# MICROSCOPY: Imaging a field of radiated light and it's source

1. Dec 9, 2012

### einfall

I have a question about observing a field of radiated light (and it's source) through a microscope, specifically vis magnification and the scale of the final image observed (the field, and also the source).

I have a slice of field intensity at the plane of a microscope's numerical aperture (i.e. all light entering the appropriate half-cone angle of 20 degrees). It's an accurate map of the intensity — it looks as expected.

The objective is 75x, and the aperture sits about 1mm from a light source (a particle) with well-known dimensions.

Since the particle sits 1mm below the plane of the field I've calculated, how do I apply the 75x magnification to both the field and the particle to reform an image with appropriate dimensions (i.e. what I see when I look through the microscope)?

The most obvious approach to me is to take the objective's focal-length, and then ray trace the field and the particle independently to the same image observation-point. Then put them back together. But this isn't giving the result I expected (particle location/dimensions don't make sense relative to the field).

A reminder of the basic optics here would be highly appreciated, as this has me irritatingly confused.

2. Dec 9, 2012

### einfall

*its

3. Dec 10, 2012

### Andy Resnick

I'm confused by your question. Specifically:

1) What do you mean by 'I have a slice of field intensity at the plane of a microscope's numerical aperture'? The numerical aperture is a measure of angle, not a plane.
2) What do you mean by 'the aperture sits about 1mm from a light source'? Which aperture?
3) What do you mean by 'how do I apply the 75x magnification to both the field and the particle to reform an image with appropriate dimensions'? Why are you distinguishing between the object (the particle) and 'the field' (whatever that means...)
4) Is this a compound or simple magnifier- a compound microscope has a secondary magnifier either/both in the eyepiece and in the main optical path (e.g. Zeiss Optivar)
5) What do you mean by 'ray trace the field and the particle independently to the same image observation-point'? Again, why are you separating the object and the emitted field?

4. Dec 10, 2012

### einfall

Thanks very much for your response - you're right, my previous question was quite unclear.

I have a micron-scale particle under lamp illumination which is also scattering laser light. I want to compute what the scattered field and particle "look like" with a specific numerical aperture under 75x magnification.

I can calculate the scattered field for any point/plane in space, so my current solution is to compute the field for a plane sitting tangentially on the particle.

I'm confusing myself because the lamp light and the scattered light both need to expand up through the solid-cone angle defined by the numerical aperture, but the scattered field is radially dependent and diverges while the lamp light is effectively collimated over this length-scale (at least I think this follows). How to go about replicating what would be "seen" on a CCD camera is the problem that's bugging me, especially because it seems like it ought to be the straightforward bit.

I know this doesn't specifically address all your questions, but I hope it clarifies the issue.

Also note: right now I'm completely ignoring the fact that the field is viewed through an aperture about 1mm above the particle, and for now I'm ignoring diffraction effects (nice Airy rings etc.)

5. Dec 10, 2012

### Andy Resnick

This is (somewhat) straightforward, if the object is illuminated with a plane wave and is significantly larger than a wavelength. In the Fraunhofer regime, the far-field is the Fourier Transform of the object. Perform the FT, truncate the spatial frequency as per the numerical aperture, and when you inverse FT (to create an image), scale as per the magnification. If the object is out of focus or there are other aberrations you must account for, when performing the forward FT you must multiply the field by the corresponding phase factor (i.e. for defocus it's exp(ikW), where W = er^2/2, where e is the residual defocus from 1/d_o + 1/d_i - 1/f = e and r is the radial coordinate of the aperture.) Goodman's 'Introduction to Fourier Optics' is the standard resource.

Don't forget to square the modulus of the final result to get something related to detected intensity. If the CCD is monochrome, the effect of sampling can be handled reasonably easily (Vollmerhausen and Driggers, 'Analysis of Sampled Imaging Systems' is an excellent resource). If the CCD is color and has a Bayer filter, then you have to take that into account.

If the object is not illuminated by a plane wave- meaning the paraxial approximation is violated- life gets difficult. An alternative approach is to use Mie scattering (http://www.t-matrix.de/ has a lot of free codes), if the object size is comparable to the illumination wavelength- but again, simply compute the far-field diffraction pattern and perform an inverse FT (after truncating the diffraction pattern as per numerical aperture).

6. Dec 12, 2012

### einfall

Andy,

Sorry for the delay responding, your response is very much appreciated and I have Goodman's Fourier Optics sitting on my desk now as a result.

I'll be even less-vague now (!) — the micron-scale particle is illuminated by a microscope lamp, and also by a 640nm diode laser. The problem I've developed a solution for is a brand of Lorentz-Mie scattering. I can compute the intensity of the scattered field at any point in space exterior to the particle with no issues.

So what I think I need to do now as-per your description (and Goodman) is:

1. Compute a 2-D grid of scattered intensity points on a plane sitting directly above the particle
2. Apply the appropriate Fourier transform to the grid of points
3. Truncate in the frequency domain as per the numerical aperture (a little unclear on this point, but I will peruse Goodman carefully)
4. Invert

All I really care about is how the field of scattered light computed via Mie theory looks. Modelling the appearance of the particle under lamp light is not important except for scale and as a sanity check that the scattered field looks sensible.

Since I want to assume perfect focus with no aberrations for now, it seems like the only "tricky" bit should be step 1 (which appears to be working fine, fingers crossed)?

Thanks again.