Digital Holography

Fresnel reconstruction grid λz/(NΔx), sensor NA and its pixel-pitch limit λ/(2Δx), resolution, depth of focus, largest reference angle and the in-line sampling distance for a digital hologram.

Model: Textbook approximation

Wavelength λ (nm)Sensor pixel pitch Δx (µm)Pixels per axis NRecording distance z (mm)Object size D (µm)

Results

Sensor size NΔx13.31 mm

Numerical aperture sin(atan(NΔx/2z))0.03326

Effective NA (pixel limit λ/2Δx)0.03326

Reconstruction grid λz/(NΔx)7.993 µm

Resolution λ/(2NA_eff)7.997 µm

Reconstructed field λz/Δx16.37 mm

Depth of focus λ/NA_eff²480.9 µm

Largest reference angle 2 arcsin(λ/4Δx)2.345°

Fresnel number (D/2)²/(λz)0.5874

Shortest in-line distance Δx(NΔx + D)/λ168.8 mm

In-line hologram sampledyes

At z = NΔx²/λ = 162.6 mm the image pixel equals the sensor pixel; closer in, the angular-spectrum method keeps Δx while the Fresnel transform gives a finer, undersampled grid.

Image pixel against recording distance

Reconstructed point (one axis)

Model

Fresnel reconstruction: Δξ = λz/(NΔx), field NΔξ = λz/Δx, point image sinc²(πx/r) with r = λ/(2NA_eff)

NA = sin(atan(NΔx/2z)), NA_eff = min(NA, λ/(2Δx)), resolution λ/(2NA_eff), depth of focus λ/NA_eff², θ_max = 2 arcsin(λ/(4Δx))

In-line sampling: (NΔx + D)/(2z) ≤ λ/(2Δx) ⇒ z ≥ Δx(NΔx + D)/λ

Scalar, paraxial Fresnel diffraction with a plane reference wave and a square sensor of N × N square pixels with 100 % fill factor assumed (Schnars & Jüptner 2002). The reconstructed point is drawn as a sinc whose first zero is the resolution λ/(2NA_eff). An off-axis hologram must also separate the twin image and the zero order, which needs a larger angle or distance; noise and the object's own speckle are not included.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Scalar paraxial Fresnel diffraction with a plane reference wave and an N × N sensor of square pixels, 100 % fill factor assumed. The NA is sin(atan(NΔx/2z)), capped at λ/(2Δx) where the pixel pitch, not the sensor width, limits the recorded fringes (below z ≈ NΔx²/λ). The reconstructed point is the sinc transform of the effective aperture. Twin-image separation in off-axis holograms, noise and speckle are not included.

Schnars U., Jüptner W. P. O. (2002). Digital recording and numerical reconstruction of holograms. Meas. Sci. Technol. 13, R85–R101. Fresnel reconstruction pixel λz/(NΔx) and the largest interference angle 2 arcsin(λ/(4Δx)). doi:10.1088/0957-0233/13/9/201

Report a problem