# Relay Imaging Problem

1. May 26, 2014

### test1234

Hi there, I've been trying to solve this problem regarding relay imaging so that I can modify my experimental setup accordingly. Please refer to the attached jpg file for the lens setup.

I am trying to image the object (the purple block in the attached jpg file) at a specific imaging plane 370mm after the final lens in my setup. The constraints in my setup include, the object has to be exactly at the focal point between the two f=200mm lenses and I only have a limited distance between the second f=200mm and f=306.8mm lenses to play with.

I tried applying the thin lens equation ($\frac{1}{s_0}+\frac{1}{s_i}=\frac{1}{f}$) twice as follows:

Starting from the object,
$s_0=200mm \\ f=200mm \\ s_i=∞ \\$

Since it is expected that the first intermediate image would be at infinity, I worked with slightly off values of $s_0=199mm$ and $201mm$ instead.
$s_i= -39800mm$ or $+40200mm\\$

Then the image of the first lens become the object of the second lens (f=306.8mm).
Again I applied the thin lens equation with
$s_0 = (1100+39800)mm$ or $(1100-40200)mm\\$
$= 40900mm$ or $-39100mm \\$

$s_i= 309.12mm$ or $304.41mm\\$

Hence, this current setup doesn't give me my intended imaging plane at 370mm. I was wondering if anyone could give some ideas regarding what I can do get the image at 370mm without adjusting the object distance from the f=200mm lens?

I tried making the distance between the second f=200mm and f=306.8mm a variable and set the final image distance to 370mm, but it resulted in a huge value of around 40m!

Would really appreciate if anyone could help please. Thanks!

#### Attached Files:

• ###### Lens system.JPG
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2. May 26, 2014

### UltrafastPED

The standard advice is to use matrix methods for "complex" systems. If you are using Gaussian beams use the Gaussian matrix method; otherwise use the simpler ray matrix method. Then you simply use the "image condition" as the constraint for the final values obtained, while using arbitrary distances between the lenses.

If you are unfamiliar with the matrix methods just do ray tracing - it will be close enough to the Gaussian when you make the adjustments during the actual setup.

You don't have the actual lenses, right? If you do, you can just fiddle ... once you have a workable design the rest is all in proper alignment and slight adjustments of the distances. Though I agree that you should calculate prior to buying the lenses.

I've used a similar setup for monitoring a thin object which was the target for a laser experiment. The collimated beam from the left is focused by the first lens onto the object (a TEM grid with some stuff on it), which diffracts the light; the diffracted light is originates at the focal point of the second lens - which collects the bits of the diffraction pattern, and in turn focuses it upon the detectors of a CCD camera (lens removed from the CCD camera as the image forms on the plane of the detector).

This two-lens system with collimated light entering from the left, and the image plane on the right, is called a 4F optical system. The object being imaged must be (a) quite small - the size of the focal spot; (b) be transparent or (c) have lots of holes - otherwise the second lens does not receive the diffraction spots.

4F setups don't require the final lens unless you need to project the image elsewhere.

BTW your drawing is incorrect; the rays from the second lens will cross at 200 mm to the right, and then expand as they approach the third lens.

PS: Also see http://www.livephysics.com/tools/optics-tools/lens-system-tool-image-distance-magnification/

Last edited: May 26, 2014
3. May 26, 2014

### test1234

Thanks UltrafastPED! =) Just a quick clarification, for the Gaussian matrix method, is it right to say that I would need to start from the leftmost collimated output and propagate it through the two focusing lenses (4f setup) before finally reaching the focusing lens at the other end?

4. May 26, 2014