Intro to Fourier Optics and the 4F correlator--- make money online

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 a complex

mathematical operation 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 wave over a little bit you would

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 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 of

having only 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 ok so now that we

have a big

the 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 going to

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 sent I actually had to get down

to about 5 or 10 microns but I don't

have a hole that small I did have a 30

micron aperture that I used with my cell

you can see the effect of 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 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 which you do

but I'll talk about that in a minute and

then another F and other focal lengths

there's another lens and in 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 want 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 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 onto 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 Fry

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 is blocked and everything in

the middle 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

can 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

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

the black on clear images but it's going

to 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

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