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Beam Astigmatism Correction: Anamorphic Prism Pair Theory, Gaussian Beams, Dispersion
I am not sure how useful this is, but a post by Planters got me thinking about anamorphic prism pairs, and ended up coming to some interesting conclusions that I have not seen discussed before.
Background:
A common problem when dealing with laser diodes is that they produce a highly astigmatic beam, which is a side effect of the way they are manufactured.
First, see http://www.photonlexicon.com/forums/...ied-discussion if you do not understand the 'fast' vs 'slow' axis notation
For so called 'single mode' diodes (which refers to the transverse mode, ex tem00, tem01, etc) the emitting structure is small compared to the wavelength of light they are emitting, which forces the intensity vs space profile to be a gaussian shape, with a certain radius rx and ry in the vertical/horizontal dimensions, which are in general not the same, but usually pretty close (within a factor of 2 or so). This poses a slight problem, because of a fundamental property of gaussian beams (and with some approximations, beams in general), that the divergence is inversly proportional to the diameter, so the axis with the smaller beam radius (so called 'fast axis') diverges faster than the other 'slow axis'. This means that by the time you get to the collimating lens, the fast axis has a larger diameter than the slow axis, which results in a collimated beam that is oval shaped. Generally for laser show applications this is 'good enough' and no one worries about correcting them, but it is possible to correct these diodes to have a 'perfect' round beam (that is to say it has a gaussian profile, diverges at equal rates in both x and y, and can be focused to a diffraction limited spot) using a short focal length lens to get the beam roughly collimated oval beam and a stretch it back into a circle using prims, etc.
For a multi transverse mode diode things are much trickier. In the fast axis the diode is (usually) thin enough that it is forced to be single mode (with a gaussian profile like with the single mode diode). However in the slow axis the emitting structure is much wider than a wavelength, ex for a XJ-A140 445nm diode the emitting area is about 1um (fast) by 40um (slow). In this case, the region is much to wide to force the laser to operate in single mode, so a lot of different modes appear, and the math works out such that in typical diodes you get a more or less constant intensity 'top hat' profile in this axis. This is a serious problem, for 2 reasons:
1. Fundamentally, it is impossible to collimate a top hat profile beam to the same divergence of a gaussian beam (it will always have a higher divergence than a gaussian beam of the same diameter for a collimated beam. For arguments sake just imagine it a bunch of gaussian beams stacked next to eachother, you can stretch it all you want but just like stacking multiple diodes together you can never get them to overlap, thus is will always be wider than the single gaussian for a given beam divergence)
2. The divergence of the beam coming out of the laser diode in this axis (slow) is much lower than that of the fast axis, which means that by the time the beam propagates to the collimating lens it will have a smaller diameter in slow axis than the fast axis. This hurts you double, because not only does the beam diverge faster because it is a top hat profile, but it is a smaller diameter than the fast axis so it diverges even more than necessary. Luckily, we can fix this problem by stretching the slow axis to have roughly the same (or maybe a little higher) diameter as the fast axis, so that the divergence is only hurt by the fundamental gaussian vs top had issue.
Implementation:
There are many ways to build a beam stretcher, lenses, fibers, light pipes, etc, but I will focus on using prisms and cylindrical lenses, because these are the 2 most common:
Cylindrical Lenses:
This system is fairly strait forward, we all know that it is possible to build a beam expander by using 2 lenses of different focal lengths,(ex http://www.edmundoptics.com/learning...eam-expanders/ ), so cylindrical lenses is a '1D' version of this. Instead of using lenses with spherical symmetry which affect the beam in both the vertical and horizontal axis, you use a lens with cylindrical symmetry, which only affects the beam in 1 axis (whichever axis the lens is in). So, by picking 2 lenses with the correct focal length ratio, you can expand the slow axis to match the fast axis quite easily.
These do have a few disadvantages however:
1. They offer a fixed magnification, the magnification is determined by the focal lengths of the lenses, so after you buy your lenses you are stick with that magnification.
2. No lens is perfect, so they will affect the intensity profile in more ways that simply stretching it--luckily if you start with a top hat profile you are already about as bad as you can get so this is not really an issue for multimode diodes
3. If they are not perfectly aligned with one another (or to a lesser degree, the axis of the diode), they will affect the fast axis as well and put an odd twist on the beam, which puts a lot of 'wings' and whatnot on it.
Anamorphic Prisms:
Another method of stretching the beam is to use so called 'anamorphic' prisms.
There is nothing fundamentally different between an anamorphic prism and the 'normal' you used as a kid to make rainbows, however they are generally anti reflection coated and made out of a glass that is chosen to have a high refractive index as opposed to dispersion (which is related the derivative of refractive index, and is what causes a normal prism to make rainbows). You can get them cut at different angles, although 30/60/90 degrees is the most common, because it gives you a little more freedom on the expansion ratio than an equilateral 60/60/60 prism.
In any case, what is neat about prisms is they stretch your bream without focusing it, so there there are none of the abberations associated with cylindrical lenses! Thorlabs has some good explanations on how this works, along with plots of typical angles see the 'beam expansion' tab at http://www.thorlabs.com/newgrouppage...ctgroup_id=149 In short, the idea is that as light enters the prism the interface between air and glass refracts the beam at an angle determined by snells law, then travels through a distance of glass, to finally be refracted again upon its exit. The key is that due to the wedge shape, different rays travel different distances through the glass, so the beam exiting the prism is stretched in the axis of the wedge! You could use the prism as-is, but it would put a funny angle on your beam, which changes depending on your desired expansion ratio, so typically people use a pair of them which has the advantage of doubling the expansion ratio (for a given angle) and leaves your beam traveling in the same direction you started in, with a horizontal shift.
There are a few downsides to prisms of course, mainly that they can be quite lossy due to the long path length in the glass (so you need a good quality glass which is very transparent at your laser wavelength) and the obvious 4 reflections (so you need good AR coatings which work at the funny angles required dictated by your desired magnification)
There is one more subtle difference however, and that is that because you are sending the beam through a prism, if there is any wavelength shift it will be converted to an angular shift! This may not seem like a serious problem, because we are using lasers which are of course 'single' frequency, but all lasers span a finite amount of wavelengths, and with diode lasers their wavelength changes with temperature and current. Not to mention that if you try to combine several diodes (ex, with knife edging) before expanding them, you now have the diode to diode wavelength variations to deal with, which can be quite large (ex, XJ-A140 diodes can vary from 435nm to 455nm, although most of them are in the 440-450nm range).
So, this poses the question, how small of a wavelength shift is enough to cause your beams to go out of alignment? To answer this, I turned to numerical simulation, using OSLO, because I had to learn it for a homework assignment. I will warn you now, it is not for the feint of heart, and the users manuals aren't all that great either.
Luckily there was an example file for a melles griot prism pair which I was able to use as a starting point, and then I modeled the prisms from the group buy by p1t8bull, and the results were surprising. For the example problem using melles griot 06GPA004 (30 degree prisms made of SF11 glass) set for 3x magnification gives a 0.1m per 100m deviation (1 milliradian) deviation going from 440nm to 450nm. Using the prisms from the group buy (30 degree prisms made of BK7 glass) set for 3x magnification gives .025m per 100m deviation (.25 milliradian) going from 440nm to 450nm.
So - The effect is measurable, although for the typical 1-5nm shift and the BK7 prisms it is not big, 0.1milliradian tops. If, however, you have no temperature regulation on your 635nm diodes and you let them warm up to the maximum 60c you can get as much as a 10nm -> .25milliradian shift, which will be noticeable but small. Also keep in mind that if you run your beams through a telescope further down the road this can be magnified by the telescope as well.
If you want to play around with the prisms, I have attached the design files I used for the 2 prism pairs.
When you open them up they are configured for a 3x magnfication, and evaluated at 440, 445, and 450nm, with a 100m distance to the image plane (which is where the plots are generated). To see the shift due to wavelength, select Evaluate->other ray analysis->lateral chromatic shift, which generates a chart of beam displacement vs distance. Note - I selected 100m to get round numbers, as configured the model does not consider any diffraction effects (is is purely ray optics), so it does not consider the effect of divergence on the beam in any way. Another useful plot to generate is evaluate->spot diagram->single spot diagram which gives you a plot of what the beam would look like (to get the magnification divide the geometrical Y size by the geometrical X size. Note - in order for this to work you have either select a single wavelength under the 'wavelength' button in the 'surface data' editor (available under lens->surface data editor if you loose it) by making all of the wavelengths the same, or setting the image distance to 0 meters by setting the 'thickness' column of the 'IMS' row to 0 in the same editor.
If you want to change the angles of the prisms, click the grey box with a 'C' on it in the surface data editor, and select Cordinates, then change the TLA box for rows 2 (first prism) and 4 (second prism). To change the prism angle you can do the same procedure for rows 3 and 5. Note - if you set the 'thickness' of 'IMS' to 0, and click the 'autodraw' button, you can see the layout of the prisms. There are 4 angles lines, which represent the surfaces of the 4 prisms (do not worry that the rays do not intersect them, however if you see rays bouncing off surfaces that shouldn't be there then something is wrong.
