Hologram Optics Bench: Recording and Replaying a 3D Image with Laser Light

Detailed abstract

Purpose

A teaching bench for classical holography. It covers the off-axis transmission hologram of Leith and Upatnieks and the single-beam reflection hologram of Denisyuk. It follows a hologram from setup to replay: split and path-match the beams, filter and expand them, light a scene in a dark room, expose and process a plate, then replay the image with the reference beam alone. Along the way it shows why holography needs a coherent laser, a stable table, matched polarizations and the right exposure.

What it can do

• Source and split: a 532 nm single-frequency DPSS laser (or a 5 mW HeNe), a timed shutter, and a half-wave plate with a polarizing cube that sets the object/reference power split.
• Reference arm: two mirrors and a spatial filter (40× objective, 5 µm pinhole). The diverging beam reaches the plate at an adjustable angle (15–45°).
• Object arm: a half-wave plate that turns p into s so the two beams can interfere, and a two-mirror trombone on a stage for path matching. A spatial filter lights a white die and a chess pawn 240 mm in front of the plate.
• Room lights: a switch toggles the room lights. With them off, only laser light remains and the scene shows laser speckle. Exposing with them on fogs the plate.
• Recording: load a plate, expose it (the shutter opens for the set time), and develop it (absorption or bleached phase for silver halide; nothing to do for photopolymer film).
• Replay: the object is removed and its arm blocked. The reference alone lights the plate, and the 3D view shows:
• the virtual image where the object was;
• the zero order;
• the conjugate real image, when it can propagate.

A viewing camera behind the plate can be moved sideways. - Panels: - Microscope patch: a true-scale 24 µm view of the plate, computed from every point of the scene, with the measured fringe period. - Scaled digital copy: the recorded hologram and its numerical reconstruction (zero order, image, twin), with refocusing in depth, left/right viewpoints, replay at another wavelength, and a sideways plate shift. - Efficiency curve: diffraction efficiency vs exposure time, or, for Denisyuk, the Bragg reflectance spectrum under white light. - Presets: standard, floated table, rigid bench without isolation, wobbly desk, HeNe with PFG-01 plates, HeNe with instant photopolymer film, Denisyuk.

Limitations

• Scaled reconstruction. A 63 mm plate holds about 10⁹ fringes, so the reconstruction uses a scaled copy: 512 × 512 samples at 3.45 µm (a camera sensor), with angles × 0.075 and the scene × 0.05 laterally. That is off-axis digital holography (Schnars and Jüptner), not the plate itself. Depth is kept at full scale so refocusing is visible. The twin image falls past the sampling limit and appears aliased; on a real plate it is not.
• Plate response. Sensitivities (VRP-M 80 µJ/cm², PFG-01 100 µJ/cm²) and resolving powers are vendor-nominal values to verify. The Litiholo exposure of 5 mJ/cm² is ASSUMED. The curves behind them are simple saturating models, also ASSUMED:
• optical density vs exposure;
• phase vs exposure;
• refractive-index modulation vs exposure.
• Scene light. The object is a Lambertian scatterer (albedo 0.7, half the illuminating power intercepted). The scene is about 180 scattering points with random phases.
• Coherence and vibration. HeNe coherence uses an ASSUMED three-mode laser with a 0.30 m cavity. Vibration and drift are treated as Gaussian path jitter averaged over the exposure.
• Not modelled: emulsion shrinkage on transmission holograms (for Denisyuk it only shifts the colour), scatter noise, reciprocity failure, aberrations from replaying with a diverging beam, and multiple reflections in the plate.

How realistic is it?

• Best case: on a floated table with a single-frequency laser and matched paths, bleached silver-halide transmission holograms reach several to ~30 % efficiency, and Denisyuk reflection holograms on VRP-M or PFG-01 give bright single-colour images in white light. Exposures here (about 1–3 s at 50 mW, about 30 s with a 5 mW HeNe) match common practice.
• Likely case: first attempts fail on stability (air currents, a desk, an exposure started too soon after touching the table), on path mismatch with a HeNe, or on forgetting the polarization. Expect dim images until those are fixed.
• Hidden burdens: a truly dark room and the right safelight; developer and bleach chemistry (and its disposal); laser safety with expanded Class 3B beams; table settling time before every exposure; scene objects glued down; and plate holders that do not creep.

The modeling: core physics and how it is handled

Core physics and how it is handled

1. Recording. The plate sees I = |R + O|² = |R|² + |O|² + 2|R||O| V cos(φ_R − φ_O). The interference term stores both the amplitude and the phase of the object wave. Fringe visibility is the product of five factors, V = V_K·V_pol·V_coh·V_vib·V_fog: - beam ratio: V_K = 2√K/(1 + K), with K = I_R/I_O; - polarization match: 1 when the object-arm λ/2 turns p into s, 0 otherwise; - coherence: V_coh = |γ(ΔL)| for the arm path difference ΔL. For the single-frequency laser it is exp(−|ΔL|/50 m). For the HeNe it is the three-mode sum |Σ a_m e^{imπΔL/L_cav}|/Σ a_m, which dips at ΔL ≈ L_cav; - stability: V_vib = exp(−½(2πσ/λ)²), with σ² = σ_vib² + (drift·t)²/12; - fog: V_fog = I_laser/(I_laser + I_room).

Handled: visibility() multiplies the five factors. ΔL comes from the actual 3D path lengths of the two arms.

2. Fringe spacing. For a reference at angle θ and object light near the plate normal, the period on the plate is Λ = λ/(sin θ + sin θ_O). The fringe planes are spaced λ/(2 sin(θ/2)). In a Denisyuk hologram the planes run parallel to the emulsion, λ/(2n cos(θ_i/2)) apart (about 170 nm).

Handled: fringes(). patch() computes the true-scale pattern on a 24 µm patch by summing every scene point's wave, which is locally a plane wave over the patch. It then measures the period from the power spectrum.

3. Irradiance and exposure. - Reference: I_R = 2P_R/(πw²), with w set by the spatial filter's divergence (0.5 mm / 4.5 mm × 400 mm). - Object light: I_O = ρ·f·P_O/(πD²), for a Lambertian scene at D = 240 mm. - Exposure: E = (I_R + I_O)·t·s(λ), plus room light when it is on. The plate's sensitivity window s(λ) is 1 inside its band and 0.01 outside.

4. Plate response and efficiency. With normalized exposure e = E/E_s: - absorption: density D = 3(1 − e^{−e/3}), amplitude transmittance t = 10^{−D/2}; - bleached: phase φ = 2.2(1 − e^{−e/1.5}); - photopolymer: a thick phase grating, η = sin²ν (Kogelnik).

Handled: efficiency() takes the first Fourier coefficient of t(e₀(1 + V cos φ)) over one fringe. This reproduces the known ceilings of 6.25 % (thin absorption) and 33.9 % (thin sinusoidal phase), and the over- and under-exposure fall-off. For Denisyuk holograms the reflection-grating forms are used: tanh²ν for phase, ≤ 7.2 % for absorption. A soft roll-off applies past the plate's nominal resolving power (ASSUMED).

5. Replay. Lighting the developed plate with R gives R·t ∝ |R|²·O (the object wave: a virtual image at the object's place) + R²·O* (the conjugate: a real, depth-inverted image) + the zero order. The conjugate leaves at sin θ₋₁ = −sin θ(1 + λ′/λ). It exists only if |sin θ₋₁| < 1, so not for θ ≥ 30° at the recording wavelength. Replaying at λ′ moves the image to distance d·λ/λ′ and tilts it by sin θ(λ′/λ − 1).

Handled: record() builds the scaled digital hologram from the scene points (Fresnel spherical waves, separable in x and y) plus a tilted reference. It applies the visibility and maps intensity through the plate response pixel by pixel. reconstruct() multiplies by the replay reference at λ′, applies a viewing window and a plate shift, and Fresnel-transforms to the chosen depth (single FFT).

6. Denisyuk colour. The reflection hologram is a stack of Bragg planes. In white light it reflects a band centred at λ_B = 2nΛ(1 − shrinkage), about λ²/(2nd) ≈ 12 nm wide for a 7 µm emulsion. Handled: bragg() uses Kogelnik's lossless reflection-grating formula.

Checks run on the model (Node)

• Fringe period: the true-scale patch gives 1.04 µm at θ = 30° against λ/(sin θ + sin θ_O) = 1.06 µm (the scene spreads sin θ_O by ±0.07). It gives 0.74 vs 0.75 µm at 45° and 1.97 vs 2.03 µm at 15°.
• Visibility limits: with K = 1 and no vibration, drift, fog or path difference, V = 1. 100 nm of rms jitter at 532 nm gives V_vib = 0.50, which matches exp(−½(2π·0.1/0.532)²).
• Point images: three isolated points reconstruct at their positions to within one 30 µm pixel. Each focuses at its own depth (100 and 112 mm), and 10 mm of defocus drops the peak to ≈ 10 %.
• Off-axis vs in-line: with the off-axis reference, 3 % of the light lands in the object region. With the reference on axis (Gabor), 98 % does: image, twin and zero order overlap.
• Other wavelength: replaying a 532 nm hologram at 633 nm focuses at 84 mm = 100·532/633 mm.
• Plate shift: shifting the plate by 69 µm moves the image by 60 µm (one-pixel rounding).
• Numerical vs analytic efficiency: the image-peak energy in the numerical reconstruction tracks the analytic η within ±25 % across these cases:
• bleached;
• absorption;
• room lights on;
• wobbly desk.

How the simulation is built

• src/ho_model.js: lasers, plates, irradiance, visibility, plate response, efficiency, the true-scale patch, the scaled record and reconstruct (FFT), and the Bragg curve. Pure functions; runs in Node.
• src/ho_app.js: 3D layout (θ-dependent reference arm, trombone, spatial filters, plate holder, scene, ghost images, viewing camera, room lights), the workflow and the panels.
• src/qkit.js: the shared optics-table kit.
• src/ho_shell.html + src/build.py: the page. python build.py writes hologram_optics_bench.html.

How to recreate it

1. Compute I_R and I_O from the laser power, split, beam size and the scene's scatter, then K and V_K.
2. Multiply by the polarization, coherence (from ΔL), vibration/drift and fog factors.
3. Compute E = I·t·s(λ) and normalize by E_s. Take the first Fourier coefficient of the plate's t(e) over one fringe to get η.
4. For images, sum the scene points' waves on a sampled hologram plane with a tilted plane reference. Apply V to the cross term, map through t(e), multiply by the replay reference, and Fresnel-transform to the depth of interest.
5. Check: fringe period vs λ/(sin θ + sin θ_O); points refocus at their depths; the in-line case overlaps the orders; λ′ moves the focus to d·λ/λ′.

Key parameters