An industrial carbon dioxide laser produces a beam of...

An industrial carbon dioxide laser produces a beam of radiation with average power of $6.00 \mathrm{~kW}$ at a wavelength of $10.6 \mu \mathrm{m}$. Such a laser can be used to cut steel up to $25 \mathrm{~mm}$ thick. The laser light is polarized in the $x$ -direction, travels in the positive $z$ -direction, and is collimated (neither diverging or converging) at a constant diameter of $100.0 \mu \mathrm{m}$. Write the equations for the laser light's electric and magnetic fields as a function of time and of position $z$ along the beam. Recall that $\vec{E}$ and $\vec{B}$ are vectors. Leave the overall phase unspecified, but be sure to check the relative phase between $\vec{E}$ and $\vec{B}$.

Key Concepts

Relationship Between Electric and Magnetic Fields

In electromagnetic waves, the electric field (E) and the magnetic field (B) are perpendicular to each other and to the direction of propagation. They are in phase, meaning that their time and spatial variations reach maxima and minima simultaneously. Moreover, their magnitudes are related by the speed of light (c) in the relation B0 = E0/c, where E0 and B0 are the amplitudes of the respective fields.

Polarization

Polarization refers to the direction in which the electric field oscillates. In an electromagnetic plane wave, the electric field oscillates in a direction perpendicular to the direction of propagation. The specification of the electric field being polarized in the x-direction means that its oscillations occur along that axis, establishing the plane of polarization for the wave.

Collimation and Propagation

Collimation implies that the beam does not exhibit any divergence or convergence, meaning it maintains a constant cross-sectional diameter over the distance of propagation. This property is essential when describing realistic laser beams in applications, and it simplifies the description to that of a plane wave, where the fields only depend on the coordinate along the direction of travel and time.

Electromagnetic Plane Waves

Electromagnetic plane waves are solutions to Maxwell's equations in free space where the electric and magnetic fields oscillate sinusoidally and propagate in a fixed direction. These waves are idealizations where the field amplitudes are uniform across planes perpendicular to the direction of propagation, allowing them to be described by simple cosine or sine functions of time and space.

Wave Parameters

The wave number (k) and angular frequency (?) are fundamental in characterizing the spatial and temporal periodicity of the electromagnetic wave. The wave number is defined as 2? divided by the wavelength (?), and the angular frequency is related to both the wave number and the speed of light by ? = ck. These parameters appear in the phase term (kz - ?t) that describes the propagation of the wave.

Transcript

00:01 Suppose we know some things about the power in a collimated beam of light, say, from a laser.

00:08 So we know the power in the beam, and we know the diameter of the beam d.

00:15 And we also know the wavelength, let's say, just for the sake of argument, that the wavelength is known as well.

00:25 And we would like to come up with expressions for the instantaneous electric and magnetic field.

00:32 In the electromagnetic wave that is carrying that energy.

00:39 How would we do that? first of all, we need to know the direction of travel of the beam.

00:46 Okay, so we are going to write down the electric field and the magnetic field, but if we know the direction of travel of the beam, then we basically know the direction of the s factor, the pointing factor.

01:02 So let's pretend that we know that that is in the z hat direction.

01:10 And so s is also in the z direction.

01:17 That is the direction of energy flow, which is perpendicular to both e and b.

01:33 And it is expressions for those that we are trying to find.

01:38 So one way to satisfy this would be if e is, polarized in the x direction.

01:49 Remember that polarization typically means the direction that the electric field is pointing.

01:57 So if we were to draw the electric field inside of this wave, it would go along the x direction and then switch to the minus x direction.

02:14 And then in order to be traveling in the positive z direction, the magnetic field would then be pointing in the y direction, which according to my drawing first comes out of the plane, and then it goes into the plane.

02:48 So yes, the cross -product between those two points to the right.

02:58 So x cross -y gives us c.

03:02 Now, if we know the power in the beam and the diameter of the beam, we configure the time -average pointing vector.

03:13 Average s, and it's both time - and space -average, is equal to the power in the laser, for example, divided by the area of the beam.

03:26 And here we are assuming a circular beam cross -section.

03:30 That's a little bit of a simplification, but we'll go ahead and do that.

03:37 And here we know the diameter, so we can figure out the area from pi r squared.

03:51 And that time averaged s is equal to, in terms of watts per meter, quite big.

04:11 Let me check my number.

04:27 Try it again until i get the same answer twice.

04:35 And let me see if i have the diameter.

04:40 It's 100 micrometers, yes.

04:42 That is an issue.

04:45 A hundred micrometers in diameter.

04:56 So the beam radius will be 50 times 10 to the minus 6 meters.

05:02 There we go.

05:11 Always good to verify that.

05:12 And so this is about 7 .64 times 10 to the 11th watts per meter squared.

05:23 And from that, we can use this to find the amplitude of the electric field.

05:36 That is the maximum amplitude, not the rms amplitude.

05:41 And then we can simply find the amplitude of the magnetic field by taking that quantity and dividing by the speed of light.

05:55 So yeah, solving we have e knot is the square root of 2 times s average over c times the permittivity of free space...

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