Halftones and grey levels explained

Some principles of halftones and the myth of grey level capability.
This information is basically true for all vendor's offerings.

A halftone dot is formed inside a halftone "cell" The cell is a grid of pixels which are turned on to form the dot. The cell begins with no pixels turned on (0% tone) and as pixels are turned on the dot grows until all the pixels within the cell are turned on and the cell is filled (i.e. 100% tone).For example.
If the cell size is 2 pixels wide by 2 pixels deep the halftone cell will contain a total of 4 pixels. As a result the following halftone dot tone values can be created:
0% = all pixels off
25% = 1 pixel turned on
50% = 2 pixels turned on
75% = 3 pixels turned on
100% = 4 pixels turned on.

So, with a 2x2 pixel halftone cell it is only possible to have 5 tone levels (grey levels). I.e. the total number of tones possible equals the total number pixels available plus one. In this case 2x2=4 4+1 = 5.
If the number of pixels is increased within the cell by making them smaller - i.e. cell size remains the same but the pixels are smaller - then the number of possible grey levels goes up.
For example:
For a 3x3 cell the number of possible grey levels is 10 (3x3=9, 9+1=10
For a 10x10 cell the number of possible grey levels is 101 (10x10=100, 100+1=101
For a 16x16 cell the number of possible grey levels is 257 (16x16=256, 257+1=257)
In a basic AM screen the dot is formed by turning on pixels starting from the center of the cell. For a basic FM screen the pixels within the cell are turned on pseudo-randomly.

So, as resolution (the "dpi" of the recording device) increases - grey levels increases. As resolution decreases grey levels decrease.

If the resolution (dpi) is fixed but the number of adjacent cells is increased (lpi, i.e. going from 100 lpi to 175 lpi) then the number of pixels available for each dot decreases and therefore the number of grey levels decreases.
This principle is captured by the classic formula:
(dpi/lpi) squared + 1 = number of grey levels

So for a 2400 dpi output device:
At 100 lpi:
2400 dpi/100 lpi = 24 squared = 576 plus one = 577 tones possible. No problem - more than enough grey levels.
But at 175 lpi:
2400 dpi/175 lpi = 13.7 squared = 188 plus one = only 189 tones possible. A big problem because when the ratio of dpi to lpi drops below 16, the number of available grey levels drops to below 256. This can result in tonal reproduction that is inaccurate and uneven, causing visible shadestepping (a.k.a. banding or contouring) in gradients. Color steps abruptly from one tone to the next without a smooth transition.

In 1984 the screening technology described in part one was the state of art for halftone screening with Postscript devices.
The only way to recoup the lack of tones as one went to higher lpis was to increase the device dpi. I.e. go from 2400 dpi to 3200 dpi or higher. The penalty was slower imaging times and increased process control required in the film workflows of the day.
However, the formula is only true for the tone represented by a single, isolated, halftone dot based on an individual halftone dot cell - something that never occurs in real production environments. So, around 1989 a new approach began to be adopted. The approach is based on the fact that we don't care about individual halftone dots. What is important is the tone represented in an area. For example, let's say that we want to see a 17% tone patch value in the presswork. However, if we cannot represent that area with individual 17% dots – because of that classic formula limitation – we can still create the 17% value by alternating 16% dots and 18% dots (this is called "dithering"). The eye (and instruments) integrate the alternating 16% and 18% dots and the result is the average value - in this example 17% – our desired tone value.
Another way to look at it is: if we constrain our halftone cell to a pixel matrix of 16 x 16 pixels then we will always have 257 levels of grey in an area irrespective of how the dots within the cell are organized. However, if we build a tone area based on multiple halftone cells – a "supercell" we can get around the grey level limitations the formula would suggest.
As one example, the highest lpi on a 2400 dpi device that I'm aware of was 1697 lpi on a poster printed with plates imaged on a Creo CtP device in 2000 by Metropolitan Fine Printers in Vancouver Canada. It won a "They said it can't be done" award at GrapExpo in Chicago.
Supercell screening gets around the grey level limitations of the classic formula by looking at a tone area (the important criteria) rather than an individual dot. As a result, since about 1995 all AM screens from all vendors adopted variants of supercell screening technology:
Agfa - ABS - Agfa Balanced Screening
Heidelberg - HDS - High Definition Screening, and later IS screening
Harlequin - HPS - Harlequin Precision Screening
Creo/Kodak - Creosettes/Maxtone
Fuji - just since 2004 CoRes screening
etc.
As a result, 2400 dpi has become the defacto standard for imaging resolution in the commercial print industry. Higher resolutions, as far as halftone screening and grey levels is concerned, provides no additional value while imposing a penalty on imaging time.

Where the various vendors distinguish themselves with their individual implementation of supercell screening is how they deal with issues such as rosette drift - the gradual shift from clear centered rosette to dot centered rosette - over the width of the plate, single channel moiré, miniscus effects as dots first touch, e.g. at the 50% point, and other nuances of halftone screening.

Once you've passed the 200 lpi frequency - the human eye can no longer resolve the halftone structure at normal viewing distances. Beyond 200 lpi the argument can be made that there is no need to be constrained to the AM halftone structure. You might as well use an FM type screen. The lithographic issues will be the same since the imaging and press issues result from the size of halftone dots - not how they are organized.