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Q & A: Spatial and spectral spread of photons.

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Most recent answer: 04/24/2013
Q:
1)A laser beam spreads out if it goes through a narrow gap or hole due to uncertainty principle, correct? but why a laser beam coming out from a "narrow hole" of a laser pointer/emitter seems to stay as a narrow beam and not to spread out? The hole of the emitter (looks quite small though)is large enough to keep the beam straight or the laser emitter/pointer has special mechanisms to prevent spread? 2)"Otherwise the component sine waves couldn't cancel outside that position range. The range of λ's gives a range of z-momenta for the photons of the beam,.. ..that range of wavelengths corresponds to a small range of colors." (http://van.physics.illinois.edu/qa/listing.php?id=21868) Does this mean that a laser beam contains(is made of? is a superposition of?) a small range of? various wavelengths(colors) from its head till tail? 0r mainly its head and tail contain different wavelengths where the wave must cancel out and disappear but the main middle part contains almost a single wavelength? And, does a very short pulse-like laser packet made of a single photon contain a single frequency or it must contain various frequencies to create a packet? (it cannot have a single frequency? due to uncertainty principle? i.e. the shorter the pulse->larger the z-momentum uncertainty?) but a single photon can have multiple frequencies?? or a ultra-short laser pulse cannot be made of a single photon but splits spontaneously into multiple photons with different frequencies? First of all, does a photon have just a single frequency or it has multiple frequencies (is made of a superposition of multiple frequencies?)?
- Anonymous
A:

Light which is confined to a small aperture does indeed spread out in accordance with the uncertainty principle. This applies to laser beams as well. If you measure the laser beam at varying distances, you will find that its size continually increases at a hyperbolic rate. In fact, take your laser out to a field at night and look at it on a screen as far away as you can, and the beam will be huge!

If you tried to hit the moon with a laser, you'd want to use a BIG beam- 8 meters (24 feet) in radius. By the time it reached the moon, it would "only" be 16 meters across. That's the smallest spot you could make on the moon. In contrast, if you shone your handheld laser at the moon, it would expand to over 60,000 meters (~40 miles) wide!

An interesting fact to notice, however, is that the uncertainty principle only fixes the minimum amount a beam can spread out. Most beams spread out much faster than Heisenberg requires. A gaussian-shaped laser beam is one of the only types of light beams that actually spreads out by this minimum amount. That is why you don't see it spreading out unless you measure carefully, or just measure its size far away from the laser. (In addition, keep in mind that a typical laser beam is over 1000 times wider than the wavelength of light. This is reasonably large, so the beam doesn't diffract very rapidly. If your laser beam were much smaller, it would diverge faster.)

As for your second question: all laser beams that can possibly be created do contain at least a small range of wavelengths. You can see this if you look at the specs of any commercial laser, for example one.

If you dimmed the laser beam down until it was just a single photon, that photon would behave statistically just like the original laser beam. This isn't completely intuitive, since any measurement on a single photon will collapse its wavefunction and yield exactly one position, momentum, wavelength, etc. As long as a photon exists, however, it is made of a superposition of different wavelengths, positions, momentums, etc. In addition, its wavefunction (and corresponding "width," or region in which it can be found) spreads out hyperbolically, just like the laser beam!

David Schmid
 


(published on 04/06/2013)

Follow-Up #1: The shape and properties of photons

Q:
Very interesting! Thank you, David, for your answers. regarding the last part of your answer:"As long as a photon exists, however, it is made of a superposition of different wavelengths, positions, momentums, etc. In addition, its wavefunction (and corresponding "width," or region in which it can be found) spreads out hyperbolically, just like the laser beam!" 1)Does this mean a photon(=a chunk/packet/unit of energy moving as a superposition of waves) gets larger and larger in width(which is in a perpendicular plane(say xy plane) to its traveling direction(say z axis)) hyperbolically like "an ever-spreading thin pancake with zero thickness(at least looks like zero thickness to any massive observer?)?" but this photon(pancake/disc etc.) keeps spreading limitlessly (in xy plane)? btw, why did you say "hyperbolically"? and what do you mean by that? Do you mean the spread(rate?)initially is greater (when closer to the photon source) than later per unit travel distance of a photon? intuitively the spread follows the inverse-square law, right? or a real photon does not "exactly" follow this law? meaning that when it is very close to the source/just emitted from it, a photon spreads slightly more rapidly?/or slowly? than when it is very far from the source? (i.e. the path of a photon is slightly off straight spontaneously?) if this is the case why? (due to space-time curvature created by the energy of a photon itself? or interaction(interference?) between all the superimposed waves of a photon? Does a photon(in gamma ray-range) with energy equal to or greater than "the Planck mass" spontaneously stop existing as a photon?? but turn into something else (another particle? or a small black hole or?)? first of all, what is the energy density and volume of a photon?) 2)but this single(?) photon pancake can spread out limitlessly (over very large distance like billions of light years in xy plane) after traveling very long distance in z direction but when it(this single photon pancake) "interacts/hits/is absorbed by" something(a (massive?) point-like particle? ordinary matter? etc.?) "it instantaneously collapses?/localizes into a point-like region"!?? This sounds strange. how could something(=single photon energy, waves? etc.)stretched/spread/diluted? over very large area can instantly comes to a point like area?(to be consistent with the upper speed limit of our universe, either the single photon energy was never spreading to begin with OR somehow the spread-out energy of a single photon can take a very (not-so obvious?) short path(s?) to localize/focus in a small or point-like area/volume when it interacts with/hits an object/particle??) btw, the xy plane(=a photon pancake with no thickness) should gradually/spontaneously take a shape of a part of sphere (something like a shape of a contact lens)as a photon travels, instead of a flat disk, correct? and what is "the minimum angle" of a single gaussian-shaped laser photon cone(with the round bottom instead of flat) at infinity(after traveling very very long distance)? i.e. "what kind of 3D shape does a single gaussian-shaped photon plane(=a traveling photon pancake/disk/contact lens etc.) draw from a source to infinity as it travels?" 3)and the spread of a photon energy/momentum in traveling direction(z-direction) changes(gets larger?) as it travels? or it stays constant? the spread in speed is zero(at constant, the speed of light, correct?) but what other components(energy density, direction, volume?, momentum spread/direction, probability density/wave, spin direction etc.) of a photon change(or do not change) over a (large) distance? 4)or does this mean, a "real(=physical)" photon travels to a definitive direction and it never spreads out to begin with? or a photon is always spreading out and never a point-like but only looks/behaves like a point when it hits/interacts with a massive particle like an electron? First of all does a photon draw/show a fine definitive path(travel) line like those seen in the detector of CERN particle collider? or only a single very energetic photon can draw a definitive thin line path? but a single low energy photon cannot?? (could it be the case that wavefunction/probability function etc. is a mathematical construct/tool that gives so far the correct predictions(close to experimental date) but does not exactly provide yet the complete process of what nature is doing??)
- Anonymous
A:
I'll try to hit all the main points. Here goes!

1. By "spreading out hyperbolically," I mean that the radius of the beam's cross section increases at a hyperbolic rate (which means it spreads out most slowly at first, and then spreads out more rapidly to an asymptotic (linear) rate. This is stated explicitly on Wikipedia: , but the best way to visualize this is to approximate the hyperbola as a cone. If you shine a laser beam over a long distance, it will spread out almost exactly like a cone. The opening angle of this "cone" gets larger as the source gets smaller.

Because we are talking about a beam, not a point source, it doesn't necessarily follow the inverse square law. In the region far away from the source (in other word, the region where the hyperbola is approximately a cone), the beam does (approximately) obey the inverse square law.

If you measure the photon on a screen far away, it will be somewhere within a disk (which gets bigger if you look further away). Its position won't be exactly along the average propagation direction, simply because the photon has a spread in momentum!

As for your question involving the Planck-mass: as far as I know, we don't actually have any idea what goes on near the Planck scale.

2. Yes, the photon can spread out indefinitely, and if it interacts with something, the wavefunction instantaneously collapses or changes. This is definitely strange, and many physicists are looking for alternative explanations. To date, however, there are no intuitive interpretations. Actually, this is one of the central questions in quantum mechanics.

The pancake visualization for the photon's wavefunction isn't a bad one, but it isn't perfect. If a photon is made of a very narrow range of frequencies, its wavefunction can be many miles long! And yes: as it travels far away, its 3D wavefunction will look more and more like a contact lens shape.

The 3D shape it traces out as it travels will be a hyperboloid of revolution; again, quite similar to a cone (at large distances from the source).

3. All photons have a spread in frequency ω and 3D wave number k, and corresponding spreads in energy hbar*ω and 3D momentum hbar*k. Since the spin is aligned along k, the spin must also have a spread.

4. Photons, like electrons and all other quantum particles, exist as probability clouds (more precisely, wavefunctions). So, just like electrons, they can be spread over large regions, or they can be quite localized. If you could make a "cloud chamber" for photons, you could see tracks like those at CERN. (Unfortunately, photons don't interact strongly with our detectors, so this is very hard. Maybe it has been done, but I'm not aware of such a detector.)

Some people do indeed think that the wavefunction is just a mathematical convenience. We certainly don't know how to measure it directly. However, there are good theoretical and experimental reasons to believe that it is the most complete description which nature will allow.

Cheers,
David

p.s. I should mention that most of my comments here apply exactly only for Gaussian beams. Since this type of beam is typical of most lasers, it is the most common and important.

(published on 04/24/2013)

Follow-up on this answer.