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Sandbox Physics

L33 · Four fields, one focus

Coherent Beam Combining

Combine four seeded amplifier channels. Dither piston phases, steer the beams and compare filled or tiled pupils on one Fourier camera.

Interactive modelCoherent Beam Combining
Feedback iteration—\text{—}
On-axis phase fidelity at current tilt—\text{—}
Second-channel piston—\text{—}
Third-channel piston—\text{—}
Fourth-channel piston—\text{—}
Exit pupil architecture—\text{—}
Phase feedback—\text{—}
Captured camera pixels—\text{—}
Selected port / input power—\text{—}
Rejected ports / input power—\text{—}
All ports / input power—\text{—}
Coherent bucket / input power—\text{—}
Incoherent reference bucket—\text{—}
Captured controller iteration—\text{—}

Physics tutorial

Phase control is only one part of combining

BackgroundIndependent seeded amplifiers can share a frequency but arrive with different spatial phases.

Why it mattersDoes locking phase repair every combining error?

Start with the essentials

Focus question
Does locking phase repair every combining error?
One-sentence intuition
Separate phase fidelity, selected-port efficiency and fixed-bucket energy.

Core mathematical model

Channel field

Ej=Aju(qj)eiΦjΦj=ϕj+kϑj⋅qjqj=r−rj\begin{aligned}E_j&=A_j u(\boldsymbol q_j)e^{i\Phi_j}\\\Phi_j&=\phi_j+k\boldsymbol\vartheta_j\cdot\boldsymbol q_j\\\boldsymbol q_j&=\boldsymbol r-\boldsymbol r_j\end{aligned}

The same piston and tilt determine both feedback and camera.

Tiled Fourier pupil

Etile=∑j=14EjIf∝∣F[Etile]∣2\begin{aligned}E_{\rm tile}&=\sum_{j=1}^{4}E_j\\I_f&\propto|\mathcal F[E_{\rm tile}]|^2\end{aligned}

Disjoint pupils conserve total power; interference redistributes the focal energy.

Filled unitary network

Ep=12∑jHpjEjHHT=4I∑pPp=∑jPj\begin{aligned}E_p&=\tfrac12\sum_j H_{pj}E_j\\H H^{\mathsf T}&=4\mathbb I\\\sum_p P_p&=\sum_j P_j\end{aligned}

The four Hadamard signs route mismatch into the rejected output ports.

Scalar dither controller

J=∣∑jcjeiϕj∣2(∑jcj)2Gj=J(ϕj+ϵ)−J(ϕj−ϵ)2ϵϕjn+1=ϕjn+3KGj\begin{aligned}J&=\frac{|\sum_j c_j e^{i\phi_j}|^2}{(\sum_j c_j)^2}\\G_j&=\frac{J(\phi_j+\epsilon)-J(\phi_j-\epsilon)}{2\epsilon}\\\phi_j^{n+1}&=\phi_j^n+3KG_j\end{aligned}

Piston correction uses sequential scalar measurements, leaving tilt unchanged.

Incoherent reference

⟨If⟩inc=s2∑j∣F[Ej]∣2s={1tiled1/2filled port\begin{aligned}\langle I_f\rangle_{\rm inc}&=s^2\sum_j|\mathcal F[E_j]|^2\\s&=\begin{cases}1&\text{tiled}\\1/2&\text{filled port}\end{cases}\end{aligned}

An ensemble average removes cross terms; it is not a noisy single frame.

Fixed bucket

ηB=∫∣ρ∣<15 μmIf d2ρPinput\eta_B=\frac{\int_{|\boldsymbol\rho|<15\,\mu\mathrm m}I_f\,d^2\rho}{P_{\rm input}}

Changing array spacing changes sidelobes even after phase locking.

Common difficulties

Brightness is new power

Typical misconceptionLocking phases creates extra optical energy.

Better mental modelThe complete unitary port ledger conserves power; tiled interference only redistributes it.

One metric fixes everything

Typical misconceptionPerfect on-axis phase fidelity means all beams are aligned.

Better mental modelThat metric is normalized at the current tilt; compare the actual camera bucket and manual alignment.

Run the experiment

  1. 01

    Close the loop

    Capture the initial array, then advance forty iterations.

    What to observe: Phase fidelity rises; the same camera records the narrower focus.
  2. 02

    Expose a failure

    Choose tilt error, lock phases, then manually align.

    What to observe: Phase fidelity can be high while bucket efficiency remains poor.
  3. 03

    Change architecture

    Compare filled, weak-channel and sparse tiled presets.

    What to observe: Filled mismatch leaves rejected-port power; tiled arrays retain sidelobes.