Schlieren images projected onto a wall (no camera)

[davekni]

Wrote a brute-force 2D simulation of my schlieren setup (of a single 0.5mm shadow-mask slit).  It models the 0.5mm light source as 1000 point sources (at 0.5um pitch), the 200mm mirror as 1000 sections (0.2mm pitch), the 0.5mm mask opening as 1000 points (also at 0.5um pitch), and the 1:1 (non-magnified) image as 1000 points (200mm image at 0.2mm pitch).  It is order n^3, so has 1E9 inner-loop executions.  Each simulation case takes 11 minutes on my home computer, or 5 minutes on the Linux server at work.  A lower-order simulation may be possible.  I opted for simple physics with no manual calculus.  The idea is to avoid mistake-prone manual integration or approximations.  Didn't want to repeat any of my flailing attempts at a simplified view.  The results generally support my last simplified view, however.

This models an ideal system.  Shadow mask has infinitely-sharp transitions from 100% to 0% transmission at it's edges.  The mirror is a perfect sphere.  (Actually a perfect circle since this is a 2D simulation.)  The light source and mask slit are on a line exactly through the mirror's center of curvature.  The simulated image is thus the best that could be achieved.  My real system with parabolic (rather than spherical) mirror, imperfect dimensions, slits farther from the mirror center, and fuzzy slit edges etc. produces less distinct images.

The simulated object is an air-stream approximation, a fuzzy cylinder (2D circle) at the mirror, with average diameter of 11mm fuzzed by 3mm.  The center 8mm diameter is uniform, with density tapering to background from 8mm to 14mm diameter.  There are 7 simulated cases.  Index of refraction difference (relative to background) varies from -3E-6 to +3E-6 in 1E-6 steps.  Light passes through the cylinder twice, before and after reflection, doubling the phase shift.  The cylinder center is positioned at the mirror surface and 20mm offset from the mirror center.

This first 7 simulations are with 500nm light and 50um overlap between the 0.5mm light-source image and the 0.5mm shadow mask opening.  Nominally 10% of the light passes through.  I haven't made any attempt to calculate absolute intensity (exact electric field strength of light source points).  Rather, I added a scale factor such that the average intensity for this nominal case is roughly 1.0.  The second set of 7 simulations is with 100um overlap, making average intensity about 2.  First plot is an overview of the 200mm image.  Second is a zoom in to the cylinder:



To get an idea of the contributions of refraction and diffraction, here are two simulations, one with 700nm light and the other from above with 500nm light.  These are of the case of a cylinder with 2E-6 increased index of refraction (colder air):


Finally, I thought I should simulate a case that is simpler to calculate, where diffraction is minimal.  Here are two 80mm wide wedges making 2E-6 radian refraction angles, one in each direction.  2E-6 radians is 3um after 1.5m light travel from mirror to slit.  With 50um overlap, 3um displacement should make a 6% change in light intensity, or 12% difference between the +3um and -3um shifts.  Here are the simulation plots verifying this:



Notice the huge diffraction spike at the left edge of the ramp image.  This is where the ramp suddenly stops, dropping back to background refraction index.  The right edge of the ramp doesn't show this diffraction.  That edge has ramped to background, so no abrupt step.

Now that I've taken time to code simulation, I'm trying to think of ways to use it to improve intuitive understanding.  Any suggestions?  Any specific cases that would be insightful?

Just in case anyone wants to pursue this further themselves, here is my ~100 line C source code (command-line).  Nothing fancy.  Zipped to fit within allowed file formats.  I use "gcc" to compile it.
schlieren.zip

[Uspring]

A nice analysis confirming the ray optics approach you used to estimate the sensitivity. You've put more reality into your calculation than I did in mine, i.e. modelling a slit instead of a knife edge and using an extended light source.  But have a look at an idealised situation, e.g. a point light source, a knife edge placed exactly through the middle of the focal point and using light ray optics. The system would be extremely sensitive to refraction, because the focal point would really be a point and light rays would only need an infinitesimal deflection to be either blocked or passed by the edge. Actually either diffraction or a finitely sized light source will blur the focal point and so limit sensitivity to refraction.

I've tried to estimate sensitivity for a point source and a knife edge using wave optics. What I found, I didn' really believe, since sensitivity seems to depend on the size of the mirror. In a way that makes sense, since the bigger the mirror, the smaller the focal point, so more sensitivity to deflection. But intuitively I think, that the object size should matter here and not the mirror size. Your calculation shows a dependency of sensitivity on wavelength. That points into the same direction as the dependency on mirror size, since a longer wavelength will also blur the focal point. This dependency is not as strong as I suspected, but you might have some blurring effects there due to the extended light source.

On the other hand your calculation does confirm the differentiation effect on the phase in the image. Gradients in optical density show up in the image.

Edit:
I've been trying to reconcile your simulation with the formulae regarding the convolution kernel mentioned previously. I finally succeeded and they come to similar conclusions as yours:

Finally, I thought I should simulate a case that is simpler to calculate, where diffraction is minimal.  Here are two 80mm wide wedges making 2E-6 radian refraction angles, one in each direction.  2E-6 radians is 3um after 1.5m light travel from mirror to slit.  With 50um overlap, 3um displacement should make a 6% change in light intensity, or 12% difference between the +3um and -3um shifts.  Here are the simulation plots verifying this:

This brings up the mental image of a 50um wide light beam, that is shifted by 3um across an edge and loses or gains intensity correspondingly. But what happens, if you use a point light source instead of an extended one?
Interestingly nothing dramatic. The sensitivity of the system seems to depend on the slit width or more precisely: The relative intensity change is close to the ratio of refraction angle to slit angle. The slit angle being the angle the slit is seen as from the object. That is what you calculate and that holds even for a point light source. In contrast to your simulation, though, the background intensity of the image is almost independent of the slit width, as long as the width is considerably bigger than the diffraction focal point size, which is object distance * wavelength / mirror diameter: 1.5 m * 500nm / 200mm ~ 4 um. I believe, the background intensity scales linearly with slit width in your case because of the size of the light source.

Here is the convolution kernel in more detail:

K(x) = (e^(j*a*x) -1) / (j*x), where a is 2*pi / lambda * d/l

d is the the slit width and l the object to slit distance. That can also be written as

K(x) = sin(a*x)/x  +  (cos(a*x)-1) / (j*x)

The first summand is a diffraction smearing term and convolution with the second term is close to a differential operator 1/(j*a) * d/dx.

The kernel is to be applied to the one dimensional image f(x), where f is a complex valued function describing transparency and phase shift. For a transparent object it would be of the form e^(j*g(x)), which always has an absolute value of 1. A wedge e.g. with refraction angle alpha would be

f(x) = e^(j*2*pi*alpha/lambda*x)

[davekni]

It has been about decades since I've used much calculus on complex numbers.  The resulting sin(x)/x and derivative functions make sense.

There is a subtle effect that I had mostly ignored.  Shows up best in the ramp graphs in reply 20 above.  There is a diffraction ring with period ~15mm, which corresponds to the 50um overlap rather than the full 500um slit width.  (You can also see some 1.5mm ring closer to the step at the left edge of the ramp, as would be expected for the 500um wide slit.)  With a single point light source, this diffraction makes the 80mm wide ramp into a blur:


These are the same two ramps used above, with +2E-6 and -2E-6 radian refraction on an 80mm wide wedge.  The slit is now 5mm wide (close enough to infinite) with 50000 samples across it (100um sample grid).  Mirror and image samples also increased, from 1000 to 5000.  (With a point light source, computation is order n^2.  That allowed much finer grids.  I also repeated the previous ramp cases with finer grids just to verify.  Took over-night, but results are within 0.1%.)  I didn't fix amplitude scaling for this point-source case, so roughly 0.1 represents no blocked light.

I haven't come up with any simplified model to explain the fuzzy result with a point source.  I did run some related cases, such as moving the slit slightly either direction, to at least somewhat verify that the simulation behaved as expected.

Here's a new version of the ramp image using a point light source.  This one includes a third case with a 80mm wide gray filter instead of the ramp.  Filter is 50% transmission with two passes, so 25% of light gets to the mirror and back through the filter.  This filter object makes a much more distinct image than does the refraction wedge.

[Uspring]

I haven't come up with a simple explanation of point source images myself. In the case of a thin wedge, the brightness change due to it depends not only on the deflection angle but also on its length. The reason for this comes from the differentiating part of the convolution kernel. It has long tails to both sides. See a plot of it here.



If you apply this kernel to a linearly growing function then the tails can contribute significantly to the integral particularly for long integrating intervals.

The amplitude change for a transparent and only phase shifting object such as the wedge can be written as:

deltaAmp ~ alpha*length/lambda ~ max optical path length of wedge in radians (as difference to the surrounding)

alpha is the deflection angle, and length the wedge length. The proportionality holds only for small deltaAmps. It maxes out around 2*pi.
The dependency of the brightness change on wedge length is particular to the point source. It is due to the diffractional spread of the focal point near the knife edge. A bigger object will have a smaller focus and consequently the dependency of brightness on deflection angle will increase.
It doesn't seem, like there is a trick to optimize sensitivity, since the optical path length is a property of the object alone. Extended light sources limit sensitivity. In that case a longish setup like yours seems optimal.