You are recording an in-line hologram with an object inscribed on a transparent sheet. Mathematically represent the recording and reconstruction of the holograms, and draw a neatly labelled diagram. What will happen if the reconstruction wavelength is two times larger than the recording wavelength. How to record a holographic converging lens? (6 marks)
Answer
In-line Holography: Recording, Reconstruction & Holographic Lens
π Recording an In-line Hologram
In Gaborβs in-line holography, both the reference wave and object wave travel along the same axis. A monochromatic coherent light illuminates a transparent object.
π’ Mathematical Representation
I(x, y) = |ET(x, y)|Β² = |ER|Β² + |EO|Β² + ER EO* + ER* EO
The interference terms hold the amplitude and phase information of the object wave and are crucial to holography.
[Laser Source] β [Transparent Object] β [Holographic Plate]
β Object Wave β Reference Wave
π Reconstruction of the Hologram
The recorded hologram is re-illuminated with the same reference wave to regenerate the original object wavefronts.
π’ Reconstruction Formula
T(x, y) β I(x, y)
This leads to:
- Zero-order term (background)
- Real image: reconstructed object wave
- Virtual image: pseudoscopic image
[Laser Source] β [Hologram Plate] β Real Image + Virtual Image
π Effect of Doubling the Reconstruction Wavelength
If reconstruction wavelength Ξ»β² = 2Ξ» (twice the recording wavelength):
- Angular deviation of diffracted light increases
- Magnification M = Ξ»β² / Ξ» = 2 β image enlarged 2Γ
- Phase mismatch β leads to blurring and distortion
This causes the reconstructed image to be larger but less accurate.
π¬ Recording a Holographic Converging Lens
To record a holographic lens, interfere two spherical waves:
- Object wave: diverging from a virtual point
- Reference wave: converging or planar
π’ Mathematical Description
ER = B β ejkrβ / rβ
Where:
- rβ, rβ: distances from the object and reference points to the hologram
- A, B: amplitudes of the waves
- k = 2Ο / Ξ»: wavenumber
The interference pattern encodes a phase profile that mimics a converging lens.
When illuminated by the reference beam, the hologram behaves like a lens focusing the light to a point.
π Summary Table
| Concept | Description |
|---|---|
| Recording | Interference of reference and object waves |
| Formula | I(x, y) = |ER + EO|Β² |
| Reconstruction | Erecon = ER Γ I(x, y) |
| Wavelength Effect | Ξ»β² = 2Ξ» β 2Γ magnification, distortion due to phase mismatch |
| Holographic Lens | Constructed via interference of spherical waves simulating focusing action |