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In my efforts to characterise a 0.67 NA microscope objective (working distance of about 15mm, effective focal length of 25mm), I have placed a 20 micron precision pinhole at the focal plane of the objective, and back illuminated the pinhole with a 767nm laser. The light coming out of the objective is focused onto the imaging plane (camera plane) with a F=1 metre lens.

By my understanding, the image field $U_i$ should be a convolution of the Point Spread Function (PSF) $h$ and the object field $U_o$, i.e.

$$ U_i(x_i,y_i) \propto \int_{-\infty}^\infty\int_{-\infty}^\infty h(x_i-\xi, y_i-\eta) U_o(\xi,\eta)\, d\xi d\eta\, , $$

where the proportional sign is for good measure as there are some coefficients in front, but that does not affect the profile. The image intensity profile is given by

$$ I_i(x_i, y_i) = |U_i(x_i, y_i)|^2\,. $$

In my case the microscope objective has a circular aperture, and that means that the PSF $h$ is given by the airy function, i.e.,

$$ h(x,y) = \frac{J_1 \left( \frac{2\pi}{\lambda} NA \sqrt{x^2 + y^2}\right)}{\sqrt{x^2 + y^2}} \, , $$

where $J_1$ is the Bessel function of the first kind, order one.

I have plotted out the intensity profile (gauss quad integral) that I expect for the current pinhole and wavelength, taking into account the magnification of the system, and I got the following plot.

enter image description here

However, this differs from my measurements taken in experiment:

enter image description here

The size of the image is approximately 20 microns (pixel size = 3.75/40 microns). The number of peaks somewhat represent what is shown from the plot above. What concerns me more is how the intensity profiles goes to zero between peaks, unlike what the theory predicted.

Is my understanding of the current theory incorrect?

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The fact that you have a dark center in your image indicates that you are not in focus. That's one problem. The diffraction pattern of a circular aperture is the well-known Airy disk, which has a bright center. I'm not sure how you came up with the intensity profile in your graph.

Another issues is that you don't know the spatial quality of your source laser. I would recommend expanding that beam with a lens and then refocussing to a spot at least twice the diameter of your pinhole. This will provide a light source (from the perspective of the pinhole output) that is as close to a pure spherical wavefront as possible. Oops, and as close to uniform spatial intensity as possible.

addendum

Somewhere in there you need to convolve the Optical Transfer Function of every lens in the system. It's certainly possible that one or another lens (and its NA as well) is degrading the pure Airy pattern. But I would certainly start by adjusting focus until the center is bright.

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