Depth of field - is the apparent 'in focus' affected by sensor size?

What do you mean by resolution because I am sure there are FF sensors with less pixels than say the APS-C sensor on a 90D?

The "classic" approach is to calculated DoF on the base of a CoC diameter that equals 1/1500th of the sensor diagonal.
Thus under "normal" viewing conditions DoF is neither entangled with sensor size nor pixel count.

Of course that changes when you scrutinize a picture at closer distance or larger magnification. Hence some people prefer to base their DoF calculations on a CoC that equals the width of a Bayer cell, i.e. twice the width of a pixel (or even just a single pixel's width in case of a monochrome sensor).

But keep in mind that there's never a sharp barrier between "sharp" and "unsharp".

Imagine two cameras setup, side-by-side, with the same model lens at the same focal length, f-stop, shutter speed and ISO, centered on the same object in the distance. One is a full-frame camera and the other is an APS-C body. You make a photo with each camera.

Let's compare the photos, specifically their respective depths of field. A common definition of depth of field is the range of distances in a photo within which the smallest detail looks acceptably sharp.

If we adopt this for the comparison, there will be a nearest and farthest point in the photo made with the full-frame camera within which the smallest details look acceptably sharp.

The photo made with the APS-C camera will be centered and focused on the same point as the first photo. Having been made at the same focal length but being APS-C, the crop factor will have produced a photo having a more narrow angle of view.

Here's the important part: if we view the second (APS-C) photo at the same size and distance - either on a computer screen or as a framed print - anything visible in the frame will be magnified in size compared with the same element in the first (full-frame) photo.

As a result, the nearest small detail in the full-frame photo that looked focused will be larger and look soft. Similarly, the most distant small detail that's sharp enough in the full-frame photo will also be larger and look soft.

Depth of field in the APS-C photo will be shallower than DOF in the full-frame photo and the factor producing that difference, is sensor size.

When you say sensor size, do you mean the physical dimensions of the sensor? If so look at Joseph James’ explanation of equivalence.

If you mean number of pixels in each dimension on sensors that are physically the same size, then at same print size the DOF is the same. At 1:1 on a screen, it is not the same, but you can make it the same by resampling to common pixel dimensions.

If I have understood you correctly, this is just an alternative view on the old question of "Is the depth of field changed by cropping an image?"

The usual definition of DoF uses a circle of confusion (CoC) that is proportional to the image size: CoC = image diagonal / R, where R is a constant. Most commonly R=1440, which gives a CoC of 0.030mm for FF and 0.015mm for MFT.

Using this definition, if you crop the image then the length of the diagonal is reduced and hence the CoC becomes smaller in proportion. Hence the DoF also becomes smaller in proportion.

In that case, the DoF does not change if you calculate the CoC in the usual way as a fraction of the image diagonal.

However, if you calculate the CoC from the pixel spacing (e.g. CoC = 1.5 x pixel spacing), then the DoF decreases as the pixel spacing decreases.

Correct. Putting more pixels on an out-of-focus element won't bring it into focus.

If anything, the camera owner may be misled into thinking the higher-resolution sensor reduces depth of field or increases noise as a result of the 100% view in an image processing app being more zoomed in on a smaller area of the scene in comparison with a 100% view of a photo of the same scene using a lower res camera. However, when the two photos are compared at the same angle of view and image size, both depth of field and noise visibility will be the same.

As I remember it, DOF is based on viewing a standard sized print or image (maybe 8X10inches) .. My remembrance is that a circle of 1/30mm or smaller (COC) are all resolved by the 20/20 human eye as a dot .... when viewed from some standard distance (maybe 18 inches) with 20/20 vision ... change any of those three parameters (print size, distance to print and vision) and the DOF changes. So a 50mm lens @ F/8 lens will give nearly the same DOF regardless of the camera it is mounted on ...

WhyNot

But not at the same print size.

How do you figure that ! the sensor is no more dense than a m43 sensor.

My interpretation of DOF would be that it is an optics problem and the COC is theoretical so the resolution of the circle would not be effected. However, in definition the DOF is somewhat subjective. A 50mm lens on a FF camera gives a certain DOF perception, on a MF it gives a wider angle of view but if the resulting print is cropped to that of that of FF then the DOF become nearly the same ...

Your conundrum is whether by better defining that circle does it changes the perceived DOF -- essentially does it better define edges as we do in sharpening .. I would think it probably does but not so much as to make any difference until the crop or size of the print becomes enough to practically enjoy that difference ... which I suspect is an enlargement that itself changes the DOF you could read on one of those on line DOF calculators ....

WhyNot

The desired CoC is an input to the DOF calculators I use.

I use two on line DOF calculators -- DOFMaster which is ancient and Photopills which is not much newer -- both require the camera (which relates to the Calculators age), the focal length, the aperture, and the subject distance and provides the DOF for those parameters .. The DOFMaster also reports the CoC! . The CoC for defining the DOF, as I understand it is 1/30mm and has to do with the eye's ability to resolve the circle but the one reported by the calculator and usually discussed is the translation of that CoC to the film or sensor plane ... and it seems increase in size with the size of film or sensor used .. but only in the third decimal point or the second for 35mm to MF and sheet film variations.)

WhyNot

I use entirely different CoCs for different purposes. Varies with content, intent, expected print size, relationship to diffractive encircled energy diameter, whether I am stacking or not, lens aberrations, etc.

DOFMaster lets you enter the CoC from 1 um through 300um.