Angleofdivergenceformula

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For a fun test, take three linear polarisers (e.g. polarised sunglasses at a store). The first set in a normal 'up' direction, the second set at right angles (across), and observe no light comes through. Now add the third at 45° at the back — still nothing, but now put that third polariser in the middle — Observe what happens, and wonder why! (Linear polarisation is not a quantum state..) See this MinutePhysics video for a demonstration.

The polarizing filter we use is actually two filters sandwiched together. The front facing filter is an ordinary polarizing linear screen. This one does the job. It mitigates reflections, cuts haze, and increases saturation. Because it darkens blue sky, white clouds are caused to stand out in bold relief. So it’s the upfront filter that does the deed.

Behind the polarizing screen is mounted a second filter called a retarder. This filter effectively de-polarizes the light. This sandwich filter is now called a circular polarizer. That’s OK because the upfront polarizing screen’s actions are not impaired and neither is the camera’s automation. If you turn this “circular” polarizer around, you effectively mitigate its ability to polarize light.

Laserbeam divergenceand spot size

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where f/2R = f/D is the f-number, f/#, of the lens. In order to make the image size smaller, we could make f/# smaller, but we are limited to f/# = 1 or so. That leaves us with the choice of decreasing R (smaller lens or aperture stop in front of the lens) or increasing s1. However, if we do either of those, it will restrict the light gathered by the lens. If we either decrease R by a factor of two or increase s1 by a factor of two, it would decrease the total light focused at s2 by a factor of four due to the restriction of the solid angle subtended by the lens.

What isbeam divergence

Understanding that my phone screen outputted polarised light, you could turn my old filter around in front of the screen and watch it go from light to dark, then light then back to dark etc... It worked both ways, so if I flipped my filter around, the effect would be the same. So I try it out with my new filter, and begin to rotate it - the effect is the same as the old one as expected. However if I flipped it over, the effect was not an increase/decrease in brightness, it actually made the screen go from warm to cool, then back to warm then to cool as you rotated it (i.e. orange to blue).

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Polarisers are fun, and display quantum mechanics effects that can confuse people. Light, and all electromagnetic waves, can be considered to have two parts, with one that is a quarter wave behind the other. If the two parts have their electric fields aligned then we have a linear polarisation (and there are many directions of alignment). If the second part has turned to the side then the overall wave effect turns in a helix (spiral). The old filter selects just for the aligned waves, while the new filters select for the ones arranged as a helix (with a particular turn direction).

So I bought a polarising filter yesterday that has a dial you can turn to increase/decrease the polarisation effect and mount directly onto the lens. I also have an old polarising filter that was one that you mounted in a rectangular filter holder. By coincidence, I'm doing polarisation in physics at school at the moment, so understand the concepts.

Beam divergenceof laser

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This application is one that will be approached as an imaging problem as opposed to the focusing and collimation problems of the previous applications. An example might be a situation where a fluorescing sample must be imaged with a CCD camera. The geometry of the application is shown in Figure 4.  An extended source with a radius of y1 is located at a distance s1 from a lens of focal length f. The figure shows a ray incident upon the lens at a radius of R. We can take this radius R to be the maximal allowed ray, or clear aperture, of the lens.

For minimal aberrations, it is best to use a plano-concave lens for the negative lens and a plano-convex lens for the positive lens with the plano surfaces facing each other. To further reduce aberrations, only the central portion of the lens should be illuminated, so choosing oversized lenses is often a good idea. This style of beam expander is called Galilean. Two positive lenses can also be used in a Keplerian beam expander design, but this configuration is longer than the Galilean design.

Now when you use the CPL in reversed orientation before a source of linearly polarised white light like a typical laptop screen, the non-perfect circular polarisation achieved after passing the quarterwave plane (which is only truly quarterwave at one particular wavelength of light) will mean that the orientation of the linear polariser behind it is not entirely unrelated to how the different wavelength portions of the white light manage to get through.

By the way, early scientists working with materials that polarized light falsely assumed that light had a + and – component, something like a magnet has a North and South Pole. They assumed the filter somehow split light into positive and negative (polarized) light beams. This was proven false but the name polarization stuck.

So best to do the phone screen trick before you buy the filter. That way you can at least tell that it has a linear component even though it says CPL on the side.

Many folk think that the linear aligned state is a 'quantum state' (i.e. countable) while in (quantum) reality it is only the two spiral states of left and right circular polarisation that are distinct countable states.

DSLRs and digital cameras with "phase detect" autofocus have problems when fed with polarised light. So a "circular polariser" or CPL filter will, after polarisation, employ a "quarterwave plate" for turning linearly polarised light (after the polarisation proper) back into something that has polarisation in two directions. This effect is not independent of the light's wavelength, so it doesn't really create fully circular polarisation for all wavelengths, but somewhat elliptical polarisation. For the purposes of phase-based focusing and metering (including analog camera TTL flash metering), this is good enough, and the photographic process itself is not affected.

You're probably comparing a linear polariser with a circular polariser. The linear polariser is a basic filter that only passes light waves polarised in a particular direction. That works either way round, and you can combine two of them to produce a variable density filter - by rotating the second polariser, it passes most of the light when the polarisation directions are the same, and a minimum amount when the polarisation directions are at 90 degrees to each other.

Lowdivergencelaser

Beam divergence angleformula

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This is a fundamental limitation on the minimum size of the focused spot in this application. We have already assumed a perfect, aberration-free lens. No improvement of the lens can yield any improvement in the spot size. The only way to make the spot size smaller is to use a lens of shorter focal length or expand the beam. If this is not possible because of a limitation in the geometry of the optical system, then this spot size is the smallest that could be achieved. In addition, diffraction may limit the spot to an even larger size (see Gaussian Beam Optics), but we are ignoring wave optics and only considering ray optics here.

The digital camera sports automation that adjusts exposure and focus. These mechanisms are likely dependent on semi-silvered mirrors. These work like mirrored sunglasses; they pass some light and reflect the remainder. When you mount a polarizing filter, it can be an impairment diminishing the effectiveness of this wonderful automation. Because mounting a polarizing filter is often desirable, we need one that will not do mischief.

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Another common application is the collimation of light from a very small source, as shown in Figure 2. The problem is often stated in terms of collimating the output from a “point source.” Unfortunately, nothing is ever a true point source and the size of the source must be included in any calculation. In figure 6, the point source has a radius of y1 and has a maximum ray of angle θ1. If we collimate the output from this source using a lens with focal length f, then the result will be a beam with a radius y2 = θ1f and divergence angle θ2 = y1/f. Note that, no matter what lens is used, the beam radius and beam divergence have a reciprocal relation. For example, to improve the collimation by a factor of two, you need to increase the beam diameter by a factor of two.

The colour shift effect is probably similar to that used in geological microscopes, where it can be used to help identify materials in thinly sliced rock samples.

Gaussianbeam divergence angle

For a digital camera with only electronic viewfinder and merely contrast-based focusing, a CPL is not actually necessary and a linear polariser will be fine.

is reduced from the original divergence by a factor that is equal to the ratio of the focal lengths |-f1|/f2. So, to expand a laser beam by a factor of five we would select two lenses whose focal lengths differ by a factor of five, and the divergence angle of the expanded beam would be 1/5th the original divergence angle.

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If you are using polarised sunglasses, you may get into trouble with the LCD screens on cameras or smartphones. When you are lucky, sunglass and screen producer have adopted polarisation conventions that differ by 45°, making the combination work reasonably either in landscape or portrait mode. Even better would be a quarterwave plate on the LCD or the polariser glasses. But I don't think that's customary.

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As an example, consider a Newport R-31005 HeNe Laser with beam diameter 0.63 mm and a divergence of 1.3 mrad. Note that these are beam diameter and full divergence, so in the notation of our figure, y1 = 0.315 mm and θ1 = 0.65 mrad. To expand this beam ten times while reducing the divergence by a factor of ten, we could select a plano-concave lens KPC043 with f1 = -25 mm and a plano-convex lens KPX109 with f2 = 250 mm. Since real lenses differ in some degree from thin lenses, the spacing between the pair of lenses is actually the sum of the back focal lengths BFL1 + BFL2 = -26.64 mm + 247.61 mm = 220.97 mm. The expanded beam diameter

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Unfortunately some manufacturers don’t understand anything about them which is why your circular polarising was mounted in a rotating bezel when there is no point in rotating it.

But there's also a polarising effect when light reflects off glass or water at an angle (how much varies with angle). - that's why rotating a polariser can often reduce reflections from water or glass.

Beam divergence anglecalculator

If s1 is large, then s2 will be close to f, from our Gaussian lens equation, so for the purposes of approximation we can take θ2 ~ R/f. Then from the optical invariant, we have

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It is often desirable to expand a laser beam. At least two lenses are necessary to accomplish this. In Figure 3, a laser beam of radius y1 and divergence θ1 is expanded by a negative lens with focal length −f1. From Applications 1 and 2 we know θ2 = y1/|−f1|, and the optical invariant tells us that the radius of the virtual image formed by this lens is y2 = θ1|−f1|. This image is at the focal point of the lens, s2 = −f1, because a well-collimated laser yields s1 ~ ∞, so from the Gaussian lens equation s2 = f. Adding a second lens with a positive focal length f2 and separating the two lenses by the sum of the two focal lengths −f1 +f2, results in a beam with a radius y3 = θ2f2 and divergence angle θ3 = y2/f2.

Unfortunately, that can cause problems when it happens in the metering/AF systems of a camera. So the manufacturers came up with the idea of circular polarisers - these are more expensive than plain linear polarisers, and combine a linear polariser (at the front) with a second layer that converts the linearly polarised light into circular polarised light - which doesn't suffer from the reflection effect. But that means they only work like linear polarisers one way round. (For details of how the second layer - a quarter wave plate - works, see https://en.wikipedia.org/wiki/Waveplate).

Since a common application is the collimation of the output from an Optical Fiber, let’s use that for our numerical example. The Newport F-MBB fiber has a core diameter of 200 µm and a numerical aperture (NA) of 0.37. The radius y1 of our source is then 100 µm. NA is defined as sine of the half-angle accepted by the fiber, which is approximate to the half-angle, so θ1 ≈ 0.37 rad. If we again use the KPX043 , 25.4 mm focal length lens to collimate the output, we will have a beam with a radius of 9.4 mm and a half-angle divergence of 4 mrad. We are locked into a particular relation between the size and divergence of the beam. If we want a smaller beam, we must settle for a larger divergence. If we want the beam to remain collimated over a large distance, then we must accept a larger beam diameter in order to achieve this.

As a numerical example, let’s look at the case of the output from a Newport R-31005 HeNe laser focused to a spot using a KPX043 Plano-Convex Lens. This Hene laser has a beam diameter of 0.63 mm and a divergence of 1.3 mrad. Note that these are beam diameter and full divergence, so in the notation of our figure, y1 = 0.315 mm and θ1 = 0.65 mrad. The KPX043 lens has a focal length of 25.4 mm. Thus, at the focused spot, we have a radius θ1f = 16.5 µm. So, the diameter of the spot will be 33 µm.

The use of linear polarisers in quantum cryptography can fool people into thinking they have broken the laws of probability (see Bell's Theorem), when in fact it is just that the probability calculation are done on a circle not a square (those 45° linear polarisers are not a classic 50:50 selection!)

As a first example, we look at a common application, the focusing of a laser beam to a small spot. The situation is shown in Figure 1. Here we have a laser beam, with radius y1 and divergence θ1 that is focused by a lens of focal length f. From the figure, we have θ2 = y1/f. The optical invariant then tells us that we must have y2 = θ1f, because the product of radius and divergence angle must be constant.