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enter image description here

(Image source: https://www.thorlabs.de/newgrouppage9.cfm?objectgroup_id=5287)

In the Shack-Hartmann wavefront sensor, the local slope of the incident beam's wavefront is measured as a displacement of the spot positions, as shown in the figure above. Many documents and paper say the wavefront gradient is computed as the ratio of the displacement and the focal length of the microlens:

$$ \frac{\partial \phi(x,y)}{\partial y} \approx \frac{\Delta y}{f_{ML}} $$

This expression makes sense at first glance, but it occurred to me that the units of the left hand and right hand seem inconsistent. The wavefront is usually measured in the unit of angle (e.g. radians), while the infinitesimal displacement ($\partial y$), spot displacement ($\Delta y$), and focal length ($f_{ML}$) are in the unit of length (e.g. meters). That means,

$$\frac{\partial \phi(x,y)}{\partial y} \quad \text{[rad/meters]}$$

and

$$\frac{\Delta y}{f_{ML}} \quad \text{[meters/meters]=[rad]}$$

I think I misunderstood something. Does anyone tell me how to interpret this formula correctly in terms of the units?

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The displacement of the wavefront is a linear distance, often measured in waves or microns, and usually described as W(x,y). If measured in waves, you can convert that to a linear distance if you know the wavelength. The slope of the wavefront is just then dW(x,y)/dy for the slope in the y direction. It is assumed the displacement W(x,y) is measured with respect to some reference surface. This is often not explicitly stated - you need to know the setup to understand what the reference surface is.

For an object at infinity, as in the picture above, the reference wavefront is a plane.

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  • $\begingroup$ Thank you for the explanation. If I were to simulate the wavefront, how would I compute it in that unit? The wavefront is often modeled as the Zernike polynomial series, which gives the phase distortion in [rad]. Let's say, it is denoted as $Z(x,y)$ [rad]. How would I get $W(x,y)$ [waves], if I know the wavelength? $\endgroup$ Commented Dec 6, 2023 at 15:43
  • $\begingroup$ Zernikes don't have units by themselves, they are just a polynomial representation. If your wavefront data has units of radians, then yes, the Zerrnikes would be radians. But your wavefront data might have units of meters, in which case the Zernikes would be in meters. To answer your question, 2*pi of phase is one wavelength. So say the wavelength is 0.6 microns, then multiply by (0.6/(2*pi)) to convert from radians to a linear distance in microns. $\endgroup$ Commented Dec 7, 2023 at 16:32

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