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# Double Slit Interference

"We only have to look at ourselves to see how intelligent life might develop into something we wouldn't want to meet."
Stephen Hawking

- We now begin a discussion of wave (physical) optics in which - in contrast to geometric optics - we explicitly consider the wave nature of light. Remember, in geometric optics light traveled in straight lines (light rays). This is a valid approximation so long as we do not consider apertures with dimensions similar to the wavelength of the light or look too closely at the edges of objects.

- So what happens when a wave passes through apertures whose size is similar to the wavelength of the wave ? First we'll consider the case of double slit interference, in which a parallel beam of incident monochromatic (containing a specific wavelength) light from the left strikes a screen with two slits, S1 and S2 , as below.

CONDITIONS for MAXIMA and MINIMA in DOUBLE SLIT INTERFERENCE

- With L >> d geometric optics predicts that two bright spots would be observed on the right hand screen immediately opposite S1 and S2 . The rest of this screen would be in shadow. What is actually observed on the right hand screen is an "interference pattern" as indicated below,

- The explanation is that each slit acts as a source of spherical waves, which "interfere" as they move from left to right as shown above.

- In the diagram at the top of the page, light reaching P from S1 and S2 will travel different distances. Assuming that the light from the two sources S1 and S2 are initially in phase, then due to the path difference S1P - S2P , at P the two waves will be out of phase. If the path difference is equal to an integral number of wavelengths the waves will interfere constructively, leading to a bright spot on the screen. Mathematically we can write this condition for maximum intensity as,

> where n can take on integer values, n = 0, 1, 2, 3... and we have assumed that θ = θ' or in other words the width of each slit is small compared to their separation (D >> a above).
> Similarly, the condition for minimum intensity at P, when the path difference is a multiple of half wavelengths, is given by,
> where n can again take on integer value, n = 0, 1, 2, 3...

- When the distance from slits to screen is much larger than the distance on the screen, D >> y above (or L >> x in first diagram), then the angle θ is "small" and we may assume tanθ is approximately equal to sinθ which is approximately equal to θ (in radians) and we may write,

> for the position on the screen for maximum intensity.
> INTENSITY DISTRIBUTION in DOUBLE SLIT INTERFERENCE

- The interference pattern shown above was first observed for visible light in 1801 by Thomas Young , the experiment is still sometimes called Young's slit experiment.

> In the above description we have assumed the incident light is monochromatic. If white light (containing all the wavelengths in the visible spectrum) is used, the maxima for the different wavelengths will occur at slightly different positions (y) on the screen. In this case an interference pattern will only be observed if the maximum - minimum separation is much larger than the separation between the maxima of the extreme wavelengths in white light (red and violet) for the same "n".

- In the above description we have shown that at certain locations on the screen there will be bright spots whereas at other locations there will be no light - the interference pattern. But exactly how does the light intensity vary as a function of position on the screen ?

> In the diagram at the top of this page the electric field from light originating at each of the slits S1 and S2 can be written,
> where each slit has the same maximum E field, E0 and φ is the phase difference due to the path difference S1P - S2P.
> Therefore the E field at P can be written,
> the product of an amplitude and a sinusoidal time varying wave. In the case of light waves the frequency of the time varying part is so large that our eyes and most instruments "see" only the ampliutde part. Actually, what we observe is the intensity, which is the square of the amplitude. The intensity observed at P is then given by,
> as shown in the red shading in the diagram above. Note that maxima of the above cosine squared function occur when φ = 2π n; this leads to bright spots on the screen.

- As we have seen, when the path difference is an integer multiple of wavelengths, the waves from the two sources interefer constructively. That is they are in phase, and as we have seen above, φ must be an integer multiple of 2π,

> Thus for small values of θ (sinθ = θ), φ and θ are proportional to each other.
> COHERENCE

- Throughout the above description we have implicitly assumed that the light waves from the two slits S1 and S2 are in phase with each other. This ensures that any phase difference in the light from the two slits is due entirely to the different path lengths the two waves travel. If the relative phase of the sources S1 and S2 is unknown and randomly varying with time then the interference pattern will also randomly change with a frequency similar to the frequency of the light. Practically this means there will be no observed interference pattern.

- The difficulty of obtaining two sources (S1 and S2) emitting waves coherently (in phase) depends on the wavelength of the electromagnetic wave.

> For long wave radiation (e.g. radio waves) a single source illuminating the two slits results in two coherent sources, since this type of radiation is typically produced in a continuous waveform, as shown at left below.
> For short wavelength radiation (e.g. light) "waves" are typically emitted by multiple individual atoms in a random incoherent manner (no definite phase between these multiple "sources". The radiation is emitted in "packets" rather than as a continuous wave. Thus a two slit configuration, at left below, will not produce an interference pattern. In order to observe an interference pattern with light the configuration at right below must be employed. The single slit to the left of the two slits ensures that light reaching the two slits is from the same part of the source and therefore in phase.
> Note that a laser beam produces a coherent light source and can be used to create an interference pattern in the left configuration.

QUANTUM LIMIT - DOUBLE SLIT INTERFERENCE

- Consider double slit interference. In the figure at right there are locations on the screen which have zero light intensity.
- Now gradually reduce the intensity of the incident light so that instead of a "continuous" wave impacting the slits we have individual "photons" incident. This is the "quantum limit", where we treat a source of light as a source of discrete "wave packets" or photons. with individual photons incident the same interference pattern is observed.
- But with photons incident one at a time it makes sense to ask, "which slit did the photon pass through ?"
- Close one of the slits, but continue illumination with individual photons. The intensity pattern observed on the screen changes with only one slit open. But if the photon passes through the lower slit, how does it know whether the upper slit is open or closed ? For the interference pattern to change it must know.
- Explanation.... The photon is an extended object, so it never really passes though one slit. Due to its extended nature it can "feel" whether the other slit is open.
- However, perhaps most intriguing, is the fact that if the two slits are illuminated with a beam of particles, for example electrons, the same interference phenomena is observed. You'd naturally expect electrons to go through one slit or the other, but with both slits open an interference pattern similar to that at left is observed. The electron is displaying "wave properties".
- This is an example of "wave-particle duality", an important consequence of the quantum theory of matter.
- For an accessible description of what's going on in the double slit experiment see what Dr. Photon has to say....

"One morning I shot an elephant in my pajamas. How he got into my pajamas I'll never know."
Groucho Marx
(in the film Animal Crackers)

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu