Optics 041 · Interference, coherence, cavities, and metrology

Coherent Wave Composer

A live signed-field landscape, true complex phasor polygons and two detector traces share one deterministic plane-wave sum. Compose two to six waves; move the probe, shift phase and optical path, and expose beats through a finite detector window.

Same-probe instantaneous and exposure intensity

I_{\mathrm{inst}},\overline I_T

$0.50$

Time / peak / pair contrast or wave count

t,I_{\mathrm{peak}},V_T,N

$50\%$

Beat cycles in exposure and phasor closure error

\Delta\nu T,\varepsilon_{\mathrm{phasor}}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Coherent Wave Composer

Number of plane waves

N

2

Adjacent-wave initial phase step

\Delta\phi

0^{\circ}

Added adjacent optical path (positive adds phase)

\Delta\ell

0\lambda_0

Adjacent-wave direction spacing

\alpha

4^{\circ}

Amplitude of every additional wave

a

1

Adjacent frequency offset (true beat rate)

\Delta\nu

0\,\mathrm{Hz}

Trailing detector exposure window

T

50\,\mathrm{ms}

Shared probe horizontal coordinate

x_p

0\,\mu\mathrm m

Pause and set two waves to opposite phase: the white phasor sum closes and the probe intensity vanishes. Try six waves, then play the hidden-beat preset: cyan still beats while gold averages flat. All views share the white probe. The upper surface is magnified signed electric field, not matter.

Ideal same-polarization plane waves, not a literal interferometer or stochastic decoherence model. Optical carrier motion is illustrative; true beat offsets and exact trailing intensity averages are quantitative. Theoretical-peak normalization is labelled; no automatic exposure or simulated mirror is used.

CONTINUE EXPLORING

From sources to phasors

Which changes move the fringes, and which reduce their contrast?

Physics tutorial

Audit the physics of Coherent Wave Composer

BackgroundLight adds as a field, not as brightness. This scalar detector model measures squared electric-field amplitude, not the full vector energy flux through an oblique surface. These ideal plane waves share one polarization. Their optical carrier is removed from the phasor frame: $S(\mathbf r,t)=\sum_{j=0}^{N-1}A_j e^{i(\mathbf k_j\cdot\mathbf r+\phi_j-2\pi j\Delta\nu t)}$ . The upper surface is a signed electric-field visualization, not moving material or a rippling mirror. It magnifies micrometre distances and replaces only the common optical carrier with a labelled slow illustrative rotation. True beat frequency offsets are retained.

Why it mattersHow do phasors, wavefronts, and detector intensity tell the same superposition story?

Start with the essentials

Focus question: How do phasors, wavefronts, and detector intensity tell the same superposition story?
One-sentence intuition: At the white probe, both phasor diagrams and the instantaneous detector intensity use exactly the same complex sum: $I_{\mathrm{inst}}/I_1=|S|^2$ . Exposure does something different: it averages intensity over time, not the fields before squaring. Cyan is instantaneous; dashed gold is the trailing exposure average. With detuning, a short exposure follows the beat while a full-period exposure hides it.

Core mathematical model

Phase, optical path and crossing angle

\begin{aligned}\theta_j&=\left(j-\frac{N-1}{2}\right)\alpha,\quad \phi_j=j\Delta\phi+j\lVert\mathbf k_j\rVert\Delta\ell\\\mathbf k_j&=\frac{2\pi(\nu_0+j\Delta\nu)}{c}(\sin\theta_j,\cos\theta_j)\end{aligned}

Adjacent waves have a uniform direction, phase and frequency step. Positive added optical path is defined to add phase at the reference plane. A half-wavelength path adds half a cycle; a full wavelength returns the same pattern. All directions lie in one plane, so a common perpendicular linear polarization is physically possible.

Exact finite-exposure detector

\frac{\overline I_T}{I_1}=\frac1T\int_{t-T}^{t}|S(\mathbf r,\tau)|^2\,d\tau=\sum_j A_j^2+2\sum_{i<j}A_iA_j\operatorname{sinc}(\pi\Delta\nu_{ji}T)\cos\!\left[\psi_{ji}\!\left(t-\frac T2\right)\right]

The rectangular window ends at the displayed time. Sinc is sine divided by its argument, with value one at zero; its sign must be retained, as must the midpoint phase. The zero-exposure control uses the continuous instantaneous limit. Steady fields are assumed before time zero, so the history can include negative times.

Two-wave extrema and measured contrast

\frac{I_{\max,\min}}{I_1}=(1\pm a)^2,\quad V_T=\frac{2a}{1+a^2}\left|\operatorname{sinc}(\pi\Delta\nu T)\right|,\quad f_{\mathrm{beat}}=|\Delta\nu|

These contrast and beat expressions are for two waves only. Two equal waves can cancel exactly; unequal amplitudes cannot. The visibility is the phase-scan or beat-envelope contrast after exposure, not a stochastic source-coherence parameter. For more waves the detector evaluates every pair rather than reusing a two-wave formula.

Many-wave closure and fixed plot scale

\sum_{j=0}^{N-1}e^{2\pi i j/N}=0,\quad I_{\mathrm{peak}}/I_1=\left(\sum_j A_j\right)^2,\quad 0\le\overline I_T\le I_{\mathrm{peak}}

At the origin, six equal waves separated by sixty degrees in phase form a closed polygon. Changing the probe changes spatial phase and opens it elsewhere. Plot ordinates and the floor map are divided by the stated theoretical peak, not the observed maximum; the physical readouts remain relative to the first wave alone.

Common difficulties

Hidden beats are not source decoherence

Typical misconceptionA flat long-exposure trace means the waves have lost their phase relationship.

Better mental modelThe inputs here are deterministic and fully coherent. Only the detector averages. At a one-second exposure and a one-hertz difference, the gold trace stays at the sum of the two individual intensities while the cyan beat and phasor rotation continue. No random phase noise is present.

Run the experiment

01
Scene 1: Two-wave fringes
Pause the two-wave preset. Set the initial phase step to 180 degrees, then reduce the secondary amplitude or add a half-wavelength optical path.
What to observe: The origin cancels only for matched amplitudes. Half a wavelength reverses the cancellation; the spatial fringes move rather than disappear everywhere.
02
Scene 2: Six-wave phasor closure
Choose the six-wave closure preset. Move the shared probe, then set phase step to zero and watch the polygon stretch into a line.
What to observe: The white endpoint and instantaneous intensity agree at every probe. Six in-phase unit waves reach thirty-six times the first-wave intensity at the origin.
03
Scene 3: Beat hidden by long exposure
Choose the hidden-beat preset and play. Shorten exposure from one second to zero; reverse the frequency difference and compare the drift direction.
What to observe: Gold stays flat at two single-wave intensity units for one complete beat exposure. Short exposure restores the modulation without changing the source.