L33 · Four fields, one focus

Coherent Beam Combining

In coherent beam combining, does locking the phases fix every error? No. Piston locking aligns each beam’s overall oscillation phase. It cannot point the beams in the same direction, equalize their powers or fill gaps across a tiled pupil. The power in the chosen output port and in a fixed focal region still have to be measured separately.

Compare the focus before and after phase control. Try tilted beams, a weak channel and two pupil layouts; follow power into the focal bucket and rejected ports.

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 $\text{—}$

Initial bucket / input power $\text{—}$

Incoherent reference bucket $\text{—}$

Captured controller iteration $\text{—}$

Explore the mechanism

In coherent beam combining, does locking the phases fix every error?

Filled / tiled pupil

a

1

Initial piston pattern scale

\phi_0

1\,\mathrm{rad}

Two-axis tilt pattern scale

\vartheta_0

0\,\mathrm{mrad}

Fourth-channel amplitude deficit

1-A_4/A_1

0

Tiled pupil center separation

d

1.8\,\mathrm{mm}

Controller gain

K

1

Phase feedback off / on

b

1

Detail: amplifiers / pupil / camera

j

0

The two acquired cameras use the same display gain. A bright on-axis phase metric can coexist with poor pointing; compare the fixed-bucket power and the rejected ports.

The fields are scalar and monochromatic at 1064 nm and pass through a 100 mm Fourier lens. The four pupils are identical Gaussians cut off by square edges. Tiled pupils do not overlap. The filled aperture uses an ideal lossless Hadamard network with four output ports, and power leaving the rejected ports is still counted. Amplitude mismatch weakens only the fourth channel. Piston and two-axis tilt follow fixed patterns for each channel; they are not samples of random noise. The controller dithers each phase both ways, measures only the on-axis signal and adjusts the three relative phases in turn. One iteration does not correspond to a real bandwidth or clock time. Manual alignment removes the imposed tilt using values the model already knows. The incoherent curve is a separate ensemble average with the same channel powers; it is not rebuilt from camera data. Bucket power is the power inside a fixed 15 µm bucket as a fraction of the transmitted input power. It is not wall-plug efficiency, and the bucket does not search for the main lobe. The pupils are sampled on a finite grid, and the drawn beam paths are schematic. Polarization, spectral mismatch, amplifier gain dynamics, turbulence and detector noise are left out. Each selection acquires the initial field, computes forty controller iterations and acquires the current field. The two images use a common fixed input-power reference of 0.04 per pixel and a display gamma of 0.38; the stored camera pixels and all power readouts remain linear. The drawn focal ray offsets enlarge tilt twelvefold. The network interior and propagation between the pupil and lens are schematic, not an independently traced optical relay. The central sensor patch shows the acquired fixed-bucket power with a common linear brightness reference of 0.3, clipped above it. Replay only animates field transport.

Start with the essentials

Focus question: Does locking the phases fix every combining error?
One-sentence intuition: Piston locking can concentrate the light, but beam pointing and unequal amplitudes still determine how much reaches the chosen focal region.

Core mathematical model

Field of each channel

\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}

Each channel has its own amplitude, piston phase and tilt. The feedback loop and the camera both work from these same values.

Tiled pupils seen through a Fourier lens

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

The pupils do not overlap, so the total power is the sum of the channel powers. Interference only moves energy around in the focal plane.

Filled aperture: a lossless four-port network

\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}

Each output port adds the four fields with its own pattern of Hadamard signs. Any mismatch between channels sends power into the rejected ports.

Dither control from one measured signal

\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}

The controller corrects the piston phases one channel at a time, using a single measured number for each step. It leaves the tilt unchanged.

Incoherent reference

\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}

Averaging over an ensemble removes the interference cross terms. The curve is that average, not one noisy frame.

Power inside a fixed bucket

\eta_B=\frac{\int_{|\boldsymbol\rho|<15\,\mu\mathrm m}I_f\,d^2\rho}{P_{\rm input}}

Changing the array spacing changes the sidelobes, even after the phases are locked.

Common difficulties

A brighter focus means more power

Typical misconceptionLocking the phases creates extra light energy.

Better mental modelNo power is created. In the lossless network, the four output ports together always carry the input power. In a tiled array, interference only moves power from place to place.

One good number means everything is fixed

Typical misconceptionIf the on-axis phase fidelity is perfect, all the beams must be aligned.

Better mental modelThe metric is normalized at the current tilt, so it can reach its best value while the beams still point apart. Check the actual bucket power on the camera, and compare it with the result after manual alignment.

Run the experiment

01
Compare the acquired spots
Choose Coherent focus. The initial camera and the camera after forty controller iterations are acquired automatically.
What to observe: The focus concentrates as the relative phases align. Both images use the same fixed brightness map; the ring marks the same fifteen-micrometre bucket.
02
Leave the beams tilted
Choose Locked but tilted and compare the phase metric with bucket power. Open Advanced controls and choose Manually remove tilt.
What to observe: A high phase metric at the current tilt can coexist with a weak bucket signal. Steering can add power to the bucket after piston locking has finished.
03
Follow the rejected power
Choose Single-port combining, then weaken the fourth channel. Compare Bright sidelobes with the tiled default separately.
What to observe: Imbalance leaves power in the three rejected ports. A sparse tiled pupil instead redistributes more of the focal pattern into sidelobes.