# From power of a laser beam to electric field amplitude

In my experiment, I use a laser beam with wavelength $\lambda=894 \text{nm}$ for some magnetic resonance experiment. Right now, I'm doing some calculation using Quantum Mechanics, which requires the amplitude of the electric field to be inserted to my Hamiltonian.

My question is basically: How do I go from a power, measured in $\mu W$ to an Electric Field amplitude $E_0$ to use it in a QM's Hamiltonian?

So I think that I need to use the Poynting vector, which is defined as:

$$\vec{S}=\frac{1}{\mu_0} \vec{E}\times \vec{B}$$

where I'm using $\mu_0$ because the interactions happen in a vacated, dilute gas cell. If I take the magnitude of the last equation I get

$$S=\frac{EB}{\mu_0}=\frac{E^2}{c \mu_0}$$

where I used the fact that $E_0=c B_0$ according to Maxwell equations.

However, the Poynting vector is energy per unit area per unit time, or power per unit area. This means that I have to know the area of my laser beam, and that's my problem now.

The laser beam in my experiment comes out of a multi-mode fiber, and then it goes to a linear polarizer, and then a quarter wave plate, where we use the latter to get circularly polarized light, and then a collimator lens. And after that it interacts with my atoms.

I know the core-diameter of my multi-mode fiber, let's call it $r_F$, but I don't know what transverse modes are produced in it. How can I know the effective area of my laser beam (call it $A_L$), so that I could substitute that in my Poynting vector and calculate the electric field amplitude of my laser beam?

I imagine that eventually I should be able to calculate $A_L$, and I can use the following relation to get my electric field amplitude, which I get from the Poynting vector definition

$$E=\sqrt{\frac{\mu_0 c S}{B}}=\sqrt{\frac{\mu_0 c P}{B A_L}}$$

where $P$ is the power of my beam.

Is my reasoning correct? Please advise. If I'm missing something in the problem, let me know.

Please note: this is a real experiment, so all practical approximations are allowed.

• Assume constant illumination over the area $A$ of the fiber, and $P=SA=E^2A/c\mu_0$? Even for a strongly peaked power-vs.-area distribution this should get you with a factor of two or so. How much precision do you need?
– rob
May 19 '14 at 11:59
• @rob That won't account for whatever happens to the beam after it leaves the fiber. The beam will spread by diffraction. May 19 '14 at 12:07
• I'm not sure about what is really your question but is your problem you don't know how to measure a laser spot width? You are using a gaussian beam, right? May 19 '14 at 12:12
• @dolan I'm not sure about it being completely Gaussian because I'm using a multi-mode fiber and Gaussian beams imply only $\text{TEM}_{00}$, right? May 19 '14 at 13:06
• Hmmm yes, I think it's surely true with good approximation, I guess... So if your problem is "how to measure a laser spot?" (in order to get the lighted surface) I know about one method. Otherwise, I don't know. But your reasoning seems ok for me... May 19 '14 at 13:24

Ok, so since your reasoning seems to be ok, the real question is : "How can one measure experimentally a laser spot width?".

In your case, since you are using a gaussian beam, an equivalent is "How to measure the waist of my laser?"

If you are working in a wealthy lab, the simpliest way seems to buy and use a CCD camera. Otherwise, an other "easy peasy lemon squeezy" (and cheaper) method is possible.

For this, you will need :

• a linear translation stage (for optic tables) with a graduate spanner
• a photodiode
• a piece of black cardboard

Step 1 : Align the photodiode with the laser.

At this point, be sure to get the maximum power from your laser.

Step 2 : Attach the piece of cardboard on the translation stage.

This blocks the laser light so that it can't reach the photodiode.

Step 3 : Place the all thing between your fiber opening and the photodiode. Step 4 : Move the translation stage with the spanner and measure the power collected by the photodiode in respect of the displacement (using the graduation).

Since your beam is gaussian, what you expect is an error function because the power that you are measuring is simply the integral of a gaussian... $$\mathcal{P}\sim\int \exp \left[ - \left(\frac{2(x-x_0)}{w}\right)^2 \right] \mathrm{d}x\;\sim \mathrm{erf}\left(\frac{x-x_0}{w/\sqrt 2}\right)$$

where $w$ is the waist.

Step 5 : Use your favorite software and do a fit. Then you have $w$.

Note that this method is not that precise. But it gives a good order of the waist, thought.

Can you simply measure the size of the spot? Your experiment is in a vacuum, so you would have to mock up the relevant parts outside of the chamber.

You can estimate the spot size by assuming the light coming out of the fiber simply diffracts as from a circular aperture, or practically equivalently, assuming a Gaussian profile (which it is not).

It really depends on how well you need to know the area/amplitude. Without knowing the actual beam profile, you aren't going to get an accurate number for the electric field amplitude, or the "area", whatever you choose that to mean, in any event. Taking a SWAG, I would guess that approximating by a Gaussian will get you within a factor of two on axis, but it can be way off in the tails.

You might try several different approximate solutions and see how they differ. If you can tolerate the spread of results, then you are good to go. If not, I can't think of anything short of actually measuring the beam profile.

I agree with dolan's answer with a few modifications.

Firstly, the CCD method is a good way to do this, but only if you are careful not to saturate the camera and FIT THE RESULTS don't just estimate a size from an image. Although the spot may have an apparent size in the image, the apparent size will be completely dependent on the contrast of whatever display system you are using.

I prefer the other method. However, I would not use a black piece of cardboard because lighting fires in an optics lab is a bad idea (black=absorbing; cardboard=flammable) and because the edge of a piece of cardboard is not very well defined on the micron scale. Instead I would use a razor blade. It has a sharp edge and it will not light on fire. If you start ablating the razor blade like I have in the past, you will get very wonky results. The only way I found to solve this problem is to use something thicker and take the hit on the accuracy of the spot size measurement. If anyone has a better solution I'd love to hear it.