Principles and methods of digital holography

Kurt Schab

What is holography?Defined as the process of holistically capturing the magnitude and phase of an incident optical (electric) field.Compare to the capture by conventional photography or the human eye -> records only I (E2)

Holographic Theory● Consider a coherent plane wave source (i.e. collimated laser beam) split into two

branches● One branch directly illuminates a recording media (film or CCD)● The other branch is scattered along its path to the recording device.

Holographic Theory● Direct reference beam

● Scattered Object Beam

● Total Field at Detector Plane

Eref=∣E ref∣ejwt

Eobj=∣E obj∣ej (wt+ϕ)

E tot=∣E obj∣e j(wt+ϕ)+∣Eref∣e jwt

Holographic Theory

● Total Field at Detector Plane

Using a square-law optical detector (CCD, photographic plate), what is recorded is intensity of the total field

● Intensity at Detector

E tot=∣E obj∣e j(wt+ϕ)+∣Eref∣e jwt

I tot=∣E obj+E ref∣2

Holographic Theory

● Total Field at Detector Plane

Using a square-law optical detector (CCD, photographic plate), what is recorded is intensity of the total field

● Intensity at Detector

E tot=∣E obj∣e j(wt+ϕ)+∣Eref∣e jwt

I tot=∣E obj+E ref∣2

I tot=E obj E obj+E ref E ref+E ref E obj+Eobj E ref

Holographic Theory

The recorded image (hologram) is modeled as a linear function of the incident intensity. For photographic film, the form is

For a CCD the bias illumination is generally dropped yielding a simpler form

h=(E obj E obj+E ref Eref+E ref E obj+Eobj E ref )β

h=h0+β I

h=β I

Holographic Theory

The hologram plane can now be treated as a transmission screen. Illuminating the hologram with a replica of the reference beam gives a transmitted source field

E s=(Eobj E obj+E ref E ref+E ref E obj+E obj E ref )βE ref

E s=hE ref

E s=β(Eref (∣E ref∣2+∣E obj∣

2)+∣E ref∣2 E obj+E obj Eref

2)

Numerical Propagation MethodsThe illuminated hologram plane can be treated as an illuminated aperture of scaled and phase shifted transmission at each point.

General diffraction theory (Huygens) states that every point on an aperture can be treated as an individual point source radiator. For re-propagation then, it is necessary to calculate the field at some observation plane away from the source plane.

Numerical Propagation MethodsThis superposition of discrete point sources propagating to an observation plane is simply a convolution of the illuminated hologram plane and the point source (PSF)

This is equivalent to multiplication in the Fourier frequency domain

E p=hE ref∗PSF

F (E p)=F (hE ref )F (PSF )

E p=F−1[F (hE ref )F (PSF )]

Numerical Propagation MethodsThe PSF is given spatially by:

and its Fourier transform is known analytically as psf (not shown here, big equation)

Both of which can easily be implemented numerically.

PSF=eikr / r

Numerical Propagation MethodsThus there are two options for propagating the field from the hologram plane:

(Impulse Function)

(Transfer Function)

E p=F−1[F (hE ref )F (PSF )]

E p=F−1[F (hE ref ) psf ]

Simulated Pt. Source IF Reconstruction

Example:

Hologram FFT Reconstruction

Experimental Holographer Setup

Experimental Holographer Setup

Experimental Transmission Screen IF Reconstruction

Hologram

FFT

Reconstruction

Sample: Metal transmission screen w/Geometric patterns ~.5mm

Works Cited

U. Schnars, W. Jueptner. Digital Holography. Springer Publishing;2005.

Goodman. Fourier Optics. 3rd Ed. Roberts and Company Publishing; 2005.