# How do you calculate the intensity of light around the focal point from a focused collimated beam of light?

Problem/Purpose of me asking this question to you people who know more than me:

So I'm doing a science project where I'm collimating a beam of light to a focus point in a light medium (water vapor or fog) and I want to calculate the intensity near it. I can't seem to find an equation that describes this problem. I want to know two things. If you know anything that can be related in solving this problem, it is much appreciated! :)

Issues/Things I need help to figure out!:

(1) If I focus a collimating beam of light with a lens, (say a hand held magnifying glass), into a relatively uniformly dispersed light medium, (water vapor or theatrical fog) Can this focus point be seen in ANY direction, (say like 5 feet away from the focus point)? Doesn't light scatter isotropically in this case? IF NOT, What is the preferred direction of scattering of the light?

(2) If an equation exists (And light scatters isotropically in the light medium used), can this equation say given the parameters, (light frequency used, index of refraction of medium, density of medium, size of collimated beam, lens dimensions used, etc) give the intensity of light in terms of the distance a person is away from the focal point? I am aware of the inverse square law, but in my case, its a bit different, isn't it?

Wouldn't my situation involve some type of directionality? How do you find the best viewpoints from the focal point in which the intensity of the focal point is most profound?

MORE relevant or related questions that need to be addressed:

• Does the particle size matter?, (the particles that make up the light medium)
• How do I determine the correct density of the given medium to produce the most profound effect, (Having the focal point illuminate as brightly as possible)
• How do I determine the right intensity of the initial column of light that is focused?
• Combining the previous two bullets, How do I determine the right combination of the density of light medium and intensity of light used to illuminate the focus point as bright as possible?

The Gist:

What I'm trying to do is to create a "point of light" inside a suspended light medium (Ideally viewable in all directions) and I'm trying to figure a way to figure this out with equations before buying a whole ensemble of things (Fog machine, light source/laser, magnifying glass or multiple lenses, etc) to test it. (If not viewable in all directions, and it has directionality, then I'm just gonna combine multiple systems pointing multiple columns of focused light in different directions to the same point in space to obtain an acceptable looking "point of light" in a medium).

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unless you have a very wide aperture you at least run into the problem that an observer looking into the focal point from the other side will get the light not absorbed from the medium into her eyes. ideally you want weak light coming from as many angles as possible to intersect at the desired point. – BjornW Jun 20 '11 at 9:40
Have you thought of holography? gametrailers.com/user-movie/real-life-3d-holograms/162171 – anna v Jun 20 '11 at 13:13
You could use a powerful enough laser to ionize the air inside the focal volume... (please don't try this at home) – ptomato Jun 20 '11 at 18:11
I have thought of holography, but I don't want to make an image that is confined within a rotating screen or something that can hurt someone. – QEntanglement Jun 20 '11 at 18:25
3D holograms do not need screens. The suggestion of Colin K in his answer seems the best for your problem. Laser light through a fiber optic coming from the ceiling into a suspended crystal ball with appropriate opacity so show as a point source. – anna v Jun 21 '11 at 4:30

1. The scatter direction depends on the size of the particle and the wavelength. Very small particles (e.g. nitrogen atoms of the atmosphere) scatter isotropically. There is still an effect on the polarisation of the scattered light (Bees use that to locate the sun if they can't see it directly).

2. Often Gaussian beams are used to describe how the intensity propagates in an optical train (a system of lenses and mirrors) http://en.wikipedia.org/wiki/Gaussian_beam. Note that this describes the intensity of a Laser with a Gaussian intensity profile (this is a good approximation for many lasers, especially if you focus them through a pinhole). If you have an extended light source you will have to add the intensities of several such beams.

Once you have a numeric intensity profile (I guess 2D is enough for your usage), you can try an exponential decay law to estimate the effects of scatter.

Your focus spot will look more intense the higher the numerical aperture of your lens is. If you use a lens that has 2.5 cm diameter and f=10cm the spot won't look as intense as with a lens that has f=3cm.

Have you thought of using fluorescence? You could use dissolve some colour in water and use laser protection goggles to see the fluorescent light. Then you don't have to cope with scattering.

You can get polygonal mirrors out of old laser printers. That way you can scan the beam in one direction. If you use a laser diode, you can modulate the intensity very fast.

I recently purchased a 405nm Laser with 120mW for \$120 from lasever.com. 120mW is very dangerous. If you don't have protection goggles or you share your space with other people don't use lasers>0.5mW!

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Anything you do involving a focused beam of light is just not going to scatter the way you would like it to. If you are using a fog machine to scatter the light, then it will make the focusing beam just as visible as the focal spot itself.

If you are attempting to produce a point of light that seems to float in space, maybe you could consider some sort of fiber optic solution. Perhaps you could illuminate a glass bead with light from a fiber. Get the glass bead sandblasted so it has a rough finish, and it will scatter light (relatively) uniformly into all directions.

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What experiments have shown that light cannot scatter in the way that I would like? I have seen two intersecting spot lights in the sky in cities at night before and where the light intersects shows a region that is brighter. I think at the focal point, light beams are intersecting too! Why won't this work? – QEntanglement Jun 20 '11 at 18:28
-1: I am sorry, but this is not true. Two photon microscopy is an extreme example, but even with just focusing, the scattering at points away from the focus might be invisible, while at the focus it is visible. – Ron Maimon Jul 23 '12 at 7:48
Lol, are you joking? Two photon effects? Really? The intensity will increase linearly as you approach the focus, so you will never be able to produce a point. A cone maybe. – Colin K Jul 23 '12 at 15:02