I was asked to chime in on an ongoing debate: When using the diffraction pattern of a laser to determine the width of a hair... should you treat the hair as a single slit, or as a double slit?
The short answer: I don't know for sure; yet it won't matter in terms of your measurement. The long answer is below.
Double slit diffraction is reasonably simple to explain conceptually. At any point on a screen that's not directly across from the spot directly between the slits, light from one slit travels a wee bit farther than light from the other slit. That extra distance traveled means that one beam is at a different spot in its wave cycle than the other beam. If that extra distance is a wavelength - or any number of full wavelengths - than the interference is constructive, and you see a bright spot. If that extra distance is a half-wavelength - or 1.5 wavelengths, 2.5 wavelengths, etc. - then the interference is destructive, and you see a dark spot. The equation for the location of bright and dark spots is derived from straightforward trigonometry, which shows that the extra distance traveled by one wave is, with commonly defined variables, dsinθ (or dx/L for small angles).
But single slit diffraction is conceptually complex. I've seen explanations that it's like double-slit diffraction, except with diffraction around the slit's edges rather than around two separate slits. That's not at all correct, though, other than as a very hand wavey hint-of-the-truth discussion with a low-level first-year student.* The derivation of the location of bright and dark spots begins with assuming an infinite number of point sources of light within the slit, and uses symmetry and trigonometric arguments to determine the locations at which most or all of these infinite point sources interfere constructively or destructively. Not simple at all.
* Sort of like the planetary model of the atom
Yet, the fundamental equations for the locations of spots is the same: dx/L = mλ. For a double slit, the central maximum is m = 0, the first dark spot is m = 0.5, the next bright spot is m = 1, and so forth. For a single slit, the central maximum is wider, such that m = 1 represents the first DARK spot away from the central maximum, m = 1.5 is the next bright spot, m = 2 the next dark spot, etc.
I do not know whether single or double slit approaches better model the behavior of a laser around a hair. Are there effectively two beams, one on each side of the hair, that travel slightly different distances as in a double slit? Or are there infinite point sources on either side of the hair, and do we need symmetry considerations to figure out where those infinite sources interfere con- and de-structively? I suspect the argument could be settled by measuring the width of the central maximum compared to the dark-fringe-to-dark-fringe distance; or by using a light meter to very precisely measure the intensity as a function of position along a screen.
What it means for the hair thickness measurement: I measure x along the screen by finding the distance from, say, one dark spot to the next to the next to the next until I reach, say, the sixth dark spot. Then I divide that distance by 6 to get a value for x.
Look at the meanings of m three paragraphs up. Whether the hair should be modeled as a single or double slit MAKES NO DIFFERENCE!!! The value of x that I measured represents the distance between dark fringes. Whether that means the distance from, say, m = 2 to m = 3, as for a single slit; or from m = 2.5 to m = 3.5, as for a double slit; doesn't matter. We can use dx/L = mλ with the measured x and m = 1. The hair-to-screen distance is L, and the wavelength of the laser is λ. Solve for d, and that's the width of the hair.
No comments:
Post a Comment