SPIM Optics 101/Theoretical basics

Geometric Optics
''This chapter will assemble some simple optical principles that should help you to align a SPIM. Most of these rules are approximations, that are typically well fulfilled for small angles. If you want a more thorough introuction, look into any optics textbook.''

Mirrors
Mirrors are shiny surfaces that reflect a beam of light. The reflection is defined by the reflection law, which states the angle of the incident and the surface normal &alpha;i equals the angle &alpha;e of the exiting beam and the surface normal. This situation is illustrated in the next figure.

Lenses and parallel light
Lenses are curved glass surfaces that refract the light and focus a beam of parallel light onto a spot, called the focus F. The distance between the lens and the focus is called focal length f. If the parallel light beam enters the lens under an angle &phi;, as in the image below, the focus F will be shifted perpendcular to the optical axis by a distance &Delta;x (Note: A shift on a flat plane is the ideal case, which is nearly true for small shifts. For real lenses and larger shifts the shifted focus will move on a sphere centered in the center of the lens). The shift is given by a simple relation:
 * &Delta;x = f&middot;tan(&phi;)

Another basic rule of optics states that:
 * Every beam path in an optical setup is reversible.

This means, e.g. for a lens, that it may either focus parallel light into a single spot, or alternatively collimate the light from a point source, which is put into the focus. In the latter case the situation above is simply reversed (i.e. flip the image horicontally): Shifting the lightsource then also changes the angle &phi; under which the light leaves the lens.

In terms of fourier optics, this finding makes a lens a fourier transformator, as it changes from a position (space) to an angle space, or back.

Imaging with a lens
In the section above, we looked at what a lens does to parallel light, or to point sources in the focal plane. These two cases are the most important ones, when it comes to understanding the optics of openSPIM. But for the sake of completeness, we will also give the "standard sketch" of imaging with a lens: Here an object of height G is positioned a distance g away from a lens (focal length ff in front of the lens. For the size of the image and the distances holds the lens formula (an approximation for thin lenses!):
 *  1/f = 1/G + 1/H

The image is magnified by a factor M:
 *  M = h/g = H/G = f/(f-G)

Telescopes
Now we combine two lenses with focal lengths f anf f' to form a telescope. A telescope is an optical assembly, where the the lenses are placed a distance f+f' apart. It can be used to expand or decrease the diameter of a beam of light. As can be seen, the size of the incident parallel light beam is expanded by a factor M (magnification):
 * M = D' / D = f' / f 

This can be understood, using the principle of reversability of optical beams (see above). The focus of the first lens can be seen as a point source for the second lens. The magnification is simply given by geometry.

To calculate the exit angle &phi;' for a given incident angle &phi;, we can use the tangent law above:
 * &Delta;x=f&middot;tan(&phi;)=f'&middot;tan(&phi;')

and therefore:
 * tan(&phi;')/tan(&phi;) = f/f' = 1/M 

Infinity Microscope
Now we take a second look at an assembly of two lenses, now forming an infinity microscope (which we use in the openSPIM to image the specimen onto the camera). Most of todays scientific grade fluorescence microscopes use this "infinity optics" principle: The whole setup looks like this: The magnification M=x'/x can again be easily calculated using the tangent law for lenses, stated above. From
 * 1) The fluorescing sample is placed in the focal plane of the objective lens. Each point (fluorophore) in the sample can then be seen as a pointsource of light. The objective lens then converts the light from each of these points (at position x) into a parallel beam of light, exiting the objective under different angles &phi;(x).
 * 2) A second lens (called tube lens) then converts these parallel beams back into an image in its focal plane, where the camera is placed. Alteratively an eyepiece can be used as a loupe to observe this image with the eye.
 * x = fO&middot;tan(&phi;(x)) and x'= fTL&middot;tan(&phi;(x))

follows that:
 * M = x'/x = fTL/fO

Note that we never used the distance between the objective and tube lens. In fact this distance does not play a major role for the imaging in an infinity microscope (If it is too long, the parallel beams expand too much and you will loose intensity in the image).

The space between the objective and tube lens is called "infinity space", because the image is focused "in infinity" here (i.e. we have parallel beams of light). The advantage of this is, that we can easily introduce any plane optical elements (fluorescence filter, dichroic mirrors ...) without disturbing the optical properties of the microscope.