Magnifying diffraction patterns and real images

[Philip Koeck]

I've been playing around with geometric optics a bit and started wondering whether it's possible to magnify both the real image and the diffraction pattern with the same lens setup.
With a single lens both become larger when the focal length is increased (as along as it stays below the object distance).
For a series of lenses I find that I can set up the lenses to produce a series of larger and larger real images, which leads to smaller and smaller diffraction patterns, or I can do the opposite.
Does anybody know if it's possible to magnify both the image and the diffraction pattern with the same sequence of lenses?

 

[tech99]

As far as I can see, the ratio of diffraction size to image size is dictated by the size of the lens, as described by Lord Rayleigh. I don't think that another lens can then retrieve additional information.

 

[Philip Koeck]

As far as I can see, the ratio of diffraction size to image size is dictated by the size of the lens, as described by Lord Rayleigh. I don't think that another lens can then retrieve additional information.

I'm not trying to retrieve more information and my question is not about a resolution limit so the diameter of the lenses should not be relevant.

I'm just wondering if there is a sequence of lenses that magnifies both the real image and the diffraction pattern.
Is it possible that the real image produced by the last lens is bigger than the object and at the same time the diffraction pattern produced by the last lens is bigger than the diffraction pattern produced by the first lens?

To clarify: When I say the diffraction pattern is bigger I don't mean it goes to higher resolution. I really just mean it's physically bigger, magnified.

 

[Drakkith]

What do you mean by 'diffraction patterns'? Do you have two different sources here?

 

[Philip Koeck]

What do you mean by 'diffraction patterns'? Do you have two different sources here?

Just one source, but the specimen diffracts the light so you do get beams in different directions.
The specimen could be a double slit or a grating or just anything, periodic or not.

The setup is this:
Parallel light (a laser beam spread out and made parallel again) hits the specimen (a transparency) from the left.
A lens is placed to the right of the specimen.
In the back focal plane of the lens you get the diffraction pattern.
If the object distance (o) is larger than the focal length you also get a real image to the right of the lens.
Here's a picture of this (but I don't want an aperture stop in the back focal plane):
https://www.researchgate.net/profile/Johannes-Stoerkle/publication/334683656/figure/fig25/AS:784604122337281@1564075608785/The-Abbe-theory-of-image-formation-which-is-based-on-Fourier-optics.png

Note: If the light is not parallel to start with things change a bit, but not dramatically, for example the diffraction pattern will not be located in the back focal plane.

Now I can use further lenses to the right to magnify the real image or the diffraction pattern step-wise.
This is more practical than using a single lens since I can get a large magnification without having a large image distance (i). (For a single lens the magnification is given by i / o.)

The question is: Is there a lens sequence which would magnify both the image and the diffraction pattern?

 

[Drakkith]

The question is: Is there a lens sequence which would magnify both the image and the diffraction pattern?

Tough to say. Unfortunately I never finished my degree in optical engineering so I doubt I'll be able to help you much.

 

[hutchphd]

If I understand your question, I believe the answer is no. The fourrier plane is effectively imaging the light source at left infinity. The image plane is imaging the grating. You are asking for telescope that will simultaneously focus at infinity and nearby.

 

[Philip Koeck]

If I understand your question, I believe the answer is no. The fourrier plane is effectively imaging the light source at left infinity. The image plane is imaging the grating. You are asking for telescope that will simultaneously focus at infinity and nearby.

The final image and diffraction pattern to the right of the sequence of lenses would be in different planes. Does that change anything?

PS: I also think the answer is no.

 

[hutchphd]

How would you access (detect) them at different planes?...use a beam splitter? Maybe (?) it could be done but you get half the light and a lot more stuff to align. Is this a thought exercise or are you contemplating something specific? I've never encountered this idea before.

 

[Philip Koeck]

How would you access (detect) them at different planes?...use a beam splitter? Maybe (?) it could be done but you get half the light and a lot more stuff to align. Is this a thought exercise or are you contemplating something specific? I've never encountered this idea before.

I don't need to actually show both planes on a screen or something.

Here's the background: In phase contrast microscopy (specifically I'm thinking of transmission electron microscopy of unstained specimens, so called weak phase objects) you can modify the diffraction pattern in order the increase the contrast you get in the real image.
For example you can insert apertures to remove parts of the diffraction pattern to give dark field and single side band contrast or you can use phase plates to phase shift some of the "beams" in the diffraction pattern. (Sorry about the weird mixture of concepts from wave and ray optics.)
The principles are the same for photons and electrons (and other waves) so you can think of a light optical system.
It's easier to modify a magnified diffraction pattern, but then you also want a strongly magnified real image on the detector or screen, which means that you have to add some extra magnification steps after you've modified the diffraction pattern and the microscope gets impractically long.
The problem is, it seems, that I first have to magnify the diffraction pattern, which has the side effect of a tiny real image, and then I have to magnify the real image with additional projection lenses.
If I could magnify both the image and the diffraction pattern with the same system of lenses the microscope could be shorter.

Note: With electrons only real images are of interest since you can't look into an electron beam and you need some sort of detector or fluorescent screen to make the image visible.

 

[hutchphd]

Thanks for the very nice and clear explanation. I have used a phase contrast optical microscopes but have only played with electron optics in undergraduate labs (very long ago) . There are a few special things that electrons can be coaxed to do but I think probably optics are optics as you say.
I clearly need to give this more thought. I will give it a try. Good Luck.

 

[Philip Koeck]

Thanks for the very nice and clear explanation. I have used a phase contrast optical microscopes but have only played with electron optics in undergraduate labs (very long ago) . There are a few special things that electrons can be coaxed to do but I think probably optics are optics as you say.
I clearly need to give this more thought. I will give it a try. Good Luck.

Thanks!
There might be some theorem saying that in any lens-system the real image and diffraction pattern after each lens are always inversely proportional in size, a bit like the mathematical property of the Fourier transform, but I haven't found anything like that.
It certainly looks like that, however.

 

[vanhees71]

Of course it depends on the geometrical setup of your obstacle, the lense, and the photo plate whether you see an interference pattern or a more or less sharp picture of the obstacle (assuming you use your coherent parallel-beam laser light).

If you put your photo plate in the focal plane all beams coming from a certain direction are mapped into one point, i.e., you map the direction of your wave vector to a point on the photoplate. The intensity at each point of the photo plate is determined by coherent superposition of the partial beams coming from the first slit and thus you see a Fraunhofer interference pattern with perfect contrast.

If you put the photo plate at the position which fulfills the lens formula, then you'll get a sharp image of the obstacle since in this cage each point of the obstacle is mapped to a certain point at the photo plate. The you don't have an interference pattern at all.

 

[Philip Koeck]

Of course it depends on the geometrical setup of your obstacle, the lense, and the photo plate whether you see an interference pattern or a more or less sharp picture of the obstacle (assuming you use your coherent parallel-beam laser light).

If you put your photo plate in the focal plane all beams coming from a certain direction are mapped into one point, i.e., you map the direction of your wave vector to a point on the photoplate. The intensity at each point of the photo plate is determined by coherent superposition of the partial beams coming from the first slit and thus you see a Fraunhofer interference pattern with perfect contrast.

If you put the photo plate at the position which fulfills the lens formula, then you'll get a sharp image of the obstacle since in this cage each point of the obstacle is mapped to a certain point at the photo plate. The you don't have an interference pattern at all.

I completely agree with everything apart from the last sentence.
A real image of a phase object is also due to interference, but interference between the diffracted waves.
That's why you can make pure phase objects visible by introducing (using a phase plate) a relative phase difference between the forward beam/wave and the waves diffracted into angles larger than zero.