Optics 054 · Interference, coherence, cavities, and metrology

Holography Studio

How can a plate that records only brightness keep the phase needed to rebuild a three-dimensional wavefront? By mixing the object light with a reference beam. Where the two arrive in phase the plate records a bright fringe, and where they are out of phase a dark one, so the phase is written into the fringe positions. Recover the complex field from those fringes and you can propagate it numerically to any depth and refocus there without recording again.

Record an object at two depths on a 3D holography bench. Recover the field from simulated intensity records, inspect sideband overlap and twin images, then refocus numerically without recording again.

Carrier / physical reference tilt

f_c,\theta_r

$0.50$

Signed digital focus / acquired records

z,N_{\mathrm{records}}

$50\%$

Known-model field error / power-conservation error in propagation

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

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Holography Studio

Wavelength used for both recording and reconstruction (reacquires)

\lambda

532\,\mathrm{nm}

Distance from the near slice to the 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 fringes, not a photograph of the object. Below the 3D bench, follow one record through its Fourier spectrum to the reconstructed field. Click Focus near H and Focus far ring: the same record shows different depths. Moving the object records a new hologram; changing the digital focus moves no hardware. Reduce the carrier until it overlaps the zero order, or raise it past the Nyquist limit. Switch to Inline to see the twin image at negative depth, then to Four phase steps to recover the phase from four intensity measurements of a stationary object. Drag to orbit. Play only animates markers along the recording path.

The object is two ideal, prescribed, band-limited coherent slices, and propagation is scalar, periodic and paraxial. The Lab does not simulate a finite aperture, real materials or a commercial camera. Occlusion, photon noise, pixel integration, phase drift and absolute radiometry are not included. The digital reconstruction is a calculation; there is no physical screen behind the camera. The display transfer is fixed, and all calculations use unclipped data. The field error compares the recovered field with the known simulated field; it is not an experimental error measurement.

Physics tutorial

Audit the physics of Holography Studio

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

Why it mattersHow can a plate that records only brightness keep the phase needed to rebuild a three-dimensional wavefront?

Start with the essentials

Focus question: How can a plate that records only brightness keep the phase needed to rebuild a three-dimensional wavefront?
One-sentence intuition: Once a complex field has been extracted from the acquired intensities, focusing is just a calculation: $U_z=\mathcal P_{-z}\{O_{\mathrm{recovered}}\}$ . Refocus the same record at the depth of the H and at the depth of the ring. Changing the original object depth needs a new acquisition; changing the 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 paths. The central spectral term is the intensity of the object plus the reference. The two displaced cross terms carry the object field and its complex conjugate. Every spectrum panel is calculated from the displayed intensity record. The reference and object light are assumed to have the same polarization and wavelength.

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 on each side, with an 8 micrometre pitch. The reference makes a whole number of fringes across the 1.024 mm width. Each prescribed object field has a circular bandwidth of ten frequency bins. After the carrier is removed, the digital circular passband must contain the object while avoiding the shifted central intensity term and the conjugate sideband. These conditions are sufficient for separation in this deliberately band-limited discrete model; they are not universal camera specifications. A filter that is too small discards information, and crossing Nyquist is clearly 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}

A positive depth means the original object was upstream of the camera, so numerical recovery propagates backward by that distance. The model uses unitary, periodic, paraxial angular-spectrum propagation and leaves out the common optical carrier phase. Both object slices use the same transverse sampling. The enlarged apparatus uses different axial and transverse scales, so it is not drawn to scale. No reconstructed screen floats behind the camera.

Inline recovery keeps a twin image

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

Only the known intensity of the reference alone is subtracted, so the unknown object-intensity term stays in. Because the prescribed object slices are real, the conjugate field comes to focus at the negative signed depth. In the inline preset, reconstructions at both positive and negative depth therefore show structure, with leftover self-interference. The result is not a clean photograph or a commercial holographic display.

Four measurements recover both 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 intensities calculated separately, not four recoloured copies of one image. Opposite steps cancel both intensity-only terms, and their signed differences recover the complex cross term. Dividing by the known nonzero reference then recovers the field at the sensor. This comparison assumes perfect quarter-turn steps and a stationary object. Noise, drift and failures with moving objects are outside the model.

Check the recovered field, not how good the image looks

\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}

The field error compares against the separately known simulated sensor field; recovery never takes that field as input. The power check tests the numerical propagation, not absolute optical energy. The depth curve shows how concentrated the light is in a region; it is not an automatic depth measurement. The record display is divided by the known reference intensity and reaches white at three, while the reconstructed intensity reaches white at 1.5. Square-root brightness and logarithmic spectra change only the display. The point-sampled periodic model leaves out finite apertures, occlusion, material response and pixel integration.

Common difficulties

One intensity recording does not automatically capture the full phase

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

Better mental modelRecovery needs a known coherent reference plus enough spectral separation, or else extra phase-stepped measurements. Inline overlap, too much tilt and a clipped bandwidth leave ambiguity or lose information. These failures are shown as they are; an ideal target image never replaces them.

Run the experiment

01
Scene 1: Off-axis record and refocus
Use the near and far focus buttons without touching any acquisition control. Compare the same recorded hologram at the two depths.
What to observe: The camera sees interference. Recognizable object structure appears only after the field is recovered and propagated.
02
Scene 2: Inline twin-image ambiguity
In Inline twin-image ambiguity, compare positive and negative focus. Then switch to Off-axis record/refocus: lower its carrier to zero, then raise it beyond 64 to see the spectral overlap and the sampling limit.
What to observe: Moving an object changes the acquisition; moving the 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 look at all four intensity records.
What to observe: A numerical check against a known model can reveal failures, but it does not certify a real experiment.

Holography Studio

Audit the physics of Holography Studio

Core mathematical model

One intensity image contains four terms

Separate the sideband without crossing Nyquist

Propagation and refocus share the same operator

Inline recovery keeps a twin image

Four measurements recover both quadratures

Check the recovered field, not how good the image looks

Common difficulties

One intensity recording does not automatically capture the full phase

Run the experiment

Scene 1: Off-axis record and refocus

Scene 2: Inline twin-image ambiguity

Scene 3: Four acquired phase steps

Related experiments and articles

Interference for measuring length and recording images