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# Spherical aberration ~ defocus balancing (M solve?)

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Hi Zemax Community,

I am reading the book, introduction to lens design with practical zemax examples.

And in the chapter 11.6, it writes

“ We now combine spherical aberration and defocus mathematically, so that we can find the position of the minimum blur(MB) spot

W = W040*ρ^4 + Wd*ρ^2

this occurs when

Wd = -3/2* W040 “

I just wondered

1. Why Wd = -3/2*W040 makes the MB minimum ,how to get the -3/2?
2. is zemax using this calculation in M solve?

Best regards

Yang

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Best answer by Akhil Dutt Vijayakumar 14 February 2024, 10:50

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Hi,

Thanks for posting in the forum.

The equation you provided is a mathematical representation of the wavefront aberration in terms of spherical aberration (W040) and defocus (Wd) terms.

The expression W = W040ρ^4 + Wdρ^2 describes the total wavefront aberration as a function of the radial distance (ρ) from the optical axis.

Here,

W: Represents the total wavefront aberration at a distance ρ from the optical axis.

W040: Represents the 4th-order spherical aberration coefficient.

ρ: Represents the radial distance from the optical axis.

Wd: Represents the defocus aberration coefficient.

To find this minimum, you typically take the derivative of the wavefront aberration equation with respect to ρ and set it equal to zero. This will give you the radial position where the aberration is minimized.

dW/dρ = 4W040ρ^3 + 2Wdρ = 0

For a non-trivial solution (ρ ≠ 0), we can divide both sides by ρ^2:

4W040 + 2Wd/ρ = 0

Multiplying both sides by ρ gives:

4W040ρ + 2*Wd = 0

And then Wd = -2W040p

Thus the minimum blur spot appears when Wd = -2W040p (Exact value)

The minimum value can be approximated to -3/2(W040p) as it adds some room for other aberrations as well. Also the term -3/2(W040p) can act as a starting value for the optimization and can do the iteration to reach the exact value.

1. There are more information on the Solve Types in the Help File: The Setup Tab > Editors Group (Setup Tab) > Lens Data Editor > Solve Types (lens data editor) > Thickness Solves

Hope that helps.

Akhil