Depth of Field  .../page 2

In our introduction the Rayleigh limit of resolution of the eye made a convenient starting point for defining a tolerance to out of focus. This limit to resolution, or limit to the detail we can see, arises from the finite wavelength of light, roughly 0.55΅m for green light (΅m’s are one millionth of a metre or one thousandth of a mm). A star, or point, object is thus focussed not to a point on a film or plate but to a disc called the Airy disc named after the astronomer George Airy who observed it when using a telescope to look at stars. The profile of an Airy disc is shown in figure 1 where its diameter d depends on the f-stop setting of the imaging lens. It can be seen as to set the minimum diameter of the circle of confusion for any imaging optics, often referred to as the diffraction limit.

At the Rayleigh limit the images of two stars are said to be resolved if they are separated by a distance equal to half the diameter d. This is when the centre of one overlaps the edge of the other and is shown in figure 2. Some resolution is possible at reduced separations but not enough to make much difference. An animation which shows two Airy discs approaching and passing inside the resolution limit can be seen by selecting Animation. At separations just less than the Rayleigh limit we cannot see the two objects clearly but rather as one ‘blurred’ together. For example if the objects are alternate black and white lines then we see a uniform grey.

In general it is not possible to convey information, detail that is, in distances less than half a wavelength and so in optical images we cannot see detail finer than about 0.25΅m even with the most powerful optical microscopes (near zero f-stop). Oil immersion objectives can improve the resolution but are beyond the scope of this introduction. You may notice that figures quoted on other websites may differ slightly from those here, that is because we have rounded some of the numbers in view of the approximate nature of the calculations. As far as we are aware these do not lead to significant errors.

d ≈ 2.4λf_stop

where f_stop is the value of the f-stop of the lens and λ is the wavelength of the light; for example, when f_stop =  8 and λ is 0.0005mm (blue/green light), d is about 0.01mm. The limit of resolution is thus half the diameter, that is 1.2λf_stop. In most practical cases the value of 0.6f_stop  ΅m will suffice, this assumes a wavelength close to 0.50΅m seen as blue-green light. The f number of the eye with respect to a screen at a distance of 250mm is 125 assuming an iris diameter of 2mm. At smaller f numbers lens aberrations start to degrade the image and reduce the resolution. Hence the approximate resolution of the eye at 250mm is 0.6 x 125΅m giving a value of 75΅m i.e.. 0.075mm, which is equivalent to 13 lines/mm or an angle of 1 minute of arc. This just about matches the spacing of the light sensitive cones near the centre of the retina.

So in figure 3 we could place cones at the minimum and two maxima of the resultant red plot, just satisfying the minimum image sampling requirements. We now have to match the value of 0.075mm to the film in the camera via the magnification given by the ratio of the print and film (or CCD array) sizes. For example if the print is 7x5, where the 7 inches is about 180mm, and the film size 35mm then the camera lens should have a resolution of 35/180 times 0.075mm, ie around 0.015mm, in this case making the diameter of the circle of confusion 0.03mm. This figure corresponds to about 60 lines per mm. For comparison a diffraction limited f-stop 8 lens produces an Airy disc with a diameter close to 0.01mm. In all probability the combination of best lens and film will not be capable of better than 100 lines/mm making the largest print size which would limit the maximum usable print size, for close viewing, to around A3.

Unfortunately this whole topic abounds with numbers and different units and it does make it all rather confusing and unintuitive. For now you might ask well how does this relate to printers that are specified in dots per inch, dpi, or pixels per inch, ppi, or lines per inch, lpi. At 13 lines/mm we need to print 330 resolved lines per inch. To do this requires distinguishable bright and darker lines (hence the use of line pairs by some authors) with a pixel each for the bright and darker part of the line pattern making 660 ppi. Each black pixel will require at minimum one black ink dot but more are required to show shades of grey. Similarly coloured lines require a minimum of 3 coloured dots. In the introductory page it was made clear that the optimum resolution of the eye was only achieved in ideal lighting conditions and for people with 20/20 vision.

Experimental evidence indicates that a figure of twice the ideal limit of resolution may be used in practice for pictures to be judged as sharp. Clearly this would prevent a picture of lines spaced at 13 lines/mm being resolved but in practice the judgment of sharpness seems to be more tolerant. In this case the diameter if the disc of least confusion can be increased to 0.06mm corresponding to 30 lines/mm. The print would now require a resolution of about 300ppi. The earliest criteria set the resolution of the eye at 0.01 inch, 0.25mm, leading to a circle of confusion at the film with a diameter of 0.09mm, currently major lens manufacturers tend to the 0.03mm value. Examination of the depth of field scale on a 1935 35mm Kodak Retina 118 camera shows that the value of about 0.03mm was used in this case.

Good quality photographic lenses are close to diffraction limited between f-stop 8 and f-stop 22 in the central part of the field, but they are usually characterised by modulation transfer function, MTF that is evaluated by measuring the contrast of line images of known lines/mm spacing. For interest a perfect lens working at the Rayleigh limit would have an MTF close to 0.1 or 10% and the equivalent lines/mm are given by 1/Rayleigh limit. This Animation shows lines with separation reducing with time and, underneath, the modulation of the lines as seen in an image. The use of two terms lines/mm and line pairs/mm can be confusing. If 20 black lines are drawn on white paper in 10mm it is easy to say there are 2 lines/mm on the paper but some people prefer to say there are also white lines between the black lines so there are really 2 line pair /mm. In the end the figures are the same, the concepts just slightly different but as we saw when considering printing the use of line pairs made it easier to work out the necessary ppi required to reproduce the image.

1/v + 1/u = 1/f

The next diagram, figure 6, shows this region in closer detail. The symbol ∆v represents the amount that the image distance changes as a result of allowing defocus around the circle of confusion. This unusual representation, ∆v, implies that it is smaller than the principal value ‘v’ and in this case the value of ∆v is very much the same each side of best focus. Strictly this is not correct and for the exaggerated size of the disc of confusion shown in figure 6 this is the case, as can be seen. However, when the diameter of the disc of confusion is only 0.03mm and v is about 50mm the approximation is for all practical purpose correct. Its use allows valid algebraic approximations and simplifications to be made. The angle of the rays to the axis 'θ' is about the same for all the rays in practice; in the figure the disproportionate size of the disc of confusion makes this look wrong.

From figure 6, tan(θ) = c/2∆v and for small angles θ ≈ c/2∆v.

The f-stop is given by the value of  f/D

But θ ≈ D/2v and since v ≈ f we can write θ ≈ D/2f and it then follows that: c/2∆v = D/2f.

Rearranging and substituting for the f-stop we get an expression for ∆v in terms of values we know, hence:

∆v ≈ cf_stop

1/u_near + 1/[v+∆v] = 1/f

where u_near is the near distance at the limit of the depth of field. Rearranging the fractions: 

u_near = [v+∆v]f/[v+∆v-f]      similarly:  u_far = [v-∆v]f/[v-∆v-f]

So far these expressions are not very useful as we do not know the value of v. However, we know the value of ‘u’, it is where we wish to focus on the object and we know the focal length of the camera lens. The lens equation can now be rearranged to obtain ‘v’ in terms of ‘u’ and ‘f’ as we have to obtain u_near and u_far  i.e.

v = uf/u-f

In line with the thinking so far it looks as if we could use the approximation v ≈ f but we have to be careful here as it is the small difference between v and f in the denominator of the earlier expressions that determines the new values of the object distances ‘u’. This time an approximation would be invalid.

After substituting for v and ∆v we obtain two straightforward approximate expressions:

u_near = uf²/[f² + ucf_stop]     and    u_far = uf²/[f² - ucf_stop]

where value of c is given by:

c =  c₀/M

and where c₀ is the diameter of the disc of confusion for the eye at the viewing distance of 250mm, taking a value somewhere between 0.15mm and 0.25mm, and M is the magnification given by the ratio of the size of the print to the size of the film or CCD array.

These are accurate to within a few percent when compared to an exact evaluation for f-stop values less than f-stop 11 or calculated far depth of field distances not exceeding about 50 metres for larger f-stop numbers. Also keep in mind that the expressions were derived using simple geometry and ignore the size of the Airy disc formed by diffraction. The possible errors are not insignificant. The diameters of the Airy discs for f-stop 8 and f-stop 16 are 0.01mm and 0.02mm respectively and would act to reduce the value of ‘c’ used in the equations. At f-stop 16 and f-stop 22 the depths of field would therefore tend to be less than calculated using geometrical optics.