A Digital Holographic Microscope for Measuring Three Dimensional Particle Distributions and Motions

J. Sheng

1. Motivation

There is a growing interests in understanding micro-scale bio-physical processes, such as the kinematics and dynamics of swimming micro-organisms, e.g. bacteria, dinoflagellates or nauplii, and their interactions with the surrounding fluids. Direct observations on such processes require suitable tools that are capable of resolving both temporal and spatial scales at the appropriate levels. The readily available candidate is the optical microscope. However, as shown in Figure, as the power of the microscope increases and the lateral resolution (1.22l/NA, where l is the wavelength and NA is numerical aperture of the entire optical system) improves, the field of view and depth of field decrease non-linearly to a very thin layer. For example, increasing the power from 10x to 40x reduces the theoretical depth of field from 12mm to 2mm, greatly limiting the size of resolvable volume. Holography, on the other hand, is capable of recording a 3D volumetric field on a single plane (hologram plane) and later reconstructing it. With the recent development of in-line digital holography for particulate flows, it is now possible to record the hologram on a digital medium, and then reconstruct the sample volume numerically. we combines in-line digital holography and a conventional microscope objective in order to circumvent the obstacles associated with the limited resolution of a digital recording medium. Using the same setup as an optical microscope, we replace the light source with a collimated, coherent laser beam, and record a stream of magnified holograms on a CCD camera. The 3D fields can be reconstructed from these magnified holograms at almost the same resolution as the optical microscopes. Reconstructed holograms of sample volumes with depth of 1 to 10 mm, containing particles ranging in size between 0.75 to 3 mm, demonstrate the efficacy of the Digital Holographic Microscope (DHM) as viable means of extending the depth of field of a microscope by almost three orders of magnitude.

2. Methodology

2.1 Optical Setup

The optical setup is very similar to a conventional transmission light microscope, but instead of using white light, we replace it with a coherent laser beam. In the current setup, we spatially filter a 3mw, He-Ne laser beam using a 25mm pinhole, expand and collimate the beam to 30mm diameter, and then illuminate the sample volume. Since the resulting intensity, 0.33mW/cm², is still too high, we use a variable Neutral Density (ND) filter to further attenuate the beam. In most of the tests, a filter of ND=1 is used, reducing the illumination intensity on the specimen to ~30mW/cm². A bright field microscope objective with proper tube length is used to imaging the optical field (hologram) onto the digital recording medium (CCD sensor). Note that the object plane is located outside of the sample volume.

2.2 Analysis of Microscopic Holography

A hologram is a record of interference between light scattered from objects, e.g. micron or sub-micron particles, and a reference beam with known phase distribution ¹⁹. One can represent the optical field at the hologram plane as

, (1)

where is the propagation vector of the reference beam and is the norm vector of the hologram plane. The first term represents the optical field of the reference beam, where the phase accounts for its angle with the scattering light, assuming that the hologram is perpendicular to the scattering axis. In the following analysis we assume that this angle is zero. The second term is the superposition of light scattered from discrete particles located at a distance z_i from the hologram plane, and produce (by being illuminated) fields with local distributions of . Thus, each particle is considered as a superposition of point sources, whose individual fields are . Using a paraxial approximation for particles much smaller than ,

(2)

If the scattering is diffraction dominated, as in in-line holography, each particle can be considered as a 2D aperture with a shape equal to its cross-section normal to the incident light. Thus, scattering from an individual particle is a convolution of a 2D aperture with the impulse response function (Eq. 2). The resulting interference intensity on the hologram plane, , is

(3)

where indicates a convolution integral. To determine the effect of the microscope objective, we model its compound lens system as a perfect thin lens. The optical field at the distance behind the lens, resulting from an optical disturbance, , where the object distance before the lens, is

where

and is the magnification. Replacing with and performing the integration, the optical field generated by the hologram at the image plane is

(4)

Thus, the image plane contains a magnified hologram plane with a phase correction that becomes unity when the magnification is sufficiently large. The intensity distribution in the image plane simply becomes a magnified hologram

. (5)

which contains the four contributors presented in Eq. 3. This true magnified hologram enables us to drastically relax the spatial resolution requirement of recording medium. Furthermore, we can use the magnification as a means of matching the desired resolution with that of the recording medium. As shown in this paper, the magnified holograms.

3. Results

3.1 Three-dimensional Particle Distribution

(a) Part of a hologram recorder using a 10X objective, containing 3.189 mm diameter particles in a 1mm deep solution. (b - d) Reconstruction of planes located 120mm, 580mm and 800 mm from the hologram plane. In focus particles appear as dark spots on the bright background. (e) A combined/compressed image containing all the particles covered by the hologram section shown in a. (f) The location of all the particles detected within the entire 1.5x1.5x1 mm³ volume, totaling 5769 particles.

3.2 Holographic Cinematography and Sub-micron Resolution

A demonstration of a cinematographic DHM. (a) Combined/compressed tracks, consisting of 5 exposures, of 3.189mm particle located within a 1mm deep sample; (b) Sample tracks, consisting of 7 consecutive exposures, of 0.75mm particles, combined over a depth of 100mm.