Tieing off another loose end...
In this topic/thread, the issue of diffraction has been mentioned several times, for example:
Rik Littlefield wrote:you can probably gain back the DOF by stopping down. The 4X larger sensor of a DSLR should allow a 4X smaller f-stop for the same degradation from diffraction.
Rik Littlefield wrote: When diffraction kicks in, you have to widen the aperture (reduce the f-number) in proportion to the shrinking sensor. This compensates for the otherwise increasing DOF at fixed f-number. As I understand the process, the result is a wash and the large and small sensors both deliver essentially the same DOF at their respective diffraction-limited apertures.
I have rechecked the theory, found some good references, and worked out a simple summary that I think is worth sharing. Here it is.
As ground rules, let's agree to hold constant the field width (subject size) and the final print size, and to evaluate DOF and diffraction in terms of resolution in that constant final print size.
First, define a few terms as follows:
M_tot = total magnification, e.g. a 25.4mm subject blown up to 254mm wide print is M_tot = 10.
M_enl = enlargement magnification, e.g. enlarging a 12.7mm sensor to 254mm wide print is M_enl = 20.
nominal f-number = aperture_width / focal_length. (This is what's marked on the lens dial.)
effective f-number = aperture_width / distance_from_sensor. (This is the nominal f-number adjusted for macro focusing.)
geometric DOF = depth of field calculated from geometric blur circles
overall DOF = depth of field (depth of detail) calculated with geometric blur and diffraction effects combined
Then, for closeup and macro work:
1. At fixed f-number, small sensors have larger geometric DOF but also larger diffraction effects.
2. Geometric DOF and diffraction effects can both be held constant by scaling the effective f-number in proportion to sensor size, e.g. effective f/16 on a 22mm sensor gives the same result as effective f/4 on a 5.55mm sensor.
3. In terms of nominal f-number, the required scaling goes as 1/(M_tot+M_enl).
4. Because of diffraction, there is a maximum overall DOF that depends only on M_tot (total magnification) and your tolerance for fuzziness. That maximum DOF does not depend on sensor size, but the f-number needed to achieve it does, following the scaling rules listed above.
5. One recommended value for nominal f-number at maximum DOF in a sharp print is 220/(M_tot+M_enl). (See http://www.modernmicroscopy.com/main.asp?article=65 )
For example a 25.4mm subject blown up to 254mm wide using a 22.2mm sensor has M_tot = 10 and M_enl = 11.44, giving nominal f-number = f/10.26. Using a 5.55mm sensor has M_enl = 45.76, giving nominal f-number = f/3.94. (The respective effective f-numbers are f/19.2 and f/4.8, a factor of 4X apart as implied by the sensor size ratio.)
This model is pretty accurate. It predicts that
small and large sensors have no differences in DOF at the same resolution, if and only if you can set corresponding f-numbers. (Noise differences are still important; see earlier posts.)
Sensor size becomes an issue for DOF when conditions require f-numbers that are achievable for the larger sensor but not for the smaller sensor. This is primarily an issue when attempting to get narrow DOF for artistic purposes. For example, it is feasible to shoot at f/2 on a 22.2mm sensor, but it is not feasible to shoot at the corresponding f/0.5 on a 5.55mm sensor because no such lens would be available. In macro work, a similar although less extreme problem may occur at moderately high magnifications. For example, imaging a 5mm subject using the 220/(M_tot+M_enl) rule would require nominal f/3.5 with a 22.2mm sensor but f/2.3 with a 5.55mm sensor.
Note that the 220/(M_tot+M_enl) rule is for producing a print at 6 line pairs/mm = 12 pixels/mm = 300 pixels/inch = 3000 pixels in 10 inches -- a very sharp print.
For lower resolution prints and screen images, the f-number can be proportionally larger, for example 5X larger to make a 600-pixels-wide screen image. For my
Christmas LED, the suggested formula 220/(M_tot+M_enl) would have called for f/4.6. Instead, I shot at f/11, choosing to more than double my depth of field at the cost of some sharpness. The finest details look crisp in a half-size image (1536x1024 pixels), but not at full-size (3072x2048 pixels).
I have found it interesting to review the history of DOF investigations. I started doing macro work on 35mm film back in the late 1960's. At that time, the bible of DOF calculations was a paper by Lou Gibson titled "Magnification and Depth of Detail in Photomacrography" (J. Phot. Scty. Amer., June 1960, 26, 34-46, currently online at
http://www.janrik.net/Papers/Gibson1960.pdf). Gibson introduced the crucial distinction between "depth of field" and "depth of detail", the former being defined in terms of geometric blur circles and the latter being defined in terms of overall resolving ability, taking into account all sources of blur including diffraction and a host of others. In Gibson's analysis (which appeared well supported by experiment), it worked out that greater depth of detail was achieved by shooting on larger film formats at smaller f-numbers.
However, Gibson's analysis assumed that significant blur is added by the film and by the enlarging process.
With digital sensors, all of the film and enlarging blurs go away, leaving camera diffraction as the one remaining issue. This turns out to provide a huge simplification. Diffraction effects and geometric blur scale together as the f-number is changed to maintain constant DOF, and the result (as outlined above) is that diffraction-limited depth-of-detail becomes independent of sensor size, given suitable lenses.
I have tried to leave behind enough links and discussion to let other people follow my reasoning and examples. Let me know if you find any mistakes or have any further concerns or questions about sensor size issues. Otherwise I'm going to move on to other matters, like actually taking some pictures instead of just thinking about taking them!
--Rik