Focusing laser or maser beams over long distances requires large aperture sizes, up to kilometers in diameter, that would be prohibitively difficult to physically create. An alternative is to use what is known as a **synthetic aperture**, where several, much smaller maser beams are coherently combined to form an _effective_ aperture of large size. Here, we will delve into the mathematics and theory to discuss how synthetic apertures work. > **Note:** In this section, we use cylindrical coordinates $(r, \theta, z)$, where $z$ denotes the optical axis and $r$ denotes the radial coordinate. ## Basic analysis Consider a single large Gaussian beam of beam width $w$ at a large distance $z \gg w$, in the following form: $ E(r, z) = E_0 \sin \left(\dfrac{n\pi}{z}\right)e^{-r^2/w^2} $ > **Note:** This is a simplified form that absorbs some variables for ease-of-understanding (for instance, $w_0/w(z)$ is absorbed into $E_0$, so $E_0 = E_0(z)$ is dependent on distance). We also ignore the radius of curvature here. Now consider two smaller Gaussian beams of beam width $w_2$ at a large distance $z$ and half the peak amplitude $E_0/2$ of the single larger beam: $ E_2(r, z) = \dfrac{1}{2}E_0\sin \left(\dfrac{n\pi}{z}\right)e^{-r^2/w_2^2} $ Since we know that larger (that is, wider) Gaussian beams diverge more slowly, we also know that $w_2 > w$ at large $z$ (that is, the smaller Gaussian beams have a much larger spot size for large $z$ than the single large Gaussian beam). However, if the two smaller Gaussian beams are coherent (identical in phase), we may combine the beams, which causes constructive interference. Thus, the combined beam takes the form of a superposition: $ E_\text{superpos}(r, z) = 2 E_2(r, z) = E_0\sin \left(\dfrac{n\pi}{z}\right)e^{-r^2/w_2^2} $ Now, let us find the power of both the original larger beam $E(r, z)$ and the beam formed by a superposition of the smaller beams $E_\text{superpos}(r, z)$ within the beam width. Since the power density of an electromagnetic wave is proportional to its amplitude, we integrate the square of the electric fields between $r = -w$ and $r = w$, giving us: $ P(r, z) \propto \int_{-w}^w E(r, z)^2\, dr \sim E_0^2\sin^2 \left(\dfrac{n\pi}{z}\right)\int_{-w}^w e^{-2r^2/w^2}\, dr $ This integral can be solved analytically as it is a [Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral), which possesses an analytical solution (although not in terms of elementary functions). The result is given by: $ P(r, z) \propto E_0^2 \sin^2 \left(\dfrac{n\pi}{z}\right)\left[\dfrac{w}{2} \sqrt{2\pi}\, \text{erf}(\sqrt{2})\right] \sim E_0^2w $ As we can see, the power of the beam within the beam width $|r| < w$ is proportional to the square of the peak amplitude $E_0^2$ and linear in $w$. Meanwhile, the power of the beam formed by the superposition of the smaller beams is given by: $ P_\text{superpos}(r, z) \propto E_0^2 \sin^2 \left(\dfrac{n\pi}{z}\right)\left[\dfrac{w_2}{2} \sqrt{2\pi}\, \text{erf}(\sqrt{2})\right] \sim E_0^2 w_2 $ Note how the the superposition of the smaller beams has a total power (within $|r| < w$) that is linear in $w_2$. Since we know that for large $z$, we have $w_2 > w$, so we find that in fact the superposition of the smaller beams actually has _more power_ concentrated in the same region, simply by virtue of constructive interference. This is the fundamental principle that leads to the surprising realization that superimposing several smaller beams creates a much larger _effective aperture size_. We call this the **synthetic aperture**, as it is not a physical aperture, but rather one formed through superposition. > **Note:** See [this Desmos visualization](https://www.desmos.com/calculator/okvmnhrenb) to be able to change the values of $E_0$, $w$, and $w_2$ and see their effect! As long as the beam combining is done in a coherent manner (such that each individual beam has identical phase), a synthetic aperture can replicate or even exceed the performance of a single beam from a large aperture. Indeed, since the size of a synthetic aperture is determined by the _baseline distance_ $D_b$ between the beam sources, which do not need to be physically-connected, a synthetic aperture-based system can effectively have _unlimited_ aperture size (for practical purposes), making it possible to create extremely focused, low-divergence beams without needing extremely large apertures. ## A more mathematically-sophisticated analysis > **Additional note:** This section references (but does _not_ copy) a derivation from the DeepSeek LLM, although the basic idea itself was original (at least in terms of *maser-based* beam-combining for ultra-long-distance power focusing, since interferometer arrays used for [very-long-baseline interferometry](https://en.wikipedia.org/wiki/Very-long-baseline_interferometry) are very widespread and based off basically the same ideas). Unfortunately, it was not possible to find the sources the model itself used/was trained on, but the prompts used (in chronological order) are as follows: > > 1. `Calculate the effective aperture of beam-combined lasers of fixed identical wavelength and explain if it is feasible to create very large apertures` > 2. `Explain if it is possible to create a synthetic aperture by beam-combining masers and the size of such an aperture considering a maser with a horn-style output coupler` > 3. `Show me mathematically how coherent superposition of two Gaussian beams creates a large synthetic aperture` > 4. `Is the distance between the horns the only factor for the synthetic aperture?` > 5. `Show me where your formula for the synthetic aperture comes from mathematically` Our previous analysis was based off heuristic arguments, and did not actually consider the _phase difference_ caused by the superposition of beams that results in constructive interference, nor compute the explicit expression for the synthetic aperture. We will now go through a more in-depth mathematically analysis to be able to derive these results. Once again, we start with the general form of a Gaussian beam in cylindrical coordinates, whose source (e.g. laser) is located at $r = L$: $ E(r, z) = E_0 e^{-(r - L)^2/w^2}e^{-i(kz + \phi)}, \quad \phi \approx k\dfrac{r^2}{2z} $ > **Note:** Here, we use complex exponentials $e^{-i(kz + \phi)}$ as opposed to sinusoids for mathematical convenience. We also use the approximation $R(z) \approx z$ for the radius of curvature of the beam in the far-field. Consider the superposition of two such Gaussian beams, one located at $L = D_b/2$ and one located at $L = -D_b/2$. The two respective beams would take the form: $ \begin{align*} E_1(r, z) &= E_0 e^{-(r - \frac{D_b}{2})^2/w^2}e^{-i(kz + \phi_-)} \\ E_2(r, z) &= E_0 e^{-(r + \frac{D_b}{2})^2/w^2}e^{-i(kz + \phi_+)} \end{align*} $ Where here, $\phi_-$ and $\phi_+$ are given by $\phi_- = \frac{k}{2z}(r - \frac{D_b}{2})^2$ and $\phi_+ = \frac{k}{2z}(r + \frac{D_b}{2})^2$ respectively (this comes from $\phi \approx k r^2/2z$ from above). Substituting the explicit forms of $\phi_-$ and $\phi_+$ gives: $ \begin{align*} E_1(r, z) &= E_0 e^{-(r - \frac{D_b}{2})^2/w^2}e^{-i(kz + \frac{k}{2z}(r - \frac{D_b}{2})^2)} \\ E_2(r, z) &= E_0 e^{-(r + \frac{D_b}{2})^2/w^2}e^{-i(kz + \frac{k}{2z}(r + \frac{D_b}{2})^2)} \end{align*} $ The superposition of the two waves is given by: $ \begin{align*} E_3(r, z) &= E_1 + E_2 \\ &= E_0 \left[e^{-(r - \frac{D_b}{2})^2/w^2}e^{-i(kz + \frac{k}{2z}(r - \frac{D_b}{2})^2)} + e^{-(r + \frac{D_b}{2})^2/w^2}e^{-i(kz + \frac{k}{2z}(r + \frac{D_b}{2})^2)}\right] \end{align*} $ Recalling that $z \gg D_b$ (that is, we analyze the superposition of the beam in the far-field), we can assume that $e^{-(r - \frac{D_b}{2})^2/w^2} \approx e^{-r^2/w^2}$ since the two beams would be so close together that they appear to be a single beam. In addition, we can use the approximation $(r - \frac{D_b}{2})^2 \approx r^2-r D_b$ and $(r + \frac{D_b}{2})^2 \approx r^2 + rD_b$, given that $z \gg D_b$, so we ignore all terms that are higher than first-order in $D_b$. With this approximation, the superposition becomes: $ \begin{align*} E_3(r, z) &= E_0 e^{-r^2/w^2} \left[e^{-i(kz + \frac{k}{2z}(r^2 - rD_b))} + e^{-i(kz + \frac{k}{2z}(r^2 + rD_b))}\right] \\ &= E_0 e^{-r^2/w^2} e^{-ik(z+\frac{r^2}{2z})} \left[e^{i(\frac{kD_b}{2z})r} + e^{-i( \frac{kD_b}{2z})r}\right] \\ &\approx E_0 e^{-r^2/w^2} e^{-ikz} \left[e^{i(\frac{kD_b}{2z})r} + e^{-i( \frac{kD_b}{2z})r}\right] \end{align*} $ Where, on the third line, we used the approximation $z + \frac{r^2}{2z} \approx z$ for large $z$. Noting that $e^{-i\phi} + e^{i\phi} = 2\cos (\phi)$ (this is readily derived from Euler's formula $e^{i\phi} = \cos \phi + i\sin \phi$), we can rewrite this as: $ E_3(r, z) = 2E_0 e^{-r^2/w^2} e^{-ikz} \cos\left(\dfrac{kD_b}{2z}r\right) $ The intensity of the superposition of beams is proportional to the square of the magnitude of $E_3$ (we must take the magnitude since $E_3$ is complex-valued), so we have: $ \begin{align*} I(r,z) &\propto |E_3(r, z)|^2 \\ &= 4I_0 e^{-2r^2/w^2}\cos^2\left(\dfrac{kD_b}{2z}r\right), \quad I_0 \propto E_0^2, \quad w = w(z) \end{align*} $ > **Note:** You can see a [live Desmos visualization](https://www.desmos.com/calculator/yvajmnrnhx) to play around the values and see what this intensity pattern looks like! The minima of the intensity pattern are located at all $\frac{kD_br}{2z} = n\pi/2$ for odd $n$, meaning that the first minima occurs at $r_1 = \dfrac{\pi z}{kD_b}$, and the distance between the first left-side and right-side maxima is given by $d = 2r_1 = \dfrac{2\pi z}{kD_b}$. Noting that $k = \dfrac{2\pi}{\lambda}$ and thus $\lambda = \dfrac{2\pi}{k}$, we have: $ d = \dfrac{\lambda}{D_b} z $ From here, we can find the beam divergence angle $\theta$ through straightforward trigonometry: $ \tan \theta = \dfrac{d}{z} = \dfrac{\lambda}{D_b} $ Using the approximation $\tan \theta \approx \theta$, which is valid considering $z \gg r_1$, we finally arrive at the result for the divergence angle, which we showed in [[A realistic space-based prototype]]: $ \theta \approx \dfrac{\lambda}{D_b} $ Meanwhile, the divergence angle $\theta$ can also be explicitly calculated for a single ideal Gaussian beam, and it is given by[^1]: $ \theta_\text{single} \approx \dfrac{\lambda}{\pi w_0} $ By comparison of these two results, we find that $\pi w_0 = D_b$, so therefore, the effective aperture size is given by: $ w_0 = \dfrac{D_b}{\pi} $ Note how this result _only_ depends on $D_b$ and not on any other parameters, not even the aperture size of the smaller beams! Thus, we have shown that a much larger **synthetic aperture** of size $w_0 = D_b/\pi$ can be created simply through superimposing several small beams, without needing to make any larger aperture. [^1]: Formula is from [this Wikipedia article on Gaussian beams](https://en.wikipedia.org/wiki/Gaussian_beam#Beam_divergence), where here we use $n = 1$ as the beam is mostly propagating through vacuum.