Optics 054 · Interference, coherence, cavities, and metrology

Holography Studio

Record two coherent object depths on a 3D holography bench. Recover the field from actual simulated intensity records, inspect sideband overlap and twins, then refocus numerically without re-recording.

Carrier / physical reference tilt

f_c,\theta_r

$0.50$

Signed digital focus / acquired records

z,N_{\mathrm{records}}

$50\%$

Known-model field error / propagation power closure

\epsilon_{\mathrm{field}},\epsilon_{\mathrm{power}}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Holography Studio

Matched recording / reconstruction wavelength (reacquires)

\lambda

532\,\mathrm{nm}

Near slice to camera (moves the object)

d_1

100\,\mathrm{mm}

Post-recording focus depth (negative shows twin)

z

100\,\mathrm{mm}

Physical depth separation of the two slices

\Delta d

35\,\mathrm{mm}

Carrier fringes across sensor (steers reference)

m

42

Reference field amplitude (noise-free model)

A_r

1.40

Digital passband radius (Fourier bins)

b

10

The camera records stripes, not a photograph of the object. Below the 3D bench, follow one record through its Fourier spectrum into a reconstructed field. Click Focus near H and Focus far ring: the same record reveals different depths. Move the object to reacquire; change digital focus without moving any hardware. Reduce the carrier to overlap the zero order, or raise it past Nyquist. Switch to Inline to inspect the twin at negative depth, then to Four phase steps to recover phase from four stationary-object intensity measurements. Drag to orbit. Play animates recording-route markers only.

Ideal two prescribed band-limited coherent slices; scalar periodic paraxial propagation. Not a finite-aperture, material or commercial camera simulator. No occlusion, photon noise, pixel integration, phase drift or absolute radiometry. A digital reconstruction is not a physical screen behind the camera. Fixed display transfer; all computations use unclipped data. Field error compares recovery with the separately known simulation field, not an experimental error measurement.

Physics tutorial

Audit the physics of Holography Studio

BackgroundA hologram does not measure phase directly. It records interference with a known reference: $I_\delta=|O+A_r e^{i(2\pi f_c u+\delta)}|^2$ . Two coherent object slices are placed at different distances in front of the camera. Their fields add before intensity is calculated; no ideal target picture is inserted into the reconstruction.

Why it mattersHow can an intensity-only plate preserve enough phase information to reconstruct a three-dimensional wavefront?

Start with the essentials

Focus question: How can an intensity-only plate preserve enough phase information to reconstruct a three-dimensional wavefront?
One-sentence intuition: After extracting a complex field from acquired intensities, focus is a calculation: $U_z=\mathcal P_{-z}\{O_{\mathrm{recovered}}\}$ . Refocus the same record at the H and ring depths. Changing the original object depth requires a new acquisition, but changing reconstruction depth does not.

Core mathematical model

One intensity image contains four terms

\begin{aligned}R&=A_r e^{2\pi i f_cu}\\I&=|O|^2+|R|^2\\&\quad+OR^*+O^*R\end{aligned}

The detector is the absorbing plane at the end of both recording routes. The central spectral term is the object-plus-reference intensity. The two displaced cross terms contain the object field and its complex conjugate. All spectrum panels are calculated from the displayed intensity record. Reference and object polarization and wavelength are assumed identical.

Separate the sideband without crossing Nyquist

\begin{aligned}L&=Np,\quad f_c=m/L\\\sin\theta_r&=\lambda f_c\\m&>2B+b\\m+B&<N/2\end{aligned}

The sensor has 128 point samples per side with 8 micrometre pitch. The reference has an integer number of fringes across its 1.024 mm width. Each prescribed object field has ten-bin circular bandwidth. After carrier removal, the digital circular passband must include the object but avoid the shifted central intensity term and conjugate sideband. These are sufficient separation conditions for this deliberately band-limited discrete model, not universal camera specifications. Too-small filters discard information; a Nyquist crossing is explicitly flagged.

Propagation and refocus share the same operator

\begin{aligned}H_d(f_u,f_v)&=e^{-i\pi\lambda d(f_u^2+f_v^2)}\\\mathcal P_d\{U\}&=\mathcal F^{-1}[\mathcal F[U]H_d]\\\mathcal P_{-d}\mathcal P_d&=\mathrm{Id}\end{aligned}

Positive depth means an original object upstream of the camera; numerical recovery propagates backward by that distance. The model uses unitary periodic paraxial angular-spectrum propagation and omits the common optical carrier phase. Both object slices use the same transverse sampling. The enlarged apparatus uses different axial and transverse scale factors; it is not to scale. There is no reconstructed screen floating behind the camera.

Inline recovery retains a twin

\begin{aligned}R&=A_r\\\frac{I-A_r^2}{A_r}&=O+O^*+\frac{|O|^2}{A_r}\end{aligned}

Only the known reference-only intensity is subtracted. The unknown object-intensity term is not magically removed. With real prescribed object slices, the conjugate field focuses at negative signed depth. Positive and negative reconstructions therefore both show structure in the inline preset, with residual self-interference. This is not a clean photograph or a commercial holographic display.

Four measurements recover the quadratures

\begin{aligned}Q&=(I_0-I_\pi)\\&\quad+i(I_{\pi/2}-I_{3\pi/2})\\O_{\mathrm{recovered}}&=\frac{Q}{4R^*}\end{aligned}

The four phase-step cards are independently calculated intensities, not four recoloured copies. Opposite steps cancel both intensity-only terms; their signed differences recover the complex cross term. Division by the known nonzero reference then recovers the sensor field. This comparison assumes perfect quarter-turn steps and a stationary object. Noise, drift and moving-object failures are outside this model.

Validate the recovered field, not a prettier display

\begin{aligned}\epsilon_U&=\frac{\|O_{\mathrm{recovered}}-O\|_2}{\|O\|_2}\\\epsilon_P&=\left|\frac{\sum|U_z|^2}{\sum|O_{\mathrm{recovered}}|^2}-1\right|\end{aligned}

Field error uses the separately known simulated sensor field as an audit reference; that field is never an input to recovery. Power closure tests numerical propagation, not absolute optical energy. The depth curve is a region concentration diagnostic, not an automatic depth measurement. Record display is divided by known reference intensity and reaches white at three; reconstructed intensity reaches white at 1.5. Square-root brightness and logarithmic spectra change display only. The point-sampled periodic model omits finite apertures, occlusion, material response and pixel integration.

Common difficulties

A single intensity record is not automatically phase complete

Typical misconceptionAny striped image can be refocused into a perfect 3D object.

Better mental modelRecovery requires a known coherent reference plus adequate spectral separation, or additional phase measurements. Inline overlap, excessive tilt and bandwidth clipping leave ambiguity or lose information. Those failures are shown rather than replaced by an ideal target image.

Run the experiment

01
Scene 1: Off-axis record and refocus
Use the near and far focus buttons without changing any acquisition control. Compare the same recorded hologram at two depths.
What to observe: The camera sees interference; recognizable object structure emerges only after field recovery and propagation.
02
Scene 2: Inline twin-image ambiguity
In Inline twin-image ambiguity, compare positive and negative focus. Then switch to Off-axis record/refocus: reduce its carrier to zero and raise it beyond 64 to inspect spectral overlap and the sampling boundary.
What to observe: Moving an object changes acquisition; moving digital focus leaves the raw record unchanged.
03
Scene 3: Four acquired phase steps
Compare Inline positive and negative depth, then acquire Four phase steps and inspect all four intensity records.
What to observe: A known-model numerical audit can reveal failure without claiming external experimental certification.