The Minkwitz statement

[mauroventura]

Just 2 week ago , reading the U.S.A. patent for Progressive lenses, I read something about the a Minkwitz theorem for the determination of the residual astigmatism of a P.A.L. outside the far/near view and outside the channel between far/near view.
Is this correct ?
Is there someon who can give me some documentation concerning this Minkwitz statement?


thanks


Mauro Ventura

[Darryl Meister]

The umbilic represents the centerline of the progressive corridor—or the channel of relatively clear vision connecting the distance and near viewing zones. The curvature down the umbilic steadily increases in order to produce a progressive increase in mean power from a point producing the lowest mean power (in the distance zone) to a point producing the highest mean power (in the near zone). The vertical distance between these two points is referred to as the corridor length of the lens. The total change in mean power along the umbilic between these two locations represents the specified Add power.

While the horizontal, vertical, and oblique curvatures at any infinitely small point along the umbilic are equal, these curvatures start to differ away from the umbilic. The surface astigmatism, which is equal to the maximum difference between these curvatures at any point, rapidly increases as you move into the lateral blending regions of the lens. In the vicinity of the umbilic, this surface astigmatism actually increases at a very predictable rate away from the umbilic. Minkwitz showed that the rate of change in surface astigmatism (A) away from the umbilic grows at twice the rate of change in mean power (M) along the umbilic. Mathematically, this can be described as:

ΔA/Δx ≈ 2 × ΔM/Δy

Where (ΔA) is the change in astigmatism, (Δx) is the lateral distance away from the umbilic, (ΔM) is the change in mean power, and (Δy) is the vertical distance along the umbilic. If the horizontal distance (Δx) to a point (A) is equal to the vertical distance (Δy) to a point (M), we can assume that the change in surface astigmatism at point (A) is approximately equal to twice the change in mean power at point (M). (In reality, this theorem applies to the instantaneous rates of change over infinitely small distances, and only in regions of the lens relatively close to the umbilic.)

[QDO1]

One could assume that between lens designs ΔA/Δx differs, and in terms of lack of astigmatisim the lower ΔA/Δx is (at the fitting cross) the better the performance the distance portion of the lens. It would be interesting to know the values of ΔA/Δx at set points relative to the fitting cross, as it would be a nice mathematical model to replace the old - Hard/soft chestnut

[Darryl Meister]

Around the umbilic, ΔA/Δx depends on ΔM/Δy, which is the rate of change in mean power along umbilic. This means that change in astigmatism around the progressive corridor will depend upon how the lens designer chooses to ramp the power up along the umbilic. Also, ΔM/Δy depends upon both the Add power of the lens and the length of the progressive corridor; shorter corridors or higher Adds will require a greater ΔM/Δy and, hence a greater change in astigmatism). This is the optical compromise that must be dealt with in short-corridor and high-Add lens designs.
Minkwitz's theorem provides us with a few useful guidelines regarding the optics of a progressive lens in the vicinity of the umbilic (progressive corridor):

1. The rate of change in surface astigmatism around the umbilic is inversely proportional to the corridor length of the lens. As the corridor length becomes shorter, the unwanted astigmatism must grow more rapidly away from the umbilic.
2. The rate of change in surface astigmatism around the umbilic is proportional to the Add power of the lens. For a simple progressive lens design, this also means that the maximum level of unwanted astigmatism in the periphery of the lens will be proportional to the Add power.
3. Because of the first two points, the size of the progressive corridor will depend upon both the length of the corridor and the Add power. This means that lenses with shorter corridor lengths or higher Add powers will by necessity provide less intermediate vision and mid-range utility.

[Darryl Meister]

I have a more thorough discussion prepared on the optics of progressive lenses at: Optics of Progressive Lenses Course.

[QDO1]

Around the umbilic, ΔA/Δx depends on ΔM/Δy, which is the rate of change in mean power along umbilic. This means that change in astigmatism around the progressive corridor will depend upon how the lens designer chooses to ramp the power up along the umbilic. Also, ΔM/Δy depends upon both the Add power of the lens and the length of the progressive corridor; shorter corridors or higher Adds will require a greater ΔM/Δy and, hence a greater change in astigmatism). This is the optical compromise that must be dealt with in short-corridor and high-Add lens designs.
Minkwitz's theorem provides us with a few useful guidelines regarding the optics of a progressive lens in the vicinity of the umbilic (progressive corridor):

1. The rate of change in surface astigmatism around the umbilic is inversely proportional to the corridor length of the lens. As the corridor length becomes shorter, the unwanted astigmatism must grow more rapidly away from the umbilic.
2. The rate of change in surface astigmatism around the umbilic is proportional to the Add power of the lens. For a simple progressive lens design, this also means that the maximum level of unwanted astigmatism in the periphery of the lens will be proportional to the Add power.
3. Because of the first two points, the size of the progressive corridor will depend upon both the length of the corridor and the Add power. This means that lenses with shorter corridor lengths or higher Add powers will by necessity provide less intermediate vision and mid-range utility.

how does this relate to the way a designer makes the progeresion wider. the 3 assumptions above seem to imply that inter'/near width is determined by add power and length alone. Clearly designs like the Technica break the rule?

[Darryl Meister]

Clearly designs like the Technica break the rule?

Not really. Technica has a very long corridor. Also, the power begins to stabilize rather slowly, almost immediately, which allows you to widen the viewing zones more.