Intro to Fourier Optics and the 4F correlator

hey everyone recently someone at work said that plain old lenses actually take the Fourier transform of the image that you put into them so at first it seems pretty weird how does a plain old piece of glass like this perform of complex mathematical operations such as a Fourier transform so I looked into it and let me show you what I found out most engineers are probably more familiar with Fourier transforms as they apply to temporal waveforms so for example if you're looking at a sine wave on your oscilloscope you might use the FFT function to see what it looks like in the frequency domain and you'll see for a sine wave you'll see a nice spike at the fundamental and for a square wave you'll see all the harmonics and for a real-world noisy waveform you'll see the noise in between all the fundamentals and harmonics there in cases like these we aren't really looking at phase information so this Fourier transform is only the magnitude so if you shifted this way of over a little bit you'd have the same exact output here because this is not showing us any phase information it's just the magnitude of the component waves that are needed to create this the same concept can be applied to things in the spatial domain so let's say we had an image that looks like this it's basically a sine wave of intensity across the x axis so every scanline is the same and they all have this sort of intensity pattern so if we were to it's possible to use the same mathematics to apply a Fourier transform to this so if we do that what you end up with is an output image with a dot here and a dot here so what's happening is is the x axis is showing us the frequencies just like in the first plot here is showing us a spike where this fundamental frequency lives and it's it's doubled across the y axis because that's how the math works out there is an excellent tutorial that I'll put a link to in the description that describes this in much more detail and does a really good job so check that out if you're interested in image trance forming using Fourier transforms similarly if we were to take the Fourier transform of this image we would have the fundamental being really bright two little dots there and then a slight decreasing in intensity more and more dots going off the x-axis and so this is again the same as this the you have decreasing intensities of the harmonics and that creates the square wave so this works in the other axis too if we were to rotate this 90 degrees and we had a square wave in the Y direction then we would just have the pattern in the Y direction here so every point on this two-dimensional plane represents a spatial frequency in the original image if you take the Fourier transform of a normal-looking picture you know a picture a photograph of a house or whatever you'll end up with something that looks like this there'll be a really really bright spot at the center and then there's just kind of a whole bunch of random looking noise around the outside and the reason for that is that pictures are very complex spatially and so if you think about all the frequencies you'd have to add together all the different spatial frequencies to come up with something that looks like a real photo you'd realize how much information is actually in the amplitude plot and again this does not include phase information just like talking about over here we're only discussing the frequency components so if you take the Fourier transform of this image and only look at the magnitude plot you actually cannot reconstruct the image fully because only only have the frequency information one of the interesting side effects that have having only a frequency information in the amplitude plot is that the location of the feature in the image doesn't matter so definitely check out that tutorial but basically taking the FFT and looking at only the magnitude the frequency output an input image looking like this would give exactly the same output as an image looking like this because the feature is the same has the same dimensions but it's at a different location in the image so this will become important later okay so now that we have a basic idea of what a fourier transform is in an image how in the world does a plain piece of glass actually make it happen there's a few important catches that I ran into so if you're gonna try this yourself definitely check this out if you search around on the internet you'll you'll quickly run into something called a 4f correlator which is sort of the quintessential fourier optics device and this does work I'll show you later I actually did get some results out of it but there's a lot of caches so saying oh well lens takes the Fourier transform is true but that you can't really make use of it except in some very limited cases so one problem is that you get the phase information out as well as the amplitude magnet information so it's in that tutorial we'll see that the phase images that you get out of a Fourier transform are really messy and it makes it such that you can't really extract anything meaningful out of it because it's just so wild I mean you can't I mean mathematically you can do things with it but if you just sort of look at it on a screen it doesn't really tell you anything so the 4f correlator is set up to only show amplitude information and it does this by using a laser which is a coherent light source so let me show you how it's set up I use the helium neon laser and then hot-glued a microscope objective to the front of it and looked at a screen while I was setting it up just to get it in exactly the right spot and then you you send the output of that through a pinhole now the idea this is called a spatial filter like if you go online and search for this stuff they'll be talking about spatial filters but really all that is it's just a pinhole and it's situated such that it's at the focal point of the microscope objective there's a chart that shows the optimum pinhole size for a given microscope objective and input beam and I which I didn't have Edmund said I actually had to get down to about 5 or 10 microns by and have a hole that small I did have a 30 micron aperture that I used with my son-- you can see the effect that the pinhole had without it there's quite a lot of spatial noise in the beam it's just not very clean and with the pinhole we have a nice smooth Gaussian distribution out there which just means that the beam is bright at the center and has a nice smooth taper out to the edges so it's really just an ideal sort of source of light next this first lens is used just to collimate the beam so the light rays are coming in at an angle and are hopefully coming out straight and I tested this pretty simply just by using a pair of calipers and making a couple marks on a projection screen and then setting up the projection screen very far away and then holding the calipers in the beam out here so if the projection if there were no optics in here and I had the projection screen way out here and I put the calipers here and knew that the distance was the same on the screen as between the caliper jaws I could move this around a bit until it was collimated because we know that the light is going perfectly straight the distance between the jaws would be the same as the distance marked on the screen this distance is not too important in the system it's because the light is collimated the rest of it is really just two lenses in a projection screen and for most of the work that I was doing I didn't even really need this part you can you can put a screen here too so what happens is you put your your input image here this is just a plain old lens F is the focal length of the lens at this plane in space you're supposed to get the Fourier transform what you do but I'll talk about that in a minute and then another F and other focal lengths there's another lens and another focal length there's the output screen so this is an image plane this is an image plane and this is the Fourier image plane since we're dealing with monochromatic light from the laser you can't really just put a photograph here unfortunately the image has to be a clear thing that just blocks out light where you don't want it and so I got a transparency just printed some stuff on it I also used this because this is actually quite opaque and it has nice sharp edges on there this is just a lid to a box of optics I also tried things like combs like this I also tried one of the best objects I tried was this very fine copper mesh now this is sort of cheating because you can actually see the diffraction pattern just looking through this light source but anyway I'll talk about some of the resulting images they got in a minute so here's how this thing works if you had nothing in the image plane let's just say it was a clear shot from this collimating optic into this first optic here all the photons are going at 0 degrees let's just say all parallel and every photon that goes into this lens it should be focused down to exactly the same point so if you put a screen here with nothing in the between these two you'll get one really really sharp bright point right at the center now if we put something in the image plane like let's say a a letter a where the light hits the edge of this pattern there will be some diffraction and the diffraction will cause the light to slightly diverge so instead of going straight out where the light hits an edge of a feature is going to be a slight divergence in angle and that divergence and angle is going to show up as something not on the spot here so any sort of a interference that you put here is going to show up as a deviation from that from that focal point so in essence all this lens is doing is focusing down the diffraction pattern from here on to a screen it's kind of annoying it's almost a little frustrating to get down to it at this simple level like for example if you go to the Wikipedia article on FOIA a optics or 4f correlator you know it's incredibly complex I mean there's just tons of equations and very little explanatory text that would actually make this you know understandable but really all that's happening is you're just focusing a diffraction pattern there's a couple of really cool tricks that you can do with this for F correlator once it's working unfortunately better than what I was able to get working tonight one of them is that you can do some primitive image processing with this so for example if you put your input image here and you put some kind of a filter here the output image what will be affected by the filter that you put here so if we put a filter that blocked out everything except the center of the of the image let's say everything outside the circle was blocked and everything in the middle is was okay then what we would have is low frequency components only getting through so remember what this means the farther are you a wave the farther away you are from the center the higher the frequency so this means no high spatial frequencies get through so it's sort of a it's a blurring filter basically another cool thing is that you can put an image here and then put a the Fourier transform of your of your target image here so let's say you're searching for a feature inside an image what you could do is put the Fourier transform of your desired image here and then kind of sort through a bunch of photographs like well not photographs unfortunately they have to be you know clear black on clear you put that here the output will change when when there's a match between your target and your source remember also that since we're in the frequency domain it doesn't really matter where in the image this feature is the rotation matters so what you have to do is actually rotate around your your target filter and keep monitoring the output to see if there's a match there scale also matters so I'm not quite sure how that's handled but you can do a really simple kind of image processing image search stuff in optics without any computers are digitizing or anything so that's pretty cool anyway so I think basically my problem is that at the lenses I have just aren't quite good enough to get a decent image I got some some halfway decent results with the copper mesh and and and sort of saw something with with the black on clear images but it's gonna need more work so I'll do a follow-up video sometime and let you know if I get anything decent okay see you next time bye

Channel: Applied Science

Views: 137,543

Keywords: fourier optics, 4f correlator

Id: wcRB3TWIAXE

Length: 13min 32sec (812 seconds)

Published: Mon Nov 12 2012