Simple determination of magnification due to recording configuration in particle field holography C. S. Vikram and T. E. McDevitt

Pennsylvania State University, Applied Research Labora­tory, P.O. Box 30, State College, Pennsylvania 16804. Received 27 June 1988. 0003-6935/89/020208-02$02.00/0. © 1989 Optical Society of America. Holography is a well-established technique for the study of

the dynamic volume of microobjects.1-3 For size distribu­tion studies, the final magnified image size due to holograph-ic4 and auxiliary (e.g., TV system) magnifications is needed. Holographic magnification with collimated recording and reconstruction beams is straightforward. Premagnification during recording and image magnification during recon­struction using lenses are also performed. However, there are applications dealing with completely lensless magnifica­tion using divergent recording and/or reconstruction beams.5-9 At the reconstruction stage, one common mode of the image analysis involves moving the hologram on an x-y-z stage with fixed reconstruction source and observation plane positions. For a divergent reconstruction beam, this results in a variable source-to-hologram distance and hologram-to-image distance but the sum (source-to-image distance) is fixed.

There are two hologram positions yielding the image of the same microobject. The one position near the reconstruction source yields higher magnification.5-9 Both positions can be jointly used for the magnification determination without knowledge of the amount of auxiliary magnification.6,7,9

In any case, at the analysis stage, the magnification m0 from the recording configuration alone as given by

must be known or determined. Here Z0 is the object-to-hologram distance, and zR is the recording source-to-holo­gram distance. ZR and z0 both are negative for the case of divergent beam recording. For divergent beams (zR ≠ ∞) and naturally unknown object distances z0 in a volume, the information about m0 is not straightforward.

Bexon6 suggested a method for determining m0. This involves obtaining the reconstruction source-to-observation plane distance so that the hologram is exactly in the middle position for the image formation. For large numbers of objects generally encountered in practical particle field ho­lography, this is a tedious and additional job. Therefore, in this Letter we present a method where m0 is easily obtainable from any position where the image is erect on the observation plane. Since one such position is already known during the ordinary readout process, the approach should be very use­ful. In fact, the approach by Bexon6 corresponds to a partic­ular case of our study. Since the particular case involves adjusting the reconstruction source-to-observation plane distance for each object plane, our general case is far simpler in practice.

Figure 1 represents a typical recording arrangement. For the in-line case χR and yR are simply zero. Similarly Fig. 2 gives the reconstruction arrangement. Again for the in-line case, xc and yc both are zero. The Zc coordinate of the conjugate (commonly used in particle field holography) im­age point is10

208 APPLIED OPTICS / Vol. 28, No. 2 / 15 January 1989

Equation (5) is simple to use. However, if m0 is known or calibrated at a distance Zc, the magnification m'0 at another distance Z'c can be determined using Eqs. (4) and (5) as

Fig. 1. Schematic diagram for recording coordinates of a particle field hologram. For the in-line case, XR and yR are both zero.

Fig. 2. Schematic diagram describing the reconstruction process with a divergent beam. Fixed reconstruction source and observa­

tion planes are used with the variable hologram position.

where μ is the reconstruction-to-recording wavelength ratio given by

In the present mode of analysis, the observation plane (i.e., the image plane) and reconstruction source (center of the divergent beam) are fixed in space. This situation is shown in Fig. 2. If Zc is the positive distance between the hologram and observation plane, then zc is a variable given by

where d is the fixed distance (positive) between the source and observation planes. Obviously zc is a negative quantity for the divergent beam case.

Now, Eqs. (1), (2), and (4) can be combined to obtain

Equation (5) can be used at any position Zc. In the meth­od suggested by Bexon,6 one must adjust zc so that |zc| = | Zc | . To do this for each object plane is very time-consuming.

We have found that for any hologram position, when the image is formed on the observation screen, M0 can be deter­mined from the corresponding values of Zc and zc. There is no need to perform the experiment during the reconstruction stage to arrange for | Zc | = |zc | = d/2. This experiment would have to be performed6 for each object plane by varying d and obtaining the image for the hologram in the middle. Obvi­ously such an approach is very time-consuming. Equation (5) requires the readily available values of Zc and zc at a fixed value of d when the image is formed at the observation plane.

This work was performed at the Applied Research Labora­tory under the sponsorship of the Office of Naval Research, Code 12.

References 1. B. J. Thompson, "Holographic Methods of Dynamic Particulate

Measurements—Current Status," Proc. Soc. Photo-Opt. In-strum Eng. 348, 626 (1982).

2. B. J. Thompson and P. Dunn, "Advances in Far-Field Hologra­phy—Theory and Applications," Proc. Soc. Photo-Opt. In-strum. Eng. 215, 102 (1980).

3. J. D. Trolinger, "Particle Field Holography," Opt. Eng. 14, 383 (1975).

4. See, for example, J. T. Bartlett and R. J. Adams, "Development of a Holographic Technique for Sampling Particles in Moving Aerosols," Microscope 20, 375 (1972).

5. M. E. Fourney, J. H. Matkin, and A. P. Waggoner, "Aerosol Size and Velocity Determination via Holography," Rev. Sci. In-strumn. 40, 205 (1969).

6. R. Bexon, "Magnification in Aerosol Sizing by Holography," J. Phys. E 6, 245 (1973).

7. R. Bexon, J. Gibbs, and G. D. Bishop," Automatic Assessment of Aerosol Holograms," J. Aerosol Sci. 7, 397 (1976).

8. R. Bexon, M. G. Dalzell, and M. C. Stainer, "In-Line Holography and the Assessment of Aerosols," Opt. Laser Technol. 8, 161 (1976).

9. W. K. Witherow, "A High Resolution Holographic Particle Siz­ing System," Opt. Eng. 18, 249 (1979).

10. R. W. Meier, "Magnification and Third-Order Aberrations in Holography," J. Opt. Soc. Am. 55, 987 (1965).

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