--- title: "Coherent Wave Composer Interactive Lab | Sandbox Physics" description: "Why do a phasor diagram, a wavefront picture and a detector reading all describe the same interference? All three come from one sum of complex fields." canonical_url: "https://sandboxphysics.com/labs/coherent-wave-composer/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/coherent-wave-composer/" last_modified: "2026-10-27" --- Optics 041 · Interference, coherence, cavities, and metrology # Coherent Wave Composer **Why do a phasor diagram, a wavefront picture and a detector reading all describe the same interference?** All three come from one sum of complex fields. The phasor diagram adds the fields as arrows, the wavefront picture shows the same sum spread out in space, and the detector reports the square of its size. A detector also averages over its exposure time, so beats between two slightly different frequencies can wash out. A live landscape of the signed field, complex phasor polygons and two detector traces all come from one deterministic sum of plane waves. Combine two to six waves, move the probe, shift phase and optical path, and reveal beats through a finite detector window. > Interactive simulation: open this page in a web browser to change the parameters and run the model. The waves are ideal plane waves with the same polarization; this is not a model of an actual interferometer or of random decoherence. Motion of the optical carrier is illustrative, while the beat offsets and the trailing intensity averages are exact and quantitative. Normalization to the theoretical peak is labeled, and no automatic exposure or simulated mirror is used. Physics tutorial ## Audit the physics of Coherent Wave Composer **Background** Light waves add as fields, not as brightness. The detector here is scalar: it measures the squared electric-field amplitude, not the full vector energy flux through a tilted surface. The ideal plane waves share one polarization, and the phasor frame removes their optical carrier: $S(\mathbf r,t)=\sum_{j=0}^{N-1}A_j e^{i(\mathbf k_j\cdot\mathbf r+\phi_j-2\pi j\Delta\nu t)}$. The upper surface shows the electric field with its sign; it is not moving material or a rippling mirror. It magnifies micrometre distances. Only the shared optical carrier is replaced, by a slow rotation labelled as illustrative, and the true beat frequency offsets are kept. **Why it matters** Why do a phasor diagram, a wavefront picture and a detector reading all describe the same interference? Start with the essentials - **Focus question**: Why do a phasor diagram, a wavefront picture and a detector reading all describe the same interference? - **One-sentence intuition**: At the white probe, both phasor diagrams and the instantaneous detector intensity come from exactly the same complex sum: $I_{\mathrm{inst}}/I_1=|S|^2$. Exposure does something else: it averages the intensity over time, rather than averaging the fields before squaring. The cyan trace is instantaneous; the dashed gold trace averages over the exposure that just ended. When the frequencies differ, a short exposure follows the beat, while an exposure lasting a full beat period hides it. ### Core mathematical model #### Phase, optical path and crossing angle $$ \begin{aligned}\theta_j&=\left(j-\frac{N-1}{2}\right)\alpha,\quad \phi_j=j\Delta\phi+j\lVert\mathbf k_j\rVert\Delta\ell\\\mathbf k_j&=\frac{2\pi(\nu_0+j\Delta\nu)}{c}(\sin\theta_j,\cos\theta_j)\end{aligned} $$ Neighbouring waves differ by the same step in direction, phase and frequency. By definition, a positive extra optical path adds phase at the reference plane. Half a wavelength of path adds half a cycle; a full wavelength brings back the same pattern. All the directions lie in one plane, so the waves can physically share one linear polarization perpendicular to that plane. #### Exact intensity for a finite exposure $$ \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