Lowdivergence laser

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A photon has a wave function that is a solution of the Schrodinger equation. Like Maxwell's equations, this is a wave equation. A photon in a cavity has the same boundary conditions as the electromagnetic wave in the same cavity. The photon's wave function is also a radially symmetric function with spherical wavefronts and a Gaussian profile.

Laser beam divergenceand spot size

There is also a quantum mechanical explanation. The simplest quantum mechanical explanation invokes the Uncertainty Principle.

The simplest description of a laser beam uses ray optics. Often it is a good approximation. In it light is a ray that follows a straight line. According to this description, there need be no divergence. This description is too simple.

A better quantum mechanical explanation shows the classical explanation is the same thing in disguise. See Interesting relationship between diffraction and Heisenberg’s uncertainty principle?

Laser beam divergenceangle formula

In order to capture the behaviour of the "central part" of a wavefront we use approximation and omit the edges to a certain extend. In the upper picture we might describe the central part as a plane wave. If instead we use spherical mirrors to generate a propagating wave, we end up with the Gaussian beam $$ E \propto exp\left( - \frac{r^2}{w_0^2 (1 + (z/z_R)^2)} \right) $$ If we include the quadratic phase correction for the wavefront and the Gouy phase the approximation improves. However, the Gaussian beam is always an approximation obtained by omitting the edges of the wave (in deriving it, we use the paraxial Helmholz equation).

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So, we can say a laser beam diverges because of fundamental physics and the nature of the spatial superposition of the coherently emitted photons from a laser.

Imagine a beam with a uniform amplitude across the cross section. The beam consists of photons. The photons pass through a circular aperture, which confines the beam cross section to a limited $\Delta x$. Because $\Delta x \Delta p \ge \hbar$, the photon must have a non-zero momentum in the direction perpendicular to the beam. The beam cannot be perfectly collimated.

Huygens principle states that the propagation is due to the generation of spherical waves, which will generate spherical waves in the next step of propagation. [Picture taken from wiki]

For example, if Millimeter (mm) value is 24 and the ppi value is 96, then Pixel (px) value will be Pixel=24*(96/25.4)=90.708661417323.

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Beam divergenceangle

The wave function is in the position basis. You take the Fourier Transform to convert to the momentum basis. The Fourier Transform of the Gaussian cross section is a Gaussian cross section. The transverse momentum of the beam is a superposition of non-zero momentum states. The beam cannot be perfectly collimated. It has the same divergence as the electromagnetic wave.

I have seen it said that diffraction causes laser beam divergence, or that a laser beam will always diverge, due to diffraction, or some variation of these statements. I understand diffraction in general, and I understand that the phenomenon applies to all waves, so I understand that it would also apply to laser beams; but it is not clear to me how it causes laser beam divergence, or why a laser beam will always diverge, due to diffraction. When trying to research to understand how diffraction causes laser beam divergence, I can't find anything that directly and clearly explains this – most results either just mention diffraction in the context of lasers without providing explanation, or mention 'diffraction-limited beams', which I think is something different to what I'm asking. So how does diffraction cause laser beam divergence, and why will a laser beam always diverge, due to diffraction?

A better description is light as a wave. To get the true beam, you must solve the classical Maxwell's equations with a boundary condition. The optical cavity of a laser must have curved mirrors to be stable. The wave solution for a cavity bounded by spherical mirrors is a Gaussian Beam. Wavefronts are spherical. "Rays" are not quite straight, but follow hyperbolic paths. The beam cross section is Gaussian. The intensity is maximum at the beam axis and falls off smoothly away from the axis.

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Laser beam divergencemeasurement

The key point is that a laser beam is a wave which propagates according to Huygens principle. Once you accept this fact the divergence follows naturally.

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Consider that each point in a wave propagates out in a circular fashion from that point (like a water ripple out from what a stone falls into the water). If a point next to that point also propagates out with the same phase (the peaks and valleys oscillate together), the two circular waves will add together to give a composite wave. As the line of “emitters” increases, the wave starts to look like a planar wave, but the edges will still propagate outwards. As the “beam” of the wave gets wider and wider from the emitters, the net effect is that the spread is less and less. It doesn’t matter if this is a light wave from a laser, a water wave, or a slit in a quantum experiment, the result is the same.

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The laser beam source is always of finite dimensions, thus it cannot provide a truly parallel beam. The laser beam is sligtly divergent by its origin.

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Laser beam divergencecalculator

I will answer this only in terms of diffraction since that is the fundamental limit of laser beam divergence. Diffraction is the spreading of the beam because of the finite width of the beam. It is a fundamental of physics. Even the uncertainty principle is an aspect of the same thing. I.e, the more you confine the location of a particle, the less you know about its direction. In a way, this all comes down to wave mechanics.

Laser beam divergenceangle calculator

This is enough to tell you that a small diameter beam will have a large divergence. If you focus a beam to a small spot, it will have a very small waist. Therefore it must have a large divergence angle.

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To my knowledge, all electromagnetic waves diffract. Since laser is a highly coherent monochromatic light created from stimulated emission, the wavelengths are all the same and the troughes overlap with the troughes and crests with the crests. This means that it would follow a perfect path of diffraction through a gap without the waves themselves cancelling each other out.

Laser beam divergenceformula

https://www.gentec-eo.com/blog/laser-beam-divergence-measurement#:~:text=Simply%20put%2C%20it%20tells%20you,beam%20on%20an%20infinite%20distance.

In the image we see that the center of the "hole" generates a "flat" wave. The diffraction is evident only in at the edges.

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In practice, the solution to Maxwell's equations has a Gaussian cross section. Apertures are carefully chosen large enough to not significantly distort the beam by truncating the edge. Even though not physically confined, the beam cross section is confined because of the Gaussian profile. The beam cannot be perfectly collimated because of the Uncertainty Principle.