Soocom1
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Zeiss has a nice write up on this very issue:
https://lenspire.zeiss.com/photo/app/uploads/2018/04/Article-Bokeh-2010-EN.pdf
4. Smaller film format with the same lens
If we remove a lens from an old analogue camera and attach it to a digital camera of the same system that has a somewhat smaller APS-C sensor, then there is a "crop factor". We do not talk about an extension of the focal length, it doesn’t exist in this case. After all, the lens does not know how much of its image circle we are capturing with our sensor. The size of the object field is reduced by the crop factor while the object-side light cones remain the same, as long as we use the same lens and do not change the aperture setting. That is why the points of the light cones may not be located so far from the focal plane if we want to maintain the same ratio of diagonal to circle of confusion. Reducing the size of the film format therefore reduces the depth of field by the crop factor.
5. Different film formats with the same object field
If we select the suitable focal length to ensure that we always display the same field with different film formats, then things go just the other way round: reducing the size of the sensor format increases the depth of field, and enlarging the sensor format reduces the depth of field, as long as we always use the same aperture setting. That is because a smaller sensor format displays the same object field with an accordingly shorter focal length. If the same f-number is used, then the entrance pupil is reduced by the crop factor and the light cones are narrower. For the same reason, medium format photographs show a significantly smaller depth of field with the usual apertures, even though the absolute diameter of the imageside circles of confusion is larger, usually 0.05 mm as opposed to 0.03 mm in 35 mm format. If the medium format lens is adapted to a 35 mm camera, then of course we have to calculate with the 0.03 mm of the smaller format. The acceptable diameter of the circle of confusion is therefore not a characteristic of the lens but rather the sensor format. A feature of the lenses is only the smallest possible circle of confusion, and this arises from the correction of the lens aberrations. At first glance we therefore observe a paradoxical characteristic whereby large formats have a smaller object-side depth of field and simultaneously a larger imageside depth of focus with the same apertures and object fields. This is also reflected in the mechanical tolerances of cameras: Large-format cameras can be built with carpenter precision, and the camera module in a mobile phone requires µm (micrometer) precision. Those are the extremes, but in SLR photography we can already see the difference between APS-C and full-frame format with regard to the requirements for focusing accuracy. It appears to be a confusing paradox at first glance, but of course it has a very simple explanation. We just photographed object fields of the same size with different sizes of image formats. If the acceptable blurriness is supposed to be the same with these different cameras, it means that the ratio of the object field diagonal and the "object-side circle of confusion" should be the same. The object-side light cones travelling from a point behind the focal plane, for example, should therefore be the same for all compared cameras. If the images have different format sizes, however, the imaging scale is different. Under these conditions, the image-side circles of confusion must therefore increase along with the scale factor. The object-side light cones can only be the same if all entrance pupils are of the same size, however. But because object fields of the same size mean longer focal lengths for larger image formats, the f-numbers must be different
https://lenspire.zeiss.com/photo/app/uploads/2018/04/Article-Bokeh-2010-EN.pdf
4. Smaller film format with the same lens
If we remove a lens from an old analogue camera and attach it to a digital camera of the same system that has a somewhat smaller APS-C sensor, then there is a "crop factor". We do not talk about an extension of the focal length, it doesn’t exist in this case. After all, the lens does not know how much of its image circle we are capturing with our sensor. The size of the object field is reduced by the crop factor while the object-side light cones remain the same, as long as we use the same lens and do not change the aperture setting. That is why the points of the light cones may not be located so far from the focal plane if we want to maintain the same ratio of diagonal to circle of confusion. Reducing the size of the film format therefore reduces the depth of field by the crop factor.
5. Different film formats with the same object field
If we select the suitable focal length to ensure that we always display the same field with different film formats, then things go just the other way round: reducing the size of the sensor format increases the depth of field, and enlarging the sensor format reduces the depth of field, as long as we always use the same aperture setting. That is because a smaller sensor format displays the same object field with an accordingly shorter focal length. If the same f-number is used, then the entrance pupil is reduced by the crop factor and the light cones are narrower. For the same reason, medium format photographs show a significantly smaller depth of field with the usual apertures, even though the absolute diameter of the imageside circles of confusion is larger, usually 0.05 mm as opposed to 0.03 mm in 35 mm format. If the medium format lens is adapted to a 35 mm camera, then of course we have to calculate with the 0.03 mm of the smaller format. The acceptable diameter of the circle of confusion is therefore not a characteristic of the lens but rather the sensor format. A feature of the lenses is only the smallest possible circle of confusion, and this arises from the correction of the lens aberrations. At first glance we therefore observe a paradoxical characteristic whereby large formats have a smaller object-side depth of field and simultaneously a larger imageside depth of focus with the same apertures and object fields. This is also reflected in the mechanical tolerances of cameras: Large-format cameras can be built with carpenter precision, and the camera module in a mobile phone requires µm (micrometer) precision. Those are the extremes, but in SLR photography we can already see the difference between APS-C and full-frame format with regard to the requirements for focusing accuracy. It appears to be a confusing paradox at first glance, but of course it has a very simple explanation. We just photographed object fields of the same size with different sizes of image formats. If the acceptable blurriness is supposed to be the same with these different cameras, it means that the ratio of the object field diagonal and the "object-side circle of confusion" should be the same. The object-side light cones travelling from a point behind the focal plane, for example, should therefore be the same for all compared cameras. If the images have different format sizes, however, the imaging scale is different. Under these conditions, the image-side circles of confusion must therefore increase along with the scale factor. The object-side light cones can only be the same if all entrance pupils are of the same size, however. But because object fields of the same size mean longer focal lengths for larger image formats, the f-numbers must be different