSURE TIME NOMOGRAM first published in the JBAA,89,3pp273-278 April 1979. <p273><p274-275><p276-277><p278>

(Again it is typical of the BAA, an organisation with a considerable balance sheet invested here there and everywhere, that they cannot find the wherewithall to digitize their journal archive, although some of it is available from NASA's ADS service)



and the "Astrophotography Exposure Calculator" designed by Glen LeDrew and sold by The Starry Room.

The APEX Ev table follows from the base 2 log scale values:


NB When B is expressed in candles/sq.ft Ev=0 corresponds to t=1s; f/# = 1; ISO = &Pi

A book based on the base 2 log APEX system was published in 1985 by Willmann-Bell Inc. "Astrophotography" by Barry Gordon, featuring the fx system of exposure determination. Astrophotography

How do I use the APEX system to calculate an exposure time?

Suppose you are photographing the First/Last Quarter Moon, high in a clear night sky, with an f/7 telescope, using either ISO100 film or a DSLR set to ISO100.

From the APEX equation:
Ev = Av + Tv = Bv + Sv
and from the APEX table we obtain the following values by reading across to the Ev column from the f/#; ISO & Brightness columns:
f/7 corresponds to Av = 5.7
B = 40 corresponds to Bv = 7.0 (B=40 is the brightness value of the First/Last Quarter Moon)
ISO100 corresponds to Sv = 5.0
from which:
Tv = Bv + Sv - Av = 7.0 + 5.0 - 5.7 = 6.3
& Tv = 6.3 corresponds to t = 1/81 secs

& you then select the closest shutter speed available; either 1/50s, 1/60s, 1/100s or 1/125s. Maybe your DSLR allows you to select 1/80s, if so, so much the better. Exposure times should be bracketed ▒1 stop at least. If the sky is hazy, or the Moon low, bracket from -1 stop to + 2 stops.

If you would care to add to this article please contact me @ chrislord brayebrook.demon.co.uk <Send Me Email>

Chris Lord

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