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Chapter 9: Problem 36

Light is emitted from an unpolarized point source. First it passes through a linear polarizer with easy transmission axis at 45 deg to the \(x\) and \(y\) axes. Then it is incident on a double slit. Each slit is covered by a linear polarizer, one slit having the polarization axis along \(\hat{x}\), the other having it along \(\hat{y}\). (a) Suppose you look at the interference pattern with the unaided eye. Do you expect to have the usual two-slit interference pattern? What do you expect? (b) Next suppose you look at the interference pattern while holding a polaroid linear polarizer in front of one eye. What do you expect to see? What happens as you rotate the polaroid in front of your eye? (c) Now suppose you look at the pattern through a circular polarizer run backward as an analyzer. What pattern do you expect to see? There are many nice variations you can make on this problem: (i) Put a righthanded circular polarizer over one slit and a left-banded circular polarizer over the other and repeat the above observations. (ii) Add a quarter- or half-wave plate just behind the slits, etc.

Short Answer

Expert verified
(a) No usual pattern; the slits output orthogonal polarizations. (b) A pattern appears when using a linear polarizer and varies with rotation. (c) No pattern is expected.

Step by step solution

01

Understand Initial Polarization

The light is initially unpolarized and then passes through a polarizer with a transmission axis at 45掳 to the x and y axes. This means the light becomes linearly polarized at 45掳 with equal components along x and y, so each component's intensity is halved from the original intensity.
02

Analyze the Double Slit Polarization

At the double slit, one slit is covered with a polarizer aligned along the x-axis, and the other along the y-axis. The initial 45掳 polarized light splits equally into intensities along these axes. As a result, each component can pass through its respective slit, with the x-component through the slit with the x-polarizer and the y-component through the slit with the y-polarizer.
03

Interference Pattern Observation (unaided eye)

Since each slit allows only one component of the polarized light to pass through, resulting waves from the slits being orthogonally polarized will not interfere. Therefore, the usual interference pattern will not be obtained, and likely no distinct pattern will be observed.
04

Effect of Holding a Linear Polarizer

When looking through a linear polarizer, it will combine the orthogonally polarized components, potentially making an interference pattern visible. Rotation of the polaroid affects the intensity of the observable pattern, as different amounts of the x and y polarized components are transmitted.
05

Observing with a Circular Polarizer

A circular polarizer converts linear polarized light into circularly polarized light. Running it backward as an analyzer over the produced light would not yield an observable interference pattern since this system combines orthogonal polarizations which can't interfere to show the usual pattern.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Polarizer
A linear polarizer is a device that harmonizes light waves to oscillate only in a single direction. When unpolarized light passes through a linear polarizer, it emerges with its electric field confined to a single plane. This process reduces the light's original intensity by half due to the transformation from unpolarized light, which vibrates in multiple planes, to linearly polarized light, which vibrates in just one. For instance, if a polarizer's transmission axis is set at a 45掳 angle, the emerging light will align with that angle, retaining equal intensity components across the x and y axes. As a result, only specific light orientations are allowed to pass through, fundamentally altering how light interacts with subsequent optical elements, like a double slit.
Circular Polarizer
A circular polarizer manipulates light differently than a linear polarizer. It transforms linearly polarized light into circularly polarized light, where the electric fields rotate in a circle. This rotation creates a helical structure of light, appearing the same from any rotational angle. In essence, it's composed of a linear polarizer followed by a quarter-wave plate, a specialized device that introduces a phase shift between the components of light. When used in reverse, a circular polarizer operates as an analyzer, evaluating the polarization state of incoming light. By blending orthogonal polarizations鈥攚hich typically would not interfere鈥攁 circular polarizer can change how we perceive interference patterns in experiments. However, it cannot generate a standard interference pattern from orthogonally polarized light.
Interference Pattern
Interference patterns occur when light waves overlap and combine, forming regions of constructive and destructive interference. Constructive interference happens when the crests and troughs of waves align, amplifying the resulting wave. Conversely, destructive interference happens when crests meet troughs, leading to cancellation. In an interference pattern, this interplay creates alternating bright and dark fringes, typically seen in setups like the double slit experiment. The nature of these patterns depends largely on the coherence and polarization of the light. For light with mixed polarizations, as through orthogonal polarization axes, no clear interference pattern emerges since these segments do not combine effectively. Observers may use techniques, such as additional polarizers, to induce visible patterns even where they wouldn鈥檛 naturally occur.
Double Slit Experiment
A classic demonstration of wave interference, the double slit experiment, involves light passing through two closely spaced slits to create an interference pattern. This experiment highlights the wave nature of light, showcasing how light waves can merge to form a series of light and dark bands on a screen.
  • Constructive interference: When waves align in phase, they intensify the resulting wave.
  • Destructive interference: When waves are out of phase, they cancel each other out.
With polarized light, this phenomenon can manifest differently. If each slit polarizes light orthogonally, they produce waves that won鈥檛 interfere visibly without additional aids like linear polarizers. The double slit experiment reveals nuances in wave behavior but hinges significantly on the kinds of light and polarizers involved. When analyzed with different polarizing elements, observers can transform not just the intensity but the very presence of recognizable interference patterns.

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Most popular questions from this chapter

Use a pinhole in a piece of aluminum foil illuminated by a broad source to study the floaters. When you are looking at one, try to wipe it off by blinking. Can it be wiped? "Roll" your eyes once or twice and then watch the floaters swirl! Now try to find if they are nearer to the pupil or to the retina: vary the distance of the pinhole from your eye. The size of the circle of light changes. (Make a sketch to aid in explaining why.) Any object at the same location as the pupil would change apparent size by the same ratio as does the projection of the pupil. (Why?) Anything at the retina (or near it) would not change apparent size at all. (Why?) What do the floaters do? Are they closer to the retina or to the pupil? Now try to estimate their length and diameter. To estimate the diameter, compare them with a human hair (yours) held in front of your pupil between the pinhole and the pupil. For this purpose, you needa very small pinhole, smaller than you can easily make with a pin. Crinkle up a piece of aluminum foil, then smooth it. Look for an accidental small pinhole. (You can tell if it is small -less light intensity gets through.) Now look at a hair. You should see its shadow and see nice diffraction fringes at its edge. Compare its size with that of a floater. Are they finer than a hair? (Note: A human hair has a diameter about \(\frac{1}{20} \mathrm{~mm}\), i.e., \(50 \mu\) (micron). A typical red blood cell has a diameter of 5 or \(6 \mu\).)

Your eye and brain do not Fourier-analyze light (the way your ear Fourier- analyzes sound). It takes some practice, but you can recognize the difference between a color due to monochromatic light and a color due to a mixture of wavelengths. Psychologically, "white" is a "color." However, your diffraction grating tells you it is the whole visible spectrum of wavelengths. \((a)\) Look at things through your purple filter, which passes red and blue but absorbs green. (b) Look at two separated white light sources-line sources or incandescent bulbsthrough your diffraction grating. Vary your distance from the two sources until the Ieft-hand first-order spectrum of the right-hand bulb can be superposed with the right-hand first-order spectrum of the left-hand bulb. Then you can superpose any two wavelengths and see what "psychological" cqlor results. In order to be superposing two "pure" wavelengths, you should use two line sources (i.e., display lamps). One gets beautiful colors. Try it! (Joseph Doyle suggested this experiment.)

Near field and far field. How far away should you be from a double slit of slit spacing \(0.1 \mathrm{~mm}\) irradiated with visible light in order to use the far-field approximation without making use of a lens? How far should you be from two microwave antennas having spacing \(10 \mathrm{~cm}\) and emitting \(3-\mathrm{cm}\) microwaves to use the far-field approximation?

Burn a piece of toilet paper and look at it through your diffraction grating (held, as always, close in front of one eye). Notice the beautifully clear "first-order flame." This shows that the soft yellow light is almost monochromatic, with very little "white light" color spectrum due to hot carbon. The yellow that you see is the by now familiar (we hope) sodium doublet of wavelengths 5890 and \(5896 \AA\). Now that you recognize "sodium yellow," light an ordinary paper match and look at it with your grating. Most of the light is "hot carbon yellow," which is not really yellow but a complete "white" color spectrum. But look closely! In the yellow part of the hot carbon spectrum, down low next to the cardboard, where the flame is "blue" looking-below the blindingly bright hot carbon spectrum-do you see a crisp, clear little monochromatic match flame? If you don't, try again! Now burn other things and look. You may well conclude that everything is made of salt or is at least contaminated by it.

Pour some table salt on a wet knife or spoon (one that you don't mind ruining). Set the knife in the flame of a gas stove. Look at the yellow flame through your diffraction grating (this is easiest at night in a darkened room). Notice that the first-order (and higher-order) images of the yellow sodium flame are as sharp and clear as the zeroth-order "direct" image. That is because the yellow light is a "spectral line" having narrow bandwidth. (Actually the yellow light from sodium is a "doublet" of two lines with wavelengths 5890 and \(5896 \AA\).) Now look at a candle. In zeroth order, it does not look terribly different from the sodium flame; they are both yellow. But in the first-order diffraction image, the candle is very much spread out in color, whereas the sodium remains sharp. The "yellow" of the candle, which is due to hot particles of carbon, has a wavelength spectrum extending over (and beyond) the entire visible range. Here are other convenient sources of sharp spectral lines; look at them through your grating: Mercury vapor: Fluorescent lamps, mercury-vapor street lights, sunlamps. (A sunlamp is convenient in that it screws directly into an ordinary 110 -volt AC socket. It is probably the cheapest source of mercury-vapor spectral lines; the cost is about \(\$ 10 .\).) Neon: Many advertising signs. Neon has a profusion of lines; you see "many signs." A cheap broad monochromatic source is a G.E. bulb NE-34 which screws directly into a 110 -volt AC socket (the cost is about \(\$ 1.60)\). Others are a "circuit continuity tester," which plugs into any wall receptacle and which costs about \(\$ 1\) (at a hardware store), and a neon "night light." Strontium: Strontium chloride salt (available at a chemical supply house for about 25 cents \(/ \mathrm{oz}\) ); dissolve a little in a few drops of water and put it in the gas flame on your ruined spoon. The wavelength of the red line is a famous length standard. Copper: Copper sulfate; availability and technique as for strontium chloride. It gives a beautiful green color. Hydrocarbon: Look at your gas flame in the first-order spectrum. There are a sharp, clear blue image and a sharp, clear green image. The "blue" color of the flame is therefore due to one or more almost monochromatic spectral lines.
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