To be able to realize space-based power beaming, it is necessary to minimize the divergence of laser beams over long distances. Unfortunately, the **diffraction limit** means that even an idealized single laser beam will diverge at a rate proportional to its wavelength and inversely-proportional to the initial beam radius $w_0$, as given by: $ \theta = \frac{\lambda}{\pi w_{0}} $ > **Note:** see [[Ideal laser beam divergence]] for more details and a derivation of this result. For most lasers, we can make the approximation that $w_0 \approx a$, where $a$ is the aperture size of the laser. This means that lasers with small apertures (and thus initially narrow beams) are at a major disadvantage, as their divergence $\theta$ is incredibly-high, even in the ideal case. To get around this issue is a difficult challenge; after all, we certainly cannot break the laws of physics! However, we can "cheat" by taking inspiration from [interferometry](https://en.wikipedia.org/wiki/Interferometry). At Project Elara, we have developed a technique - one inspired by existing technologies, but to our knowledge is novel (or at least developed independently) in its specific form - that allows us to functionally focus a beam to narrower than the diffraction limit, while not breaking it _per se_. The key to this technique is **interference**. Like all electromagnetic waves, Gaussian beams can interfere with each other. A superposition of two Gaussian beams from sources (i.e. lasers) located at different positions can *constructively* and *destructively* interfere, giving us a combined beam that has peaks and troughs. We will call this combined beam the **synthesized beam**, and the net effect of the superposition is that the power is concentrated between within a central lobe at the center of the beam, as shown below: ![[synthesized-beam-profile.png]] _The computed intensity profile of the synthesized beam from two sources spaced 2 meters apart. The calculations are present in `notebooks/gaussian-beam-calculations.ipynb`_ Indeed, we will show that the synthesized beam has a divergence angle given by: $ \theta = \frac{1}{z}\sqrt{ \frac{\lambda}{2}z } $ Which is a result that is *completely independent* of $w_0$, meaning that it applies for any two lasers, even ones with small apertures! Indeed, it is this result - if experimentally verified - that will ultimately allow Project Elara's space power beaming to be made practical. We will now show how we derived this result. > **Note:** The synthesized beam does not break the diffraction limit because it only alters the beam's **intensity profile**, not the divergence of the individual beams themselves, or of the combined beam either. ## Mathematical derivation The Gaussian beam is given (in cylindrical coordinates) by: $ E(r, \phi, z) = E_0 \frac{w_0}{w(z)} \exp\left(-\frac{r^2}{w(z)^2}\right) \exp \left(-i\left(kz + \frac{kr^2}{2R(z)}\right)\right) $ Where we assume that the optical axis (the axis along which the beam propagates) is the $z$ axis, while $r$ is the radial distance from the center of the beam. We consider two Gaussian beams separated by distance $L$ that are aimed at a common spot far away from both lasers. The respective beams then take the forms: $ \begin{align*} E_1 = E(r_+', z) \\ E_2 = E(r_-', z) \end{align*} $ Where each beam is pointed at angle $\theta_0$, the respective positions of the lasers are at $r = r_\pm$, and: $ r_\pm' = z \sin \theta_0 \pm \left(r \mp \small\frac{L}{2}\right)\cos \theta_0 $ However, since the two lasers are aimed at a spot that is very far away, their beams must be near-parallel to the optical axis; otherwise the two beams would cross long before they reach their intended target. This tells us that $\theta_0 \approx 0$, and thus: $ r_\pm' \approx r \mp \frac{L}{2} $ The synthesized beam then takes the following form: $ \begin{align*} E(r,z) &= E_{1} + E_{2} \\ &= \frac{E_{0} w_{0}}{w(z)}\left[ e^{- i \left(\frac{k \left( r + \frac{L}{2} \right)^{2}}{2 R{\left(z \right)}} + k z\right)} e^{- \frac{(r + \frac{L}{2})^{2}}{w^{2}{\left(z \right)}}} + e^{- i \left(\frac{k \left( r - \frac{L}{2} \right)^{2}}{2 R{\left(z \right)}} + k z\right)} e^{- \frac{(r - \frac{L}{2})^{2}}{w^{2}{\left(z \right)}}} \right] \end{align*} $ In the far-field limit, where $z \gg L$, then it is possible to use the approximation $r \pm \frac{L}{2} \approx r$. Thus $(\pm \frac{L}{2} + r)^2/2R(z) \approx r^2/2R(z)$ and $(\pm \frac{L}{2} + r)^2/w(z)^2 \approx r^2/w(z)^2$ so the sum simplifies to: $ E(r,z) = \frac{2E_{0} w_{0}}{w(z)} \exp\left( -i\left( \frac{kr^2}{2R(z)} + kz \right) \right) e^{-r^2 / w^2(z)} $ Now, to find the physical field from the complex-valued solution, we take the real part of the electric field and discard the imaginary part, giving us: $ E(r, z) = \dfrac{2E_0 w_0}{w(z)} \cos \left(\frac{kr^2}{2R(z)} + kz\right) e^{-r^2/w(z)^2} $ We can then find the intensity profile of the beam as follows $ I = \dfrac{|E(r, z)|^2}{2\eta} = \left[\frac{2E_0 w_0}{w(z)} \cos \left(\frac{kr^2}{2R(z)}\right) e^{-r^2/w(z)^2}\right]^2 $ Where $\eta \approx 377\ \Omega$ is the _impedance of free space_ (or more generally, the impedance of the medium, but we are considering only vacuum for now). We can simplify this by defining: $ I_0 = \frac{2}{\eta}\left(\dfrac{E_0 w_0}{w(z)}\right)^2 $ Thus our final result is: $ I(r, z) = I_0 \cos^2 \left(\dfrac{kr^2}{2R(z)}\right)e^{-2r^2/w(z)^2} $ Recall that $\cos^2(\theta) = 0$ at $\theta = n\pi/2$ where $n$ is an odd integer. Solving for $\frac{\pi}{2} = \frac{kr^2}{2R(z)}$ with the far-field approximation $R(z) \approx z$ gives us the positions of the first minima, which are located at the edge of the central lobe. This therefore tells us that the radius (half-width) of the **central lobe** is given by: $ r_{min} = \sqrt{\dfrac{\lambda}{2}z} $ > **Note:** The diameter $d$ of the central lobe is simply twice this, that is, $d = 2r_{min} = \sqrt{ 2\lambda z }$. We can verify that this is correct with a simple plot, where we combine the analytical (approximate) solution we derived from evaluating the exact solution for a Gaussian beam numerically: ![[synthesized-beam-profile-2.png]] _Location of the first minimum is given by the black dashed line. Note how it corresponds perfectly with our result._ We may then calculate the divergence angle $\theta$. Since we have $\tan \theta = r_{min}/z$ and $\tan \theta \approx \sin \theta \approx \theta$ for large $z$ (and thus the small-angle approximation applies) then we have $\theta \approx r_{min}/z = \sqrt{\dfrac{\lambda}{2}z}/z$. Note how this result is remarkably **completely independent** of the aperture widths of the two individual lasers. We have thus reached our desired result: creating a **single synthesized beam** that has a divergence angle that does not depend on the apertures of the individual lasers used to create it. The percentage $r_{p}$ of power concentrated in the central beam lobe at distance $z = D$ is given by: $ r_{p} = \frac{\displaystyle \int_{-r_{min}}^{r_{min}} I(r, D) dr}{\displaystyle \int_{-\infty}^\infty I(r, D) dr} $ We find that in the limit $z \to \infty$, $r_p \to 1$, meaning that as we go further and further from the source, more and more power is concentrated in the central lobe (although the width of the central lobe does also grow proportional to $\sqrt{z}$). This is very useful because it means that the beam's power falls off quickly outside the central lobe, meaning that power transmission is much safer, and thus we can use ground receivers that are fairly small (and easy to deploy); even if we are unable to capture the full width of the beam, the > **Note:** Please see [this interactive demo](https://www.desmos.com/calculator/yvajmnrnhx) showcasing the results we have discussed so far. ## Other formulae for similar beams We can combine the results we arrived for coherent beam-combining with some classic formulae for diffraction-limited beams. For instance, the minimum-diffraction beam that can be made by a perfect lens forms an [Airy disk](https://en.wikipedia.org/wiki/Airy_disk). The corresponding formula for its divergence is given by: $ \theta_{\text{airy}} \approx 1.22 \frac{\lambda}{d} $ Where $d$ is the diameter of the aperture. Note that for short wavelengths ($\lambda \ll 1$) the formula we derived for the Gaussian beam and that for the Airy disk agree quite closely[^1]; this is not surprising, however, given that Airy disks are very similar in principle to Gaussian beams. This formula is also applicable for [astronomical interferometers](https://en.wikipedia.org/wiki/Astronomical_interferometer) used for [Very-Long Baseline Interferometry](https://en.wikipedia.org/wiki/Very-long-baseline_interferometry), although in some cases the factor of $1.22$ is often dropped to just one, giving the formula[^2]: $ \theta_{\text{VLBI}} \approx \frac{\lambda}{B} $ Where $B$ is the *baseline distance* (the separation between individual telescopes). Unlike our result, neither of these formulas depend on the distance $z$ from the source; however, they are quite similar and illustrate a common property of beam divergence, albeit in different contexts. [^1]: Interestingly, the results for the Gaussian beam and Airy beam roughly agree up to to around $\lambda = 0.5$ (in other words, up to $\text{50 cm}$), which is well into the microwave range. You can check this by plotting the value of $f(\lambda) = |\sqrt{\lambda/2} - 1.22 \lambda|$ (the factor of $d$ or $L$ is unimportant because $d$ and $L$ are mathematically equal, they just represent different physical things). [^2]: This formula is taken from the [Wikipedia article on angular resolution](https://en.wikipedia.org/wiki/Angular_resolution#Telescope_array). It appears that astronomers are satisfied with an order-of-magnitude result for the angular resolution.