The setup for the recording of digital holograms is depicted in Fig. . The 532 nm coherent light from a miniature pulsed YAG laser (Continuum Minilite) operating at about 2 mJ per Q-switched 10 ns pulse is used for hologram recording. The laser output, spatial-filtered and collimated, is split into reference and object beams in an interferometer based on the Mach-Zender configuration. A magnified image of an object specimen is projected onto the CCD camera, as well as a similarly magnified reference beam. A pair of similar microscope objectives, either (20× 0.4NA) or (40× 0.65NA) depending on the desired lateral magnification, are placed in the two optical branches to match the curvatures of the two wave-fronts. The camera (Sony DFW-V500) has an array of 640 × 480 pixels on 4.7 × 3.6 mm2 active area, with 8-bit gray scale output, and the electronic shutter speed of the of the camera is usually set at 1/30 sec. A digital delay generator (Stanford Research DG535) triggers both the laser and the camera at a repetition rate of 20 Hz. An IEEE1394 cable connects the camera to the desktop computer, which processes the acquired images and calculates the holographic diffraction using a number of programs based on LabVIEW® and MatLab®.
Apparatus for digital holography experiments. BS = Beamsplitter, L1; L2 = Microscope objectives, DDG = Digital Delay Generator, REF = Reference beam, OBJ = Object beam; H = Hologram plane;
The hologram which is created by the interference between the object and reference fields, |H|2 is recorded by the camera and consists of;
|H|2 = |O|2 + |R|2 + O*R + OR* (1)
where O and R are the object and reference fields, respectively. Separate exposures and subtraction of the zero-order background |O|2 and |R|2 helps to reduce the zero order term significantly. Once |O|2 and |R|2 are removed the holographic twin image O*R + OR* terms remain. A slight angle is introduced between the object and the reference beams by tilting the beam splitter BS2 in order to separate the conjugate twin images, O*R + OR* for an off-axis hologram. The object is moved back a distance z from the focal plane of the CCD to create the out of focus image of the object.
Once the hologram has been captured the optically diffracted field is numerically propagated by use of reconstruction algorithms to calculate the optically diffracted field [19
]. Here we briefly describe some of the commonly applied methods of reconstruction.
Huygens convolution method
In the convolution process [23
] the reconstructed complex wave-field E
) is found by
and the convolution is
= F-1[F (E0)·F (SH)]
where SH is the Huygens PSF
The whole process requires three Fourier transforms, which are carried out using the FFT algorithm. The pixel sizes of the images reconstructed by the convolution approach are equal to that of the hologram. In order to achieve as high a lateral resolution as possible, one keeps the object-hologram distance as short as possible, but the discrete Fourier transform necessitates a minimum distance such that:
where ax is the frame size and nx is the number of pixels in ax. At too close a distance, the spatial frequency of the hologram is too low and aliasing occurs. Normally the object is placed just outside this minimum distance.
Fresnel transform method
The Fresnel transformation is the most commonly used method in holographic reconstruction. The approximation of spherical Huygens wavelet by a parabolic surface allows the calculation of the diffraction integral using a single Fourier transform. The PSF can be simplified by the Fresnel approximation as:
and the reconstructed wave-field is:
The pixel resolution Δx of the reconstructed images determined directly from the Fresnel diffraction formula will vary as a function of the reconstruction distance z as:
where N is the number of pixels and Δx0 is the pixel width of the CCD camera. As with the Huygens convolution method there is a minimum z distance requirement set by Equation 5.
Angular spectrum method
The angular spectrum method of reconstruction has the significant advantage that it has no minimum reconstruction distance requirement as is this case for the Fresnel and convolution methods[24
]. If E0
;0) is the wave-field at plane z
= 0, the angular spectrum A
;0) at this plane is obtained by taking the Fourier transform:
;0) = E0
where kx and ky are corresponding spatial frequencies of x and y. Fourier-domain filtering can be applied to the spectrum to block unwanted spectral terms in the hologram and select a region of interest corresponding only to the object spectrum. Subsequently the wave-field E0(x0, y0;0) can be rewritten as the inverse Fourier transform of its angular spectrum,
;0) = A
The new angular spectrum at plane z, A(kx, ky;z) is calculated from A(kx, ky;0) as
A(kx, ky;z) = A(kx, ky;0) exp[ikzz] (11)
The reconstructed complex wave-field of any plane perpendicular to the propagating z axis is found by
) = A
Two Fourier transforms are needed for the calculation in comparison to the one needed by the Fresnel transform. However once the field is known at any one plane, only one additional Fourier transform is needed to calculate the field at different values of z. This method allows frequency-domain spectrum filtering to be applied, which for example can be used to block or remove the disturbance of the zero order and twin image components. A signficant advantage of the angular spectrum is that there is no minimum z reconstruction distance requirement.