How to calculate achievable flashlight beam divergence

[joema]

Assuming a typical white-light focusing flashlight (e.g, Mag-Lite) with a parabolic reflector focused to produce a convergent/divergent beam, how do I calculate the smallest achievable spot size (i.e, smallest beam cross sectional area) at a given distance?

How does this vary with reflector diameter, source size, focal length, distance to target, and beam focus (whether source is inside/outside the focal point)?

I can't just use angular field = arctan (source dia. / focal length), as the beam can be initially convergent then divergent.

Observationally it seems to involve mirror diameter since a large WWII searchlight beam converges for a long distance before it diverges.

Looking for the right formula.

 

[krab]

You are correct; it depends upon source diameter. The limit is in fact given by diffraction. See alsohttps://www.physicsforums.com/showthread.php?t=105043&highlight=beam". Roughly, the minimum angle of divergence is the wavelength divided by the source size. So it is easier to focus small wavelength (high frequency) than large. Radio waves are so large that they go all over.

 

[joema]

...Roughly, the minimum angle of divergence is the wavelength divided by the source size...

Thanks for the response, but I'm still struggling to understand this. There must be more involved than wavelength and source size, since those same parameters will produce different divergence in a 1 cm dia. penlight vs a 2 meter dia searchlight.
I'm familiar with Rayleigh criterion, but am unsure how or if it applies in determining the achievable divergence of a white light noncoherent source feeding a parabolic mirror (car headlight, flashlight, searchlight, etc). Any other suggestions?

 

[krab]

Sorry, I assumed you were asking how small you can focus it with required optics. Actually, you are asking how small given ONLY the flashlight reflector. In that case, there is a conservation law that can help.

If you plot light rays in phase space (that's position and angle), the beam coming from the flashlight filament occupies a certain area in this space. This area is conserved. Example: you have a flashlight with a 1mm filament, and a reflector that captures almost all angles of light rays leaving the filament (that's 2pi in one transverse dimension). You want to know minimum size a_min 10 metres away. At that distance, the angles are a_min/10metres. Multiply by a_min. This is equal to 1mm*2pi.

{a_{\rm min}^2\over 10\,\mbox{metres}}=2\pi\,\mbox{mm}

This gives a_{\rm min}=250 mm.

BTW, this is true only if the spot size is large compared with the reflector size. If such is not the case, then the angles are given by reflector size divided by distance.

 

[joema]

Krab, thanks for the reply, but I'm still having trouble following this.

I wanted to know how to calculate the "beam waist" size and and beam waist distance from flashlight, given source size, reflector size, focal length and focal ratio.

I studied the equation you posted, but I don't think it gives this. The only input parameters it accepts are source size and distance to target, not reflector size. Don't you at least need that? Let me know if I misunderstood.

 

[krab]

You maybe missed the last BTW sentence. In fact, spot size and reflector size add in quadrature.

 

[joema]

I've studied the equation you posted, along with "the angles are given by reflector size divided by distance", and there are several things I don't understand:

- The equation doesn't consider reflector size, only source size and distance to target. It seems a 1 mm source in a 1 meter reflector would produce different beam angles than a 1 mm source in a 1 cm reflector, even if spot size was large compared to reflector size for both cases (i.e, longer distances).

- I don't understand how to calculate the angles just from "reflector size divided by distance". E.g, a 2 cm reflector at 5 cm distance. That is 2/5 or 0.4. Is that degrees, radians? Do you take the arctan of that? Is it a divergent or convergent angle? Is it the half angle or whole angle?

- Is there a threshold between the two methods, say, if distance is 5x the reflector diameter?

- I still don't understand how to calculate the distance to and size of the beam waist. Maybe if I understood the "reflector size divided by distance" method I would.

I'm trying hard to understand this, sorry if I'm slow.

 

[krab]

Sorry if I was a little terse. Here is a fuller explanation, complete with formulas you can use directly.

Imagine you have a very small source (say the filament of a bulb) of size a. This source is at the focus of a parabolic reflector, which is a distance d from the vertex of the deflector. The deflector diameter is D. Then the spot diameter S a distance L away is

S=\sqrt{{a^2L^2\over d^2}+D^2}

If you have a maglite, you notice that you can adjust the parameter d to achieve different focusing, so you can optimize the spot size for a particular L. In that case,

S=\sqrt{{a^2L^2\over d^2}+(1-Lw)^2D^2}

where w is a focusing parameter (specifically, it is the amount by which you change d, divided by (2d)^2). Notice that this gives minimum spot size when Lw=1. In that case, the spot size is simply given by

S={aL\over d}

Both d and D are proportional to deflector size, so you can see that a larger deflector does indeed give a smaller spot.

 

[joema]

Krab, thanks for the detailed response.
I've run some numbers through the 1st equation, but it doesn't appear to accurately predict what I empirically measure. E.g, a Surefire U2 fixed focus LED flashlight.

Dimensions:
reflector dia (D): 30 mm
source size (a): 4 mm
focal length (d): about 2 mm
distance to target (L): 1000 mm

All of the above are accurate measurements except for focal length d, which can only be visually approximated.

Running the numbers through the 1st equation gives a value S (spot size) of 2000 mm, which is way wrong.

This flashlight has a measured beam angle of about 6 degrees, so at 1000 mm the spot diameter S is (tan (3) * 1000) * 2, or about 100 mm, vs the 2000 mm predicted by the equation.

To make the equation produce a value S of 100 mm requires a value d of 41 mm, which cannot possibly be correct. The reflector focal ratio is super short, probably 0.06, and visually the source is very close to the vertex. Therefore focal length d must be very short, probably about 2-3 mm.

Something about the equation isn't right, or I've made a mistake. Any ideas?

 

[krab]

No way the source size is 4mm. I looked up a few LED bulbs and the light emitting area is typically 1mm, in some cases as small as 0.06 mm.

 

[joema]

Krab, thanks for being so thorough and taking the time.

However this flashlight uses a Lumileds Luxeon V Star LED, which has an emitter die diameter of roughly 4mm. Below is the datasheet (PDF):

http://www.lumileds.com/pdfs/ds40.pdf

Also I can decrease the output of the flashlight and see the actual size of the emitter die, which is about 4mm diameter.

You're right most general purpose LEDs have vastly smaller emitter areas. However those used in powerful flashlights are typically larger, varying from about 1.5 mm to 4mm diameter (actual emitter area).

E.g, I have another one with an approx 1.5 mm die diameter Luxeon III LED, which has a reflector diameter of 19mm. It also has an approx. 6 degree beam angle.

You're also right the actual emitter surface (die) is smaller than the transparent material encasing it. However with the above flashlights I can see the actual emitter die at low output, so there's no doubt of the approx. source size.