You need to separate the shadestepping (banding) in the original PShop image vs shadestepping in the halftone reproduction.
If the original PShop image contains banding then the typical halftone screen (133-200 lpi) will likely render the banding faithfully.
The formula: – [output resolution/screen frequency] (squared) + 1 = The number of possible gray shades – only refers tho the tone values available in a single halftone cell. For example, let's assume a halftone cell that's 2 x 2 pixels. That means that the halftone dot can only have 0, 1, 2, 3, or 4 pixels turned on - thus:
So for each halftone cell, the number of possible gray levels is: 2x2 squared = 4 + 1 = 5 possible levels of gray, in the above case 0%, 25% 50%, 75%, and 100%. Increasing the dpi, i.e. keeping the halftone cell size the same, but dividing it up into smaller pixels will give more gray levels. E.g., if the cell size is 3x3 then there are 10 possible gray levels, a 4x4 cell gives 17 gray levels and so on.
However we do not see single halftone dots but areas of multiple dots instead - and this gives us a way to work around the gray level limitations imposed by the formula.
For example, here is an area made up of (2x2 pixel) halftone cells from the original example.
I can achieve a 50% tone. However, I'm limited to a 75% tone as the next darker one or a 25% as the next lighter one. Big jumps in tone. However, by "dithering" adding a dot or removing a dot from some cells within the tone area I can increase the levels of possible grays:
In the above sample, I can go from just 3 levels: 25%, 50% and 75% to 5 levels: 25%, 40%, 50%, 58%, and 75%.
This dithering strategy to get around gray level limitations has been used by most screening technologies since about 1990. It has also meant that 2400/2540 dpi has become the defacto standard dpi of imaging systems for both film and plate setting even though the dpi/lpi formula suggests that the resolution is not high enough to give you all the gray levels required to avoid shadestepping in gradients.
best, gordo