Sketch of the electric field of a laser beam

[spareine]

I am trying to sketch the electric field E in snapshot of a linearly polarized laser beam. Is it correct that the E vectors bend from vertical to longitudinal near the cylindrical surface of the beam, and that all field lines within a half wave segment are closed loops?

 

[Gordianus]

The electric field lines do not bend at the "top" or the "bottom". The electric field is stronger at the center and weakens as you move from the center to the border, but it always points in the same direction.

 

[spareine]

The electric field cannot weaken along a field line, if the electric field is vertical everywhere. $\nabla \cdot \mathbf{E} = 0$ in the absence of electric charges (Gauss' law).

 

[davenn]

I am trying to sketch the electric field E in snapshot of a linearly polarized laser beam.

does this help ?

 

[sophiecentaur]

The electric field cannot weaken along a field line, if the electric field is vertical everywhere. $\nabla \cdot \mathbf{E} = 0$ in the absence of electric charges (Gauss' law).

I can't reconcile that with a narrow beam in free space and each E field line around a radiating dipole follows a curve in a plane. The field strength in any particular elevation will follow a Cos θ law for a short dipole. A collinear / coaxial array will produce a more vertically directional pattern but, being the sum of dipole patterns, why would the E vector direction change? The same thing should apply for a very narrow beam, wouldn't it?

BTW The diagram at the top looks more like the fields in a circular waveguide (conducting walls). That wouldn't be the same boundary conditions as for a narrow beam in free space.

 

[spareine]

does this help ?

I don't think polarization is the essence of my question. For me, the advantage of specifying the laser beam is linear polarized is that it is slightly simpler to describe and draw than other options.

The essence of my question is that I am wondering about the profile of the electric field on cross sections of the laser beam, and what the field lines look like, given that somehow they should be closed loops because of Gauss' law.

 

[spareine]

I can't reconcile that with a narrow beam in free space and each E field line around a radiating dipole follows a curve in a plane. The field strength in any particular elevation will follow a Cos θ law for a short dipole. A collinear / coaxial array will produce a more vertically directional pattern but, being the sum of dipole patterns, why would the E vector direction change? The same thing should apply for a very narrow beam, wouldn't it?

BTW The diagram at the top looks more like the fields in a circular waveguide (conducting walls). That wouldn't be the same boundary conditions as for a narrow beam in free space.

I am afraid I do not understand your answer. My question is about a laser beam in free space, outside the optical cavity. I don't see how or why radiating dipoles are present in that laser beam in free space.

 

[davenn]

I am afraid I do not understand your answer. My question is about a laser beam in free space, outside the optical cavity. I don't see how or why radiating dipoles are present in that laser beam in free space.

the confusion is because the drawing you supplied implies light in a waveguide ... optical fibre, NOT free space

 

[spareine]

Ok, but I am actually more interested in a laser beam in free space. Gordianus suggested the laser beam in free space is identical to an infinitely wide plane wave, multiplied by a gaussian-like beam profile in the cross sectional plane. My objection against that suggestion is that it makes $\nabla\cdot\mathbf{E}$ nonzero. Do you think this objection is invalid?

 

[tech99]

The electric field cannot weaken along a field line, if the electric field is vertical everywhere. $\nabla \cdot \mathbf{E} = 0$ in the absence of electric charges (Gauss' law).

I think there is a distinction between a Field LIne, which is a contour of equal field strength, and a Line of Force, which is the path taken by a positive charge when placed in the field. Field LInes will run circularly around a charge, whereas lines of force will project radially from it.
Lines of Force become further apart as we depart from a charge, so they indicate a weakening field. The original diagram seems to show Lines of Force, but describes them as Field Lines. The field is not entirely contained within the main beam, but at the top of it the field lines wander away into the rest of the page, and finally met up at the bottom. When using Lines of Force, the strength of the field is often shown by using small arrows of varying length.
Regarding polarisation, there are geometrical distortions arising from any source, and it differs depending on the type of radiator. Most usually, the polarisation is "correct" on the principle axes and then bends away to some extent off axis in the 45 degree planes.
I think the diagram from Davenn depicts the fields using arrows proportional to field strength, rather than showing the shape of the field.
(My apologies for using capitalisation to emphasise the terminology).

 

[sophiecentaur]

I am afraid I do not understand your answer. My question is about a laser beam in free space, outside the optical cavity. I don't see how or why radiating dipoles are present in that laser beam in free space.

EM waves are EM waves, whatever wavelength. Why would the E vector of light waves be any different from the E vector of RF waves? They don't know what generated them so they must behave the same way. I was, of course, discussing the far field pattern.

 

[spareine]

I think there is a distinction between a Field LIne, which is a contour of equal field strength, and a Line of Force, which is the path taken by a positive charge when placed in the field. Field LInes will run circularly around a charge, whereas lines of force will project radially from it.

I meant field lines in the sense that is common in electricity and magnetism: lines that follow the direction of the vector field (wikipedia). They can only form closed loops if there are no sources and sinks.

 

[tech99]

I meant field lines in the sense that is common in electricity and magnetism: lines that follow the direction of the vector field (wikipedia). They can only form closed loops if there are no sources and sinks.

In such a case, there is no need for the lines to turn over at the edge of the beam, they just continue outside the main beam but weaker..

 

[spareine]

In such a case, there is no need for the lines to turn over at the edge of the beam, they just continue outside the main beam but weaker..

Could you draw that field line pattern? Wouldn't it be radial with a source at r=0? But there is no source at r=0.

 

[berkeman]

Could you draw that field line pattern? Wouldn't it be radial with a source at r=0? But there is no source at r=0.

I think the confusion comes because you are trying to use too "big" of a probe to measure the E-field at various points in the beam. Cosnider using a probe that is much smaller than the diameter of the beam. As you move that linearly polarized probe from the center of the beam toward the edge, you measure the Gaussian drop-off of the amplitude of the linearly polarized E-field. The polarization remains the same, just the amplitude drops off.

Like if the beam were aimed vertically in the plot below of E-field squared amplitude versus radius...

http://www.wavelength-tech.com/images/Top%20Hat%20Profile.jpg

 

[spareine]

The problem I was trying to address, as mentioned in post #9, is more elementary. It does not involve measurement probes. Multiply a linearly polarized plane wave with a gaussian beam profile. The result is that $\nabla\cdot\mathbf{E}$ is nonzero everywhere (except at the axis of the beam). That is incompatible with free space, where charges are absent.

 

[tech99]

Could you draw that field line pattern? Wouldn't it be radial with a source at r=0? But there is no source at r=0.

This is my field diagram if it opens for you:-
https://www.flickr.com/photos/58337586@N08/35537703754/in/dateposted-public/

 

[spareine]

I see, so all field lines are closed loops around several side lobes. Thanks!

 

[sophiecentaur]

. My objection against that suggestion is that it makes ∇⋅E nonzero.

It's been a long time since I used those operators but I don't think your inference can be correct. If you were right, it would mean that every antenna / radiator would have to be isotropic, I think. And none are.
I found this link and it has a diagram which accounts for what happens to the E field at the edges of the beams. It gives a ∇.E = 0 because the field vectors follow a curve in the other sense from the one you have drawn in your diagram.

 

[tech99]

I see, so all field lines are closed loops around several side lobes. Thanks!

The circles in my diagram just indicate the nominal position of the main beam and some suggested sidelobes.The sidelobes are 180 degrees out of phase with the main beam.

 

[sophiecentaur]

I see, so all field lines are closed loops around several side lobes. Thanks!

I would say that's not what is going on. If you look at the link on my last post you will see the 'loops' in the antenna fields are not in the plane of the advancing wave front but they double back at the edges of the beam and join to field lines of the previous and next wave front. It is easier to look at the coarser structure of an antenna pattern than to try to draw on the scale of light waves. That also allows ∇.E to be zero without disturbing the polarisation at the edges. The field lines spread out, signifying a lower field at the edges.

 

[anorlunda]

The electric field cannot weaken along a field line, if the electric field is vertical everywhere. ∇⋅E=0∇⋅E=0\nabla \cdot \mathbf{E} = 0 in the absence of electric charges (Gauss' law).

You can not apply classical equations to quantum phenomena. A single photon has E and B fields but zero charge. Lasers won't lase except for quantum mechanical behavior of bosons.

QED (quantum elctrodyanmics) is the real theory. With simplifying assumptions, QED reduces to Maxwell's Equations. With simplifying assumptions, Maxwell's equations yield to circuit analysis where we can apply Kirchoff's Laws. But you can't go backward applying Kirchoff's Laws or Maxwell's Equations to QED.

 

[Gordianus]

You can not apply classical equations to quantum phenomena. A single photon has E and B fields but zero charge. Lasers won't lase except for quantum mechanical behavior of bosons.

QED (quantum elctrodyanmics) is the real theory. With simplifying assumptions, QED reduces to Maxwell's Equations. With simplifying assumptions, Maxwell's equations yield to circuit analysis where we can apply Kirchoff's Laws. But you can't go backward applying Kirchoff's Laws or Maxwell's Equations to QED.

Laser modes are derived from classical electromagnetic theory without resorting to QED. Siegman, in "Lasers", derives the paraxial wave equation and shows the gaussian mode is the simplest solution to that equation. Everything within the framework of classical electromagnetism.

 

[berkeman]

It seems like we should focus on a flashlight beam for now to answer the OP's question, IMO. It's an interesting question, and my previous reply may be wrong based on the resonse @tech99 -- I'll post a follow-up in a bit...

 

[berkeman]

The circles in my diagram just indicate the nominal position of the main beam and some suggested sidelobes.The sidelobes are 180 degrees out of phase with the main beam.

Interesting, so each linear polarizer that a linearly polarized beam would pass through would add more loss than just the impedance discontinuity reflection? The 180 degree polarization shifted components are attenuated? Does passing a linearly polarized Gaussian beam through multiple linear polarizers change the Gaussian shape due to more attenuation at the outsides of the beam where you show a different E-field polarization characteristic?

Maybe I'm not understanding this phase shift that you are graphing...?

 

[tech99]

Interesting, so each linear polarizer that a linearly polarized beam would pass through would add more loss than just the impedance discontinuity reflection? The 180 degree polarization shifted components are attenuated? Does passing a linearly polarized Gaussian beam through multiple linear polarizers change the Gaussian shape due to more attenuation at the outsides of the beam where you show a different E-field polarization characteristic?

Maybe I'm not understanding this phase shift that you are graphing...?

I have shown a general case of a beam of EM radiation from an aperture. There are different sorts of illumination of the aperture, which may be dipole-like, loop-like or a mixture and they have different cross polarisation patterns. But in general we expect the field lines at the aperture to be curved due to the characteristics of the source, and this translates into the radiation far field. For linear polarisation, it creates side lobes lying in the 45 degree planes. There are also cross polarised field components in the main beam, but they cancel to zero on-axis.
If the beam is passed through a linear polariser, it will remove the cross polarised sidelobes and will also remove the cross polarised components in the main beam. So there is a small loss of energy. At large angles from the axis of the polariser, geometry appears to dictate that cross polarised components will again be observed in the 45 degree planes.
In all these studies, it is important to be careful to understand the type of axes used if large angles are involved.

 

[sophiecentaur]

You can not apply classical equations to quantum phenomena. A single photon has E and B fields but zero charge. Lasers won't lase except for quantum mechanical behavior of bosons.
QED (quantum elctrodyanmics) is the real theory. With simplifying assumptions, QED reduces to Maxwell's Equations. With simplifying assumptions, Maxwell's equations yield to circuit analysis where we can apply Kirchoff's Laws. But you can't go backward applying Kirchoff's Laws or Maxwell's Equations to QED.

Sounds a good argument on the face of it but you would have to explain where the 'limits' of your statement apply. When does a Laser Beam become indistinguishable from a tight, conventional monochromatic beam? Even a laser beam beam will have a certain coherence length and a wider bandwidth than the source at a distance.
Your claim that "Lasers won't lase" is to do with the source and not with what happens to the beam, once it has been launched. How much will the Boson behaviour dominate over the basic EM field behaviour?

linear polarizers change the Gaussian shape due to more attenuation at the outsides of the beam

Yes. The term "linear polarisation" implies a plane wave. If the wave is constrained then the edges are not 'truncated' in a simple fashion (hence the OP topic). It's a matter of geometry. I still hold that the field pattern at the edge will have the same form as the diagram in my above post (here it is)

I can see no reason why that picture is irrelevant to the situation for a very narrow beam. It shows how the fields are less at the edges without any charge being involved. So, off axis, you are seeing E fields that are not actually parallel with the field at the centre - that's cross polarisation and a linear polariser will remove some of the energy.

 

[tech99]

Can you explain in the diagram how the field arrows can sometimes point towards or away from the source when we have a transverse wave? I don't think these arrows would be detected by an E-field sensor, such as a short dipole, positioned radially from the source.Thank you.

 

[davenn]

Can you explain in the diagram how the field arrows can sometimes point towards or away from the source when we have a transverse wave? I don't think these arrows would be detected by an E-field sensor, such as a short dipole, positioned radially from the source.Thank you.

who / what are you referring to ?
quoting would have been good

 

[tech99]

who / what are you referring to ?
quoting would have been good

My apologies, not very scientific! I was referring to the diagram in Post 27 from sophiecentaur.

 

[Justintruth]

Consider a vector field E:
Ex=e^(-x^2)
Ey=0
Ez=0

What is the divergence at x=1?

Imagine a unit cube centered around the point x=1, y=0, z=0

Clearly their is no flux through the planes perpendicular to y or z.

There is flux through the planes perpendicular to x. Specifically there is flux through the plane x=1.5 and x=.5 for this unit cube.

Also the flux out of the cube is not equal to the flux into the cube as the vectors at x=1.5 are not the same magnitude as the ones at x=.5.

e^(-1.5^2) *Area of a side < e^(-.5^2)*Area of a side

Now imagine another cube around the point x=1, y=0, z=0 but instead of a unit cube it is a cube with a side length of .5

All of the facts stated for the first cube are true for the second. But the flux into this cube is less because the flux on the x=1.25 side of the cube got closer to the flux on the x=.75 side

e^(-1.25^2) *Area of a side < e^(-.75^2)*Area of a side

Now imagine another cube around the point x=1, y=0, z=0 but instead of a unit cube it is a cube with a side length of .25

All of the facts stated for the first cube are true for the third...

e^(-1.125^2) *Area of a side < e^(-.875^2)*Area of a side

see how the flux into the cube keeps going down?

Etc

If you look at the difference between the total flux entering each of these cubes and then take the limit as the process continues then the LIMIT of the flux into the cube will be zero. So the divergence is zero.

Now imagine the same electric field but with one difference. For x <=1, Ex=e^(-x^2) and for x > 1 Ex= -e^(-x^2) (note the minus sign is the difference)

Do the same process. Now the divergence is not zero at x=1 but zero everywhere else.

Just because the amplitude of the E field is a Gaussian and therefore that a cube surrounding any point on the x-axis other than zero has non-zero flux into the cube does not mean that the limit of fluxes from successively smaller cubes does not approach 0.

Does that do it? Did I answer your real question? How about this?

Let len =length of the cube / 2

Limit of e^(-(x-len)^2)-e^(-(x+len)^2) as len -> 0 =

e^(-x^2) - e^(-x^2) =0 so no divergence.

Or this:

Consider a coulomb field of a point particle. The divergence of the electric field is everywhere 0 except at the point where the charge is.

Or this:

The divergence of the gradient of a scalar Gaussian field is everywhere zero but at the peak of the Gaussian

-but-

the divergence of a vector field where the vector magnitudes are Gaussian is undefined. You need to specify the directions of the vectors to get the divergence.

 

[Spinnor]

I am trying to sketch the electric field E in snapshot of a linearly polarized laser beam. Is it correct that the E vectors bend from vertical to longitudinal near the cylindrical surface of the beam, and that all field lines within a half wave segment are closed loops?

I don't mean to resurrect an old thread, which I bumped into while Googling, but to complete the thread I thought I would point out that this is very similar to a problem from Jackson's Classical Electrodynamics, from the 1975 edition, problem 7.20, page 333,

With a little thought one should be able to convert the above solution of the fields of a circularly polarized wave into the fields of a linearly polarized wave. It is interesting to plot with Mathematica. If I still had a copy I would plot it again, anyone else willing and able? Knowing that the divergence of both the magnetic and electric fields is zero we know all field lines form closed loops.

 

[Spinnor]

If I still had a copy I would plot it again, anyone else willing and able?

Mathematica lent me a copy for 2 weeks, that was nice of them. After an embarrassingly long time I came up with a plot of the electric field of the equation above with y=0 and with the above equation converted into a linearly polarized beam. The magnetic field is the same, just rotated 90 degrees about the x axis.

Field lines end and I can not figure out how to force them to continue, seems there is some maximum density that I can not override for now. Remember the divergence of the electric field is zero with no sources, field lines should form continuous loops. Also note the overall vertical scale should be stretched so that the beam is many wavelengths wide as per problem 7.20 above. Any suggestions on how to improve the above plot? I tried way too long to use VectorPlot3D but could not get any good results.

 

[tech99]

The magnetic field is the same, just rotated 90 degrees about the x axis.

If we assume a line source as an example, I thought the magnetic lines would be concentric rings around it.

 

[Spinnor]

If we assume a line source as an example, I thought the magnetic lines would be concentric rings around it.

No sources, electric field is of a laser beam in vacuum, somewhat distorted vertically.

 

[sophiecentaur]

Can you explain in the diagram how the field arrows can sometimes point towards or away from the source when we have a transverse wave? I don't think these arrows would be detected by an E-field sensor, such as a short dipole, positioned radially from the source.Thank you.

Sorry for picking this up so late. I must have been on holiday. The crucial part of your question is "when we have a transverse wave?" The term Transverse only applies for a plane wave that's infinitely wide or in the centre of a beam. At the edges of a directed beam, the E field has to bend away from the normal to the propagation direction so it's no longer 'transverse' but has a longitudinal component.
You can show this with a dipole probe placed at various locations around an HP transmitting dipole. The theory can be seen to work.

 

[Cutter Ketch]

A laser beam is just the superposition of a very large number of dipole radiators all in phase and nearly on axis, but distributed along the axis (backwards in time from a particular point in the phase front) and also distributed transversely according to the size of the gain medium and the intensity distribution in the cavity. From the point of view of a particular point in the beam, the resonant cavity causes there to appear to be another gain medium's worth of emitters one round trip further away in time and another and another as far back as either the pulse length, or the coherence length whichever is shorter. Here are 3 dipole emitters lined up along an axis, and one wavelength out of phase with each other.

you can see that each has a dipole field where the electric field is curled in the longitudinal direction at the edges. You can also see how they add up on axis but begin to interfere off axis. In reality there isn't a single line of emitting atoms, so there is a transverse distribution, and the transverse distribution of emitters generally isn't uniform. Diffraction plus the apertures in the cavity favor modes so that the transverse intensity in the cavity isn't uniform and so the participation of the emitters is also not uniform in the transverse dimension. However, all of that can be thought about later. For right now just imagine having a bunch of dipole emitters emitting in phase stretching back in time. The field at a plane somewhere away from the laser is the superposition of the fields of many of these emitters stretching back in time for the length of the gain medium and transversely across the gain medium AND from multiple passes through the gain medium so that it looks like there was a train of emitting volumes.

The net effect is approximately a plane wave which falls off quickly in intensity from the axis as the dipole fields interfere. All of the fields within the high intensity part of the beam are approximately transverse to the beam axis and polarized vertically (assuming the emitters all were aligned, usually enforced by a polarizer in the cavity throwing away any emissions that weren't) However in detail the electric field retains some curvature and far away from the axis where interference has added up to zero amplitude, each dipole field has components of electric field parallel to the axis.

So, yes the electric field from each emitter must curl around like a dipole field, but superposition and interference allow that to be negligible in the intense part of the beam.

 

[Spinnor]

So, yes the electric field from each emitter must curl around like a dipole field, but superposition and interference allow that to be negligible in the intense part of the beam.

So what I plotted is correct or incorrect given the caveats?

 

[Cutter Ketch]

So what I plotted is correct or incorrect given the caveats?

Although you had the correct idea that Maxwell's equations won't allow an E field to be truly be 100% transverse polarized, I think I would have to say your drawing was incorrect.

To put it another way, forget about lasers. You believe dipole fields are allowed. There is no fundamental reason why a long line of dipoles all in phase shouldn't be allowed, and in that construct you can't avoid the fact that by interference the intensity is axial and the electric field is polarized and transverse. Whether or not you believe that construct represents a laser, you have to believe that it does not violate Maxwell's equations. The only possible conclusion is that the electric field can be well polarized and highly transverse everywhere within the intense region of a beam.

 

[Spinnor]

I think I would have to say your drawing was incorrect.

The function I tried to graph was taken from a problem from Jackson's Electrodynamics. He gave the result, the problem was to prove the result. I just used the result Jackson gave. Because the transverse electric field of a beam of finite width most go to zero away from the beam and using the fact that the divergence of the electric field is zero I don't think any other result works, I think the electric field must have the form that I plotted? Granted the field plot must be stretched vertically but that does not change the basic form.

 

[sophiecentaur]

There is no fundamental reason why a long line of dipoles all in phase shouldn't be allowed,

That needs to be re-stated, actually. Their phases should be progressive along the length of the array so that the passing EM wave is in step with the radiation from each element. It's the basis of many directional antenna arrays!

 

[Cutter Ketch]

That needs to be re-stated, actually. Their phases should be progressive along the length of the array so that the passing EM wave is in step with the radiation from each element. It's the basis of many directional antenna arrays!

Yes, that’s better.