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## Casual Observation ofDiffraction and Interference of Light

I've always found interference effects in light highly entertaining, but it also seemed disappointing that obvious visual effects due to interference are so rare.

Of course, they're demonstrated in high school physics labs, and anyone who's done much astrophotography has probably seen diffraction patterns from stars, but both of those take some fairly fancy equipment -- they're certainly not something you can typically see just walking down the street.

Holograms are, of course, an example of interference effects -- but they're anything but simple, and the relationship between a simple diffraction pattern and a 3-d photograph is anything but obvious to most of us.

And rainbows sometimes show a colorless "diffraction bow" inside the main bow -- but these are not common (at least where I live), and proving that the "diffraction bow" is really caused by diffraction, let alone understanding exactly why it appears where it does, is not an especially simple or obvious exercise.  Similarly, during a total solar eclipse one may see dark bands on the ground which are apparently due to diffraction -- but this is extremely rare, and most of us will never see it.

There's one major reason why we rarely see diffraction or interference patterns:  The sun is too fat.  It subtends about 1/2 degree in the sky, which means that the edges of shadows -- the angle between the umbra and penumbra -- spreads at an angle of 1/2 degree, also.  This means shadows cast by the Sun are too fuzzy to show visible diffraction effects, and when the Sun shines through closely spaced holes or slits, the "fuzzing" of 1/2 degree again tends to mush out the pattern.

All this can leave one feeling that diffraction of light is such a subtle effect, and the dimensions of the waves are so small, that save in certain very unusual circumstances, one will never see a "gross" diffraction or interference pattern without using special, precision-made equipment.

I was enormously amused to find that this is not true.

### Diffraction from Window Blinds

Long ago, I worked in a high rise office building down town.  There were venetian blinds in the window of my office.  One day, when the Sun was on the other side of the building (not shining in my window), I noticed an annoyingly bright spot shining into my eyes from a parking lot next door.  It was the Sun, reflected from a shiny chrome bumper (figure 1).  Before closing the blinds to block it out, I happened to glance at the shadows the blinds were casting on the opposite wall.

Each slat was casting a sharp, sharp shadow -- far sharper-edged than the shadows they cast when the Sun shone in directly.  Of course, the reason was that the image of the Sun, reflected on the curved surface of the bumper, appeared far smaller than the directly viewed image of the sun would have.   From my office, the Sun's image subtended a tiny angle, much smaller than 1/2 degree.  In other words, the sunlight coming in the window was very "well collimated".  And, much to my amazement, I noticed that the shadow of each slat had a series of dark and light bands along its edge.  The sunlight was forming a diffraction pattern on the wall!  It was clearly visible to a casual glance, no special equipment needed.

 Figure 1 -- Placement of sun, bumper, and building:

Unfortunately I didn't photograph the pattern on the wall, and I no longer work next to a parking lot, so that situation isn't likely to come up again.  But I recently ran across something nearly as surprising.

### Curtains

A few years ago, I was staying in a hotel room, and noticed that the street lights from the next block looked "funny".  They appeared "broken up", almost like I was seeing multiple images of them.  A little experimenting revealed it was the curtains doing it.  The curtain material was inexpensive, very coarsely woven stuff, made of some shiny white fabric.  Since the dark colored window screens weren't doing the same thing, my guess at the time was that it was the effect of reflections from the shiny white threads.  I dismissed it as odd but not very interesting.  I was wrong, as I later learned!

More recently, we moved into a different house, and the former owners left behind some inexpensive coarsely woven curtains something like the ones at the hotel.  One night I happened to notice the same effect when I looked at a lamp in front of a house down the street, and I decided to figure out what really was causing it -- I was no longer sure it was just reflections from the threads.  And, indeed, it was not!

First test:  How did the image behave as I moved around -- how did it change?   Answer:  It didn't change; it didn't depend on where I stood or how close I was to the curtain.  Its size and position remained fixed relative to the image of the lamp, whether I was close to the curtain or on the other side of the room.  Strange -- if this was some reflection from the threads, it's not how I would have expected it to act!

Second test:  Was it an illusion of some sort?  Answer:  Nope -- a camera sees the effect, too.  Figure 2 shows the lamp photographed "directly", with the curtain pulled to one side.

 Figure 2 -- Lamp (telephoto shot, enlarged):

Figure 3 shows the same lamp, photographed at the same scale, with the curtain in place.  It's a high contrast subject and the pattern in the photo is far more limited than what I can see with bare eyes, but none the less it's a striking image.

 Figure 3 -- Lamp shot through curtain (same scale as figure 2):

Third test:  How did the pattern size depend on the spacing of the threads?  If it's really a diffraction (or interference) pattern, then moving the threads closer together, and thus making the holes between them smaller, should expand the pattern.

 Figure 4 -- Curtain Mesh:
I turned the curtain, so I was looking through it diagonally, thus effectively shrinking the holes.  Sure enough, the pattern "spread out" as I did so -- it got larger.

At that point I was convinced that this had to be an interference pattern of some sort.  But the holes in the curtain still seemed too large for me to believe it ... perhaps a measurement was in order.  I took a few pictures of the curtain material with a couple of rulers (see example in figure 4, using a 64th inch steel rule).  Counting the holes and comparing with the ruler, we find the holes appear to be on 0.33 mm centers.  To my eyes, looking at a blowup of the photo, it looks like the threads are about 1/2 the width of the holes, which makes the holes about 0.22 mm square.

With the camera placed about 5 feet from the window, the angle subtended by one hole would be about 0.22/1524 = 0.000144 radians.  The angular separation of two holes would be about 0.33/1124 = 0.000217 radians.

Before I could go much farther, I also needed to know how "big" the image of the lightbulb really was -- that is, I needed to know what angle it subtended.  In principle I could determine that from the photos I'd taken, but I don't know off hand what the angular width of an image is on this camera, and I'd need to run some careful tests to find out.  So, I took the direct approach instead.  I went out the next morning and paced off the distance from the window to the lamp post; it was about 60 paces, or about 150 feet.  The bulb is roughly 2 inches in diameter.  Dividing out, the angular diameter of the bulb, viewed from the window, is about 0.00111 radians.  The lightbulb's image is about five times as large as the separation between two holes -- i.e., if we could see the fabric of the curtain in figure 3, we would see five threads running each way across the central bright area of the picture where the lightbulb should be.  (And, not incidentally, the apparent diameter of the bulb is about 1/8 the apparent diameter of the Sun -- consequently, it's likely to produce far sharper, more visible diffraction and interference patterns than direct sunlight would.)

The next question which came up is whether the light passing through one hole in the curtain would actually spread enough to form an interference pattern with the light from the neighboring holes.  It would spread due to diffraction -- but how much?

From here on I'll be treating the holes as slits, and assuming that what I've got is two "crossed" grids of slits.  The slits are 0.22 mm wide, on 0.33 mm centers.

 Figure 5 -- First null in diffraction pattern:
I want to know the angle at which the first null in the diffraction pattern cast by a single slit occurs, as this will tell me how wide a cone the light will spread into after passing through a hole in the curtain.  The first null in the pattern will be the point at which the light from the middle of the slit exactly cancels the light from one edge; as we continue on across the slit, each portion cancels the light from one "half-slit" to the left.  This implies that, in figure 5, the path length from the observer to the right edge of the slit must be exactly 1 wavelength longer than the path to the left edge of the slit.
The angles are grossly exaggerated in the figure.  In fact, θ is very small, so the slit width and the separation of the left and right edges of the "outgoing light" beam can be taken as identical.  Let's define some things:

(1)

The first null will therefore be at

(2)

and the angle subtended by the central maximum in the diffraction pattern from one hole will be 0.0055 radians, or about five times the apparent size of the lightbulb.  That's more than enough "spread" to allow for the interference pattern we're seeing.

Now, what should an interference pattern from the curtain look like?  A grid of bright spots seems right: it will be the intersection of two patterns of lines, one vertical and one horizontal.  We can presumably figure out the spacing of the lines by treating the curtain as a grid of slits rather than an array of holes.

Let's just consider two slits.  In fact, let's be sloppy about it and just consider two "line sources", separated by distance Δ.  There's a maximum in the center of the pattern, of course.  The next maximum will be where the two sources are once again in phase; that will be at the angle λ/Δ.  Amusingly, the diagram shown in figure 5 describes this case nicely as well, if we just eliminate all the "middle" arrows in the "Incoming Light" rays!

Our holes are on 0.33 mm centers, so we have Δ = 0.33 mm, and the first maximum in the pattern will be at

(3)

A little fiddling with diagrams of multiple slits seems to show that adding more slits increases the sharpness of the central maximum but won't change the locations of the first side lobes in the pattern.  So, we can reasonably use this angle for our estimate of where the lobes on either side of the central bright spot should fall.

 Figure 6 -- Figure 3 again, with guides:
We know the bulb diameter is 0.0011 radians, and we can measure the size of its image in figure 2.  From that, we can determine how large each of the features we've been discussing should appear in the photographs -- and that, after all, is the whole point of this exercise!  In figure 6, we've superimposed a diagram of our "predicted" features on a copy of figure 3. The blue ring shows the size of the lightbulb as measured in figure 2 (which is thoroughly obscured by the interference pattern!).   The yellow square shows the location of our computed "lobes" in the interference pattern, and the magenta circle shows the boundary of the central maximum of the light shining through one hole in the curtain.

These features agree quite well with the observed pattern, particularly considering the rather imprecise measurements which went into them.   From this I conclude that it is, indeed, an interference pattern caused by the mesh curtain.

Finally, we show two more photos of the lamp, taken during the afternoon with the sun reflecting from the curved surface of the lamp, making a small but intense bright spot something like the bright spot on the car bumper I described earlier.   These are shown at the same scale as the other images.  Figure 7 shows the lamp with with the curtain pulled to one side, and figure 8 shows the lamp photographed through the curtain; the bright spot of sunlight reflected from the lamp is responsible for the intense pattern of nine bright spots.  In the photo as a whole, a number of light-colored features appear "doubled" or "ghosted"; before doing this exercise, I never would have realized that was caused by interference as a result of the light passing through the fine mesh of the weave.

 Figure 7 -- Lamp in the afternoon: Figure 8 -- Through the curtain:

Page created on 7 Sep 2007