I am trying to understand how optical flats are used to measure the surface flatness of an optic.
See this image from Edmund Optics note on optical flats.
In this image I will point out 4 surfaces.
- #1 - top surface of Optical Flat
- #2 - bottom surface of Optical Flat
- #3 - top surface of Test Piece
- #4 bottom surface of Test Piece
Suppose the Test Piece in this case is an uncoated parallel glass window. My understanding of the measurement is that one illuminates the setup shown in the figure with plane wave illumination from above (directed downwards so that it first passes through the optical flat and then the test piece. Now focus on surface #2 and #3. Some of this plane wave will reflect off of surface #2 and exit the way it came in (about 3.7% of incident power). Some more of this plane wave will reflect off of surface #3 and reflect the way it came in (about 3.4% of incident power)*.
The light reflected off of surface #3 had an optical path length slightly longer than that of the light reflected off of surface #2. The path length difference is a 2D function $2d(x,y)$ over the surface between the two optics where $d(x,y)$ is the distance between the two surfaces. If the optical flat is perfectly flat that $d(x,y)$ is a topographic map of the surface of the Test Piece plus possibly a tilted plane representing the wedged air gap between the two optics.**
When the light from surface #2 and surface #3 leave the system (travelling upwards) the two beams interfere with eachother. Whenever $2d(x,y)$ is an integer multiple of $\lambda$ there will be constructive interference and whenever $2d(x,y)$ is a half-integer multiple of lambda there will destructive interference. This interference leads to the bright and dark fringes which are seen and used to characterize the image.
1) The discussion above only focused on the reflected light from surfaces #2 and #3 interfering. But what about light from surface #1 and #4? Basically as much light is going to come from those surfaces as from #2 and #3 and then in principle all 4 of these beams should interfere in some possibly complicated way depending on the surface deviations of each. Why is this not a consideration in all the explanations of this measurement that I see?
1a) One possible explanation is that the coherence length of the source is shorter than the thickness of either optic meaning that you won't see interference from these other reflections, just a diffuse background reducing the contrast of the desired interference between #2 and #3. However, it seems that highly monochromatic light, such as a HeNe with a coherence length of at least meters, is used for these measurements. What gives?
1b) Possibly one of the surfaces of the optical flat is AR coated so that light doesn't reflect. However Edmund optics' spec sheets indicate both surfaces of their flats are coated. Furthermore, the measurement technique doesn't specify that the secondary surface of the optic under test must be AR coated. Again, what gives?
2) I don't understand why there is often a wedged air gap between the two optics? If the two optics are optically contacted wouldn't there be no air between and the only deviations present would be a result of surface deviations between the two optics? Why would an air gap persist and not be close by gravity for example? In the image above it is obvious that the optical flat should just fall and close the gap between the optical flat and test piece.. why would there be a steady state wedged air gap and is this in fact desirable to make the flatness measurement more straightforward?
*Actually my understanding is that at every surface interface between glass and air there is generally about 4% reflection unless there is an AR coating. My understanding is that optical flats are uncoated. This means the reflected light off of surface #2 will be 0.96*0.04*0.96 = 3.7% of the original power and the reflected light off of #3 will be 0.96*0.96*0.04*0.96*0.96 = 3.4% of the original power. Note that the reflected light off surface #1 will but 4% of the original power and the reflected light off surface #4 will be 3.1% of the total power. There are also beams which are reflected multiple times but these will all be suppressed in power by at least $0.04^2 = 0.0016$ compared to the promptly reflected beams.