# Diifference between temporal coherence and transverse/longitudinal spatial coherence.

## Main Question or Discussion Point

I'm bit confused about using terms temporal coherence and transverse/longitudinal spatial coherence when speaking about electro-magnetic radiation.

I have understood that temporal coherence basically means how monochromatic light is. But I don't understand why temporal coherence is related monochromatic light.

And what is the difference between transverse spatial coherence and longitudinal spatial coherence?

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Andy Resnick
Coherence is a statistical concept.

Temporal coherence is given by the frequency spread of the light:

$$\Delta t \leq \frac{2\pi}{\Delta\omega}$$, which can be converted into a length:

$$\Delta l = \frac{2\pi c}{\Delta \omega}$$

and refers to the maximum path difference in a Mach-Zender interferometer.

The coherence area (transverse coherence length, using your terms) is:

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

$$\Delta \omega$$ is the freqency spread of the radiation
$$\Delta \Omega$$ is the solid angle of the source
$$\overline{\lambda^{2}}$$ is the mean wavelength of the radiation

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Look at the image i have provided , Source L emits light waves which travel in all direction , when the wave train has length greater than the distance FE then temporal coherence is the effect as the phase relationship b/w two points on the wave can be predicted .

Above F , point O lies normal to F , the phase of the waves will be constant or same at both points F & O then spatial coherence is the effect .

http://img149.imageshack.us/img149/756/41797049fc3.th.jpg [Broken]

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Temporal coherence is given by the frequency spread of the light:

$$\Delta t \leq \frac{2\pi}{\Delta\omega}$$, which can be converted into a length:

$$\Delta l = \frac{2\pi c}{\Delta \omega}$$

and refers to the maximum path difference in a Mach-Zender interferometer.
So do you mean by the feaquency spread for example the full width at half maximum of freaquency spectrum or some kind other width of freaquency spectrum?
So basically $$\Delta l$$ tells us the length after which lightwaves initially in-phase are out of phase. Doesn't this $$\Delta l$$ now describe the longitudinal spatial coherence.

So there is basic relationship between temporal coherence and longitudinal spatial coherence:

$$l_{coh}=t_{coh}c$$ ?

The coherence area (transverse coherence length, using your terms) is:

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

$$\Delta \omega$$ is the freqency spread of the radiation
$$\Delta \Omega$$ is the solid angle of the source
$$\overline{\lambda^{2}}$$ is the mean wavelength of the radiation
So transverse spatial coherence means just size of means coherent area?

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Andy Resnick
You got it exactly, except for:

"...the length after which lightwaves initially in-phase are out of phase."

If a collection of waves has a known phase relationship, regardless of whether it's in or out or changing, the light waves are mutually coherent. Remember, coherence is a statistical concept. The coherence length is the distance in which knowledge of the phase relationship is lost. One may no longer confidently predict phases of other waves based on measurements of a single wave.

Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

It didn't quite get through my head...
can someone explain in greater detail what spatial and temporal coherence mean...??

Andy Resnick
Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

Temporal coherence relates to the frequency spread of the source; spatial coherence relates to the size of the source.

Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

ok....
so I calculate the temporal coherence for a source, by the (delta)t expression you gave above...
and, how spatial coherence depends on the size of the source....

Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

I have a confusion, please refer to the two drawings attached.

In the first drawing, a light goes from left to right, passes a hole in a non-transparent card, and the diffraction pattern on the screen is the fourier transform of the hole.

In the second drawing, I tilted the card with a hole. Now if I assume the longitudinal coherence length is LC (assume LC is very small), and if I tilted the card with hole large enough so that the horizontal distance between top edge and bottom edge of the hole is larger than the longtudinal distance (meaning for the light hitting the top and bottom edge of the hole, the phase relation is lost), will the diffraction pattern on the screen still simply the fourier transform of the hole?

You got it exactly, except for:

"...the length after which lightwaves initially in-phase are out of phase."

If a collection of waves has a known phase relationship, regardless of whether it's in or out or changing, the light waves are mutually coherent. Remember, coherence is a statistical concept. The coherence length is the distance in which knowledge of the phase relationship is lost. One may no longer confidently predict phases of other waves based on measurements of a single wave.

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Andy Resnick
Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

I have a confusion, please refer to the two drawings attached.

In the first drawing, a light goes from left to right, passes a hole in a non-transparent card, and the diffraction pattern on the screen is the fourier transform of the hole.

In the second drawing, I tilted the card with a hole. Now if I assume the longitudinal coherence length is LC (assume LC is very small), and if I tilted the card with hole large enough so that the horizontal distance between top edge and bottom edge of the hole is larger than the longtudinal distance (meaning for the light hitting the top and bottom edge of the hole, the phase relation is lost), will the diffraction pattern on the screen still simply the fourier transform of the hole?
I'm not sure, but it appears you are combining aspects of a Young interferometer and a Michaelson interferometer. The fringe visibility will now depend on *both* the longitudinal and transverse coherence of the beam. If the transverse coherence is always larger than the aperture, the fringe visibility will probably vary as the longitudinal displacement of the top with respect to the bottom exceeds the coherence length.

Putting in some semi-random numbers, the visible light from a lightbulb has a coherence length of about 19 microns. At what angle do you need to tilt your 10-micron diameter pinhole to obtain that kind of path difference?

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Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

Right, for my question, I was assuming large transverse coherence length.

What if I assume the pin hole is 1mm, and the longitudinal coherence length is 10um, I just need to tilt the card by a small angle so that the horizontal distance between top and bottom of the pinhole is larger than the longitudinal coherence distance.

The consequence of tilting the card is (1) changing the time that the light hits different place inside pinhole; and (2) the time that it takes for the light to go from the pinhole to the screen. (3) however if both the screen and the source are fixed, the path different for the light to reach, eg. the center of the screen is still within the longitudinal coherence length. thus interference should still be there. The pattern might be slightly different, but would not smear very much because of this issue of longitudinal coherence length. before and after tilting the pinhole card, the path different is still smaller than the longitudinal coherence length, which is the key factor.
Thanks.

I'm not sure, but it appears you are combining aspects of a Young interferometer and a Michaelson interferometer. The fringe visibility will now depend on *both* the longitudinal and transverse coherence of the beam. If the transverse coherence is always larger than the aperture, the fringe visibility will probably vary as the longitudinal displacement of the top with respect to the bottom exceeds the coherence length.

Putting in some semi-random numbers, the visible light from a lightbulb has a coherence length of about 19 microns. At what angle do you need to tilt your 10-micron diameter pinhole to obtain that kind of path difference?

Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

I got a question about pictures of coherence too. In this Wikipedia diagram:

http://en.wikipedia.org/wiki/File:Spatial_coherence_finite.png

I understand the coherence length, since after that length the waves look different. And within the coherence length you can define a wavelength too.

I don't understand the spatial coherence though. How you can you read off the spatial coherence area from such pictures? I have some formulas for the spatial coherence area, but they require that you know things like the size of the source, the angle it subtends, etc. This picture is just a wave with no information about the source. How are you supposed to interpret the coherence area in the picture? Would it be like this: if you know the phase at the top of the area, then you know the phase at the bottom of the area? If that's the case I think I can extend that area downwards as the whole top half looks periodic to me over the coherence length.

Also, does anyone know how to draw such wavefronts? Do you just say at time=10, I'm going to draw a line connecting every point in space that has the maximum amplitude? At time=11, I'm going to do the same thing again. Does anyone know of a free computer simulation program that shows you an animation of such wavefronts? I've seen programs that have spherical waves coming out of each source, and when they overlap the colors change, but not one that connects every point in space that has maximum amplitude.

Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

I attached a drawing defining the transverse and longitudinal coherence length, hope it'll help.

the transverse coherence is kind of the degree of the flatness of wavefront, which you can tell from the picture. for a planar wave, it is super flat, thus having a very long transverse coherence. for the picture in the wiki link, the defined Ac is reasonable.

the shape of the wavefront depends on the source size, shape, and distance from the source, etc..

I got a question about pictures of coherence too. In this Wikipedia diagram:

http://en.wikipedia.org/wiki/File:Spatial_coherence_finite.png

I understand the coherence length, since after that length the waves look different. And within the coherence length you can define a wavelength too.

I don't understand the spatial coherence though. How you can you read off the spatial coherence area from such pictures? I have some formulas for the spatial coherence area, but they require that you know things like the size of the source, the angle it subtends, etc. This picture is just a wave with no information about the source. How are you supposed to interpret the coherence area in the picture? Would it be like this: if you know the phase at the top of the area, then you know the phase at the bottom of the area? If that's the case I think I can extend that area downwards as the whole top half looks periodic to me over the coherence length.

Also, does anyone know how to draw such wavefronts? Do you just say at time=10, I'm going to draw a line connecting every point in space that has the maximum amplitude? At time=11, I'm going to do the same thing again. Does anyone know of a free computer simulation program that shows you an animation of such wavefronts? I've seen programs that have spherical waves coming out of each source, and when they overlap the colors change, but not one that connects every point in space that has maximum amplitude.

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Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

I attached a drawing defining the transverse and longitudinal coherence length, hope it'll help.

the transverse coherence is kind of the degree of the flatness of wavefront, which you can tell from the picture. for a planar wave, it is super flat, thus having a very long transverse coherence. for the picture in the wiki link, the defined Ac is reasonable.

the shape of the wavefront depends on the source size, shape, and distance from the source, etc..
In the first picture you have for temporal coherence, although there is complete destruction of the two amplitudes at the coherence length, if you wait awhile longer to twice the coherence length, you'll get the same pattern again. Do we still say that the wave is temporally incoherent after the coherence length, when if you just wait twice the coherence length, you get a repeating pattern again? This is a consequence of adding two waves to get another periodic wave, instead of performing an integral over a bandwidth to get a finite wavetrain.

I like the second picture. You can clearly see that at the spatial coherence length, the crest of B hits the trough of A (this occurs on the axis). Still, the picture is of two waves, and you have to do the adding mentally in your head. How would you combine the two waves into one wavefront? I think that's what's confusing me. The wavefront (the lines in the picture) of A and B individually correspond to crests. When you add them to get a single wavefront, what's the relationship between all points on the wavefront? Do they all have the same amplitude?

Andy Resnick
Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

I got a question about pictures of coherence too. In this Wikipedia diagram:

http://en.wikipedia.org/wiki/File:Spatial_coherence_finite.png

I understand the coherence length, since after that length the waves look different. And within the coherence length you can define a wavelength too.

I don't understand the spatial coherence though. How you can you read off the spatial coherence area from such pictures? I have some formulas for the spatial coherence area, but they require that you know things like the size of the source, the angle it subtends, etc. This picture is just a wave with no information about the source. How are you supposed to interpret the coherence area in the picture? Would it be like this: if you know the phase at the top of the area, then you know the phase at the bottom of the area? If that's the case I think I can extend that area downwards as the whole top half looks periodic to me over the coherence length.
I don't understand the picture either. The best way I think of to describe spatial coherence relies on a Young interferometer type of setup, where different parts of the wavefront are made to interfere.

Re: Diifference between temporal coherence and transverse/longitudinal spatial cohere

can anyone tell me what is the difference between spatial or temporal coherence