AcademyDiffraction
Academy
Holography
Level 1 - Physics topic page in Diffraction.
Principle
Holography records phase information by interfering an object wave with a coherent reference wave. When the developed hologram is illuminated, diffraction reconstructs a wavefront similar to the original object wave.
Notation
\(E_o\)
object-wave electric field
\(\mathrm{V\,m^{-1}}\)
\(E_r\)
reference-wave electric field
\(\mathrm{V\,m^{-1}}\)
\(I\)
recorded intensity
\(\mathrm{W\,m^{-2}}\)
\(\Delta\phi\)
phase difference
\(\mathrm{rad}\)
\(\lambda\)
recording wavelength
\(\mathrm{m}\)
\(p\)
fringe spacing
\(\mathrm{m}\)
Method
Derivation 1: Record interference, not only brightness
The photographic or digital recording stores an intensity pattern made by adding fields before squaring.
Field superposition
\[E=E_o+E_r\]
Recorded intensity
\[I\propto|E_o+E_r|^2\]
Derivation 2: Keep phase in the cross term
The interference term depends on the phase difference between object and reference waves.
Intensity with phase
\[I=I_o+I_r+2\sqrt{I_oI_r}\cos\Delta\phi\]
Phase from path
\[\Delta\phi=\frac{2\pi\Delta r}{\lambda}\]
Derivation 3: Reconstruct by diffraction
Illuminating the hologram with a reference-like beam diffracts light so that one diffracted wave reproduces the original object wavefront.
Symmetric two-beam fringe spacing
\[p=\frac{\lambda}{2\sin\alpha}\]
Grating-like reconstruction
\[p\sin\theta=m\lambda\]
Rules
Hologram recording intensity
\[I=I_o+I_r+2\sqrt{I_oI_r}\cos\Delta\phi\]
Phase from path difference
\[\Delta\phi=\frac{2\pi\Delta r}{\lambda}\]
Two-beam fringe spacing
\[p=\frac{\lambda}{2\sin\alpha}\]
Examples
Question
Two equal recording beams arrive in phase. What relative recorded intensity is produced compared with one beam alone?
Answer
Equal in-phase fields double the amplitude, so intensity becomes
\[2^2=4\]
times one beam alone.Checks
- A hologram records an interference pattern, not a direct picture of the object.
- Coherent light is needed so the phase relationship remains stable while recording.
- Reconstruction is a diffraction process.
- Changing reconstruction wavelength can change the reconstructed angle or scale.