# Lots of questions on wave optics

Why if you shine a flashlight on the wall, the circle of light is bigger than the opening of the flashlight (the aperture)?

If I attach a pipe (of the same radius as the opening of the flashlight) to the aperture so that light has to go through the pipe, then geometrically the circle of light on the wall should be as big as the pipe diameter.

I'm thinking it could be diffraction, that you treat the aperture of the flashlight as a circle filled with continuous sources, and the equation for how big the light should appear on the wall should go roughly around:

$$R=\frac{\lambda}{D} L$$

where R is the radius of the light formed on the wall, lambda the wavelength of light, D the diameter of the aperture, and L is the distance to the wall. But just an order of magnitude estimate doesn't produce a reasonable answer, as it'll take a huge L to get a big R.

But strictly speaking that equation only holds for coherent light sources, when the sources in the aperture are uniformly in phase. For noncoherent light, shouldn't R be infinity, since there can be no interference effects? Each source in the aperture propagates spherical waves that can reach any point on the wall and won't interfere with neighboring sources.

I'm also a little bit confused about spatial coherence. Two noncoherent finite-size light sources (or even just one), cannot interfere (with itself in case of one). Yet why is it said that a light bulb has a coherence area, albeit small? I know that coherence area can be taken to mean how far two slits have to be to produce interference fringes from a source (incoherent or coherent), but it is really the two slits that cause interference and not the incoherent sources. So when you give incoherent light a coherence area, does it just mean that it'll produce interference if two slits are within the area? Or is there a more general definition of coherence area that doesn't involve double slits?

Also, given that the wavelength of light it very small, shouldn't the light just go straight through the slits geometrically and form no diffraction at all, so that the trajectory of the light from the source would look like a less than sign < )?

Also, just collimating an incoherent source shouldn't produce spatially coherent light, right? Within a cross-section of the beam, all the phases at each point in the cross-section should still be random? I heard in a laser each point in the entire cross-section is in phase. Is this the difference between a laser and collimated light passed through a colored piece of glass? What can you do with a laser that you can't do with monochromatic collimated light? What's so good about having all the light in a cross-section have uniform phase?

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Claude Bile
With negligible diffraction (i.e. the ray-optic limit) and an isotropic (non-directional) source, the solid angle subtended by the aperture will be equal to the solid angle subtended by the illuminated region.

In the ray-optic limit, the equation governing the size of the illuminated spot will be completely independent of the wavelength. Also, there is no reason why R and D as you have derived them should be inversely proportional to one another (i.e. surely the bigger the aperture, the bigger the projected spot?).

In the wave limit though, you do actually get an inverse relation between the output aperture of a source and the projected spot, due to the effect of diffraction.

Claude.

Andy Resnick
<snip>

I'm also a little bit confused about spatial coherence. Two noncoherent finite-size light sources (or even just one), cannot interfere (with itself in case of one). Yet why is it said that a light bulb has a coherence area, albeit small? I know that coherence area can be taken to mean how far two slits have to be to produce interference fringes from a source (incoherent or coherent), but it is really the two slits that cause interference and not the incoherent sources. So when you give incoherent light a coherence area, does it just mean that it'll produce interference if two slits are within the area? Or is there a more general definition of coherence area that doesn't involve double slits?

<snip>

Also, just collimating an incoherent source shouldn't produce spatially coherent light, right? Within a cross-section of the beam, all the phases at each point in the cross-section should still be random? I heard in a laser each point in the entire cross-section is in phase. Is this the difference between a laser and collimated light passed through a colored piece of glass? What can you do with a laser that you can't do with monochromatic collimated light? What's so good about having all the light in a cross-section have uniform phase?
Spatial coherence is related to the size of the source:

$$\Delta A = \frac{\lambda^{2}}{\Delta\Omega}$$

Where $\Delta A$ is the coherence area, $\lambda$ the mean wavelength, and $\Delta\Omega$ the angular size of the source. Note that spatially coherent light can have a large waveband: starlight is highly spatially coherent. Sunlight has a spatial coherence of about 0.5mm, IIRC.

Young's double slit experiment measures the spatial coherence, because the interference fringes result from the (original) wavefront interfering with itself. Thus, the slit separation is a direct measure of the coherence area. These interferometers are used to measure the diameter of stars.

In the case of a large source, the coherence area is small and can be thought of as a way to characterize how ignorant different points on the source are of other (spatially separated) source points.

Lasers have a large coherence time (the spectral spread is small) but generally have a small coherence area: speckle.

I can increase the spatial coherence of any source by using a spatial filter- passing the light through a small pinhole means the light now acts as if from a source the size of the pinhole.

In the ray-optic limit, the equation governing the size of the illuminated spot will be completely independent of the wavelength. Also, there is no reason why R and D as you have derived them should be inversely proportional to one another (i.e. surely the bigger the aperture, the bigger the projected spot?).

In the wave limit though, you do actually get an inverse relation between the output aperture of a source and the projected spot, due to the effect of diffraction.
I guess my confusion is when one is supposed to use the ray-optic limit, and when one is supposed to use the wave limit. Normally I'd say that the ray-optic limit occurs when the wavelength is much smaller than any apertures. But light is so small, it's hard to imagine when you can't use the ray-optic limit, yet diffraction is usually discussed with light waves and not something like radio waves!

Even more confusing is no matter the wavelength, this equation: $$R=\frac{\lambda}{D} L$$ where D is the size of the aperture and R is the spread a distance L away, should be true no matter what approximations you are using, since it is fundamental: that is what a line source creates, and no matter how big D is or how small $$\lambda$$ is, there will be a line of sources at D, and the angular spread will be $$\theta=\frac{R}{L}=\frac{\lambda}{D}$$ (the angular width of the 0th order beam occurs when the size of the source D times the sine of the angle equals 1 wavelength).

So to me the diffraction theory should always give correct results, since it is more general than ray-optics. But with diffraction theory you get that the spread is inversely proportional to aperture diameter, which as you mentioned seems to be the opposite of ray-optics.

Andy Resnick said:
In the case of a large source, the coherence area is small and can be thought of as a way to characterize how ignorant different points on the source are of other (spatially separated) source points.
I can't seem to interpret it that way. For a spatially incoherent source (say a finite size light bulb), each point on the source is independent of each other point. To me every point on the source is always ignorant of the other points. If you pass the light from the light bulb through two slits, then the intensities on a screen beyond the two slits from two different points on the light bulb are still additive. It just so happens that if the slits are so close together, then the phase difference at each of the two slits coming from each point on the incoherent source is practically the same, and so the minima of one intensity pattern from one spatial point won't fall on the maxima of another spatial point. But this is not interference of amplitudes, but of intensities.

It seems to me that spatial coherence means the ability for intensities to interfere, not amplitudes. Or more accurately the non-ability of intensities to interfere. So the more spatially coherent a source is, the less ability it has for the intensities to smear each other out.

Andy Resnick said:
Lasers have a large coherence time (the spectral spread is small) but generally have a small coherence area: speckle.

I can increase the spatial coherence of any source by using a spatial filter- passing the light through a small pinhole means the light now acts as if from a source the size of the pinhole.
That to me is confusing. So a laser acts like it came from a large source? Or rather, two slits have to be placed close together to observe fringes when using laser light? That to me is confusing too, since you usually think of a laser as a very thin beam, so of course if the beam is not thick enough to even envelop both slits, then you won't get any interference.

If you place a laser beam through an aperture, the aperture will act as a source. Will this source be a random source like a light bulb, or will there exist definite phase relationships among points in the aperture? In other words, if you take a cross-section of the beam, is there a definite phase relationship among points in the cross-section?

Thanks.

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But strictly speaking that equation only holds for coherent light sources, when the sources in the aperture are uniformly in phase. For noncoherent light, shouldn't R be infinity, since there can be no interference effects? Each source in the aperture propagates spherical waves that can reach any point on the wall and won't interfere with neighboring sources.

//some my understannding - coherence included transverse coherence and longitudinal coherence. transverse coherence is perpendicular to the light travel direction and is determined by the source size. longitudinal coherence is basically how monochromatic the light is. for the light from flashlight, I guess the wavelength distribution is faily wide, thus very short longitudinal coherence.
The smaller the pipe diameter, the longer the transverse coherence. //

I'm also a little bit confused about spatial coherence. Two noncoherent finite-size light sources (or even just one), cannot interfere (with itself in case of one). Yet why is it said that a light bulb has a coherence area, albeit small? I know that coherence area can be taken to mean how far two slits have to be to produce interference fringes from a source (incoherent or coherent), but it is really the two slits that cause interference and not the incoherent sources. So when you give incoherent light a coherence area, does it just mean that it'll produce interference if two slits are within the area? Or is there a more general definition of coherence area that doesn't involve double slits?
//coherence may change. before and after slits, the coherence length can be very much different. transverse coherence depends on the source size or slit size if light pass a slit. //

Also, given that the wavelength of light it very small, shouldn't the light just go straight through the slits geometrically and form no diffraction at all, so that the trajectory of the light from the source would look like a less than sign < )?

Also, just collimating an incoherent source shouldn't produce spatially coherent light, right? Within a cross-section of the beam, all the phases at each point in the cross-section should still be random? I heard in a laser each point in the entire cross-section is in phase. Is this the difference between a laser and collimated light passed through a colored piece of glass? What can you do with a laser that you can't do with monochromatic collimated light? What's so good about having all the light in a cross-section have uniform phase?[/QUOTE]

//seems to me, before talking about the phase, better make sure the light is very monochromatic. otherwise, the phase can not be the same anywhere along the beam. //

Andy Resnick
<snip>

I can't seem to interpret it that way. For a spatially incoherent source (say a finite size light bulb), each point on the source is independent of each other point. To me every point on the source is always ignorant of the other points.
For a perfectly (spatially) incoherent source, that is correct- but a finite sized-source is not perfectly incoherent. Maybe this will help- given the wavelength of emitted light, source points separated by less than a wavelength are able to interact with each other.

Alternatively, consider starlight. Stars appear to us as points, meaning the wavefront is flat. We cannot distinguish the different source points on the star- the star appears as a point- so the spatial coherence is very high. What happens between the surface of the star and our eyes is rather complex, but described by the propagation/evolution of the coherence matrix (Mandel and Wolf's "Optical Coherence and Quantum Optics" is the authoritative text).

If you pass the light from the light bulb through two slits, then the intensities on a screen beyond the two slits from two different points on the light bulb are still additive. It just so happens that if the slits are so close together, then the phase difference at each of the two slits coming from each point on the incoherent source is practically the same, and so the minima of one intensity pattern from one spatial point won't fall on the maxima of another spatial point. But this is not interference of amplitudes, but of intensities.
Close- You are ignoring the reason why the slits have to be close together to get interference patterns. Remember, coherence is a *statistical* property of the light. Temporal coherence means the wavefront interferes with itself at a later time (perhaps via a path difference), while spatial coherence means different parts of the wavefront interfere with each other. For a wavefront that varies stochastically (either because there are lots of colors, or lots of independent source points), the coherence defines how well you can predict the value of the wavefront *there and then* given a value *here and now*.

It seems to me that spatial coherence means the ability for intensities to interfere, not amplitudes. Or more accurately the non-ability of intensities to interfere. So the more spatially coherent a source is, the less ability it has for the intensities to smear each other out.
Intensities can interfere as well- that's the Hanbury Brown-Twiss effect

http://en.wikipedia.org/wiki/Hanbury_Brown_and_Twiss_effect

and I confess to not understanding it very well.

That to me is confusing. So a laser acts like it came from a large source? Or rather, two slits have to be placed close together to observe fringes when using laser light? That to me is confusing too, since you usually think of a laser as a very thin beam, so of course if the beam is not thick enough to even envelop both slits, then you won't get any interference.

If you place a laser beam through an aperture, the aperture will act as a source. Will this source be a random source like a light bulb, or will there exist definite phase relationships among points in the aperture? In other words, if you take a cross-section of the beam, is there a definite phase relationship among points in the cross-section?

Thanks.
Well, yes- the source is a large volume of gas (for a gas laser), or a chunk of crystal (for a solid-state laser), or some other bulk material. Lasers being 'thin beams' have more to do with the optical properties of the cavity than the lasing material. Certainly, diode lasers have a large divergence.

Spatial filtering can be thought of heuristically as generating a source that is only a few wavelengths across (the pinhole diameter is somewhere between 5-15 microns, usually). Thus, single-mode (transverse modes) operation is easily achieved. Again, it is instructive to observe speckle patterns, by shining the laser onto a rough surface. Raw laser beams have a lot of speckle, while spatially filtered beams have very little speckle (it's still there, but much diminished). The size of a single speckle is about equal to the transverse coherence length. Again, shine a raw beam on a surface and compare that to an expanded beam- the speckle patterns are the same. Only by spatially filtering the beam is speckle reduced.

For a perfectly (spatially) incoherent source, that is correct- but a finite sized-source is not perfectly incoherent. Maybe this will help- given the wavelength of emitted light, source points separated by less than a wavelength are able to interact with each other.
So how would you take into account that finite-sized sources are not perfectly spatially incoherent? Would you divide a light bulb into chunks the size of a wavelength, and have everything in a chunk be in phase, but let each of the chunks be random with each other? Or is this too rough an approximation?

Alternatively, consider starlight. Stars appear to us as points, meaning the wavefront is flat. We cannot distinguish the different source points on the star- the star appears as a point- so the spatial coherence is very high.
I remember reading about a Michelson-Stellar interferometer, and the key to measuring the solid angle subtended by the star is to make the distance between the slits big via mirrors.

So normally to observe interference your area between slits is small. But with the interferometer, you want your area big so that you can observe a smearing out of intensities. So here is a weird case where incoherence of a star is used to measure the size of a star, since normally a star is coherent and you have to have big separation of the slits to observe incoherence.

while spatial coherence means different parts of the wavefront interfere with each other. For a wavefront that varies stochastically (either because there are lots of colors, or lots of independent source points), the coherence defines how well you can predict the value of the wavefront *there and then* given a value *here and now*.
I'm a bit confused about the meaning of the term wavefront. In general physics books, the wavefront is often defined as equipotentials of constant phase at a certain time. In a classical mechanics book, the wavefront is defined as the envelop of a wave at a certain time. To me these two things are not necessarily the same. The latter are contours of the farthest places your wave can reach at a certain time, but the points on the envelop do not necessarily have the same phase (for example parts of the envelop could have been slowed down in a medium, so if the wavefront is meant to correspond to crests, then the part of the wavefront in vacuum is a crest but the part in water might be a trough). Are the terms wave-envelop and wave-front interchangeable?

I think I understand temporal coherence, but with spatial coherence there is still some haziness. Suppose you have a perfectly temporally coherent source in the shape of a ball, but the ball is of finite size. Aren't the wavefronts perfect spheres radiating outwards, just from spherical symmetry? There is no distortion in the wavefront: it's a perfect sphere, and not a crumpled up piece of paper. So wouldn't the wavefront be flat and not jagged like a circular saw blade?

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