How can Fourier Optics be used to analyze and explain experimental physics?

[JamesJames]

I've been reading some stuff on Fourier Optics (Johnson, Optics) and have become quite interested in how Fourier transforms can be used to explain so much. But I have stumbled on something thta is confusing me.

Two students perform a "Fourier Optics" experiment to explore Fourier transform analysis in relation to Experimental Physics. The He-Ne laser used in the experiment is focused onto a 10 micrometer pinhole with adjustable micrometer screws. Three identical lenses are provided inorder to realize 4f-focussing. They all have a focal length of 350 mm. The first lens acts as a condenser to provide a parallel uniformly illuminated beam. The second lens produces a Fourier image in the focal plane (where low pass filtering can be applied using an aperture). The third lens is used to convert the modified Fourier image again into a normal filtered inverted image.

A grid shaped ruling was placed between the condenser and imaging lens. The image was to be observed in the Fourier plane as well as in the image plane. Slits of different widths were placed in the focal plane to cut out high Fourier components first in the horizontal and then in the vertical planes.

Here’ s my question: Would it be possible to completely remove one set of lines and if the slit was rotated in the Fourier plane by 45 degrees what would be observed?

Here’ s the way I see it: A slit of certain width can used to block higher frequency components. The slit could be aligned with a vertical axis, and completely shuts out frequency components along another axis. The produced image in the Image plane would consist of only horizontal lines, since the diffractive pattern of Ronchi rulings is perpendicular to the plane of the rulings.

Next, the students inserted an image slide into the holder and a low pass filter was used to remove the graininess. Here’s what I think: I have read and understood that a low-pass filtered image would be blurred, but preserves the low frequency broad smooth regions of dark and bright and losing the sharp contours and crisp edges. Mathematically, low-pass filtering is equivalent to an optical blurring function.

How could mathematical analysis (just a broad description of the procedure) be used to explain the recorded images?

James

 

[JamesJames]

If someone could tell me why my analysis is wrong it would help a great deal.

 

[JamesJames]

Anybody? Please, it would help my understanding of the topic.

 

[JamesJames]

Guys, I have understood the thing about the lines and 45 degree angle but am confused about only the last part and would REALLY appreciate some feedback.

My initial explanation of the low pass filter thing is wrong. From what I understand regarding the image slide, the spatial frequency spectrum of this image consists of a low-frequency component representing the picture information and a high-frequency component generated by the grid. Blocking the high frequencies using a low pass filter resulted in us getting a smoother image and the discontinuous nature of the original was not observed. What can I say mathematically about this? Is there some way to elaborate on this point using Fourier transform analysis? Or something mathematical?

 

[MAAAAHK]

I've been utilizing Fouriers for a while now. Basically, the key is to import several togglers at once. It's pretty simple to juggle the reciprocating results. The Fraunhoefer diffraction typically splits them image frame, hence the need for at least three opt toggles.

 

[JamesJames]

Thanks for the suggestion but I think it is at a higher level than my understanding. Is there a simpler mathematical formulation? Or are togglers something not too complicated?

 

[JamesJames]

Regarding the grid(mesh thing) would it be correct to say that a slit of certain width (to block higher frequency components) aligned with a vertical (y-axis) completely removes frequency components along z-axis. The resulting image produced in the image plane would consist of only horizontal lines, since the diffractive pattern of Ronchi rulings is oriented in a perpendicular direction to the plane of the rulings.

Can someone confirm this for me?

 

[JamesJames]

Anything mathematical would be of great help for the low pass filter and for the above post, it does not need to the be mathematical.