Quantitative Phase Amplitude Microscopy of Three Dimensional Objects

C.J. Bellair§,+, C.L. Curl#, B.E.Allman*, P.J.Harris#, A. Roberts§, L.M.D.Delbridge# and K.A.Nugent§

§School of Physics, The University of Melbourne, Vic., 3010, Australia#Department of Physiology, The University of Melbourne, Vic., 3010, Australia

*IATIA Ltd, Rutland Road, Box Hill, Vic., 3128, AUSTRALIA

Introduction Quantitative Phase Amplitude Microscopy is a non-interferometric phase imaging technique which uses the Transport of Intensity Equation to recover unique phase maps from intensity images. This phase imaging technique is well understood when thin objects are being imaged, objects whose thickness is less than the depth of field of the imaging system. In practice, especially in the biological sciences, objects will extend outside the depth of field of the imaging system and so the intensity images will contain out of focus information. This work looks at how the out of focus information contributes to phase images and how the phase images of thick objects relate to the objects themselves.

Quantitative Phase Amplitude Microscopy can be described using three dimensional optical transfer functions [Barone-Nugent et al 2002]. The optical transfer functions were used to theoretically predict the behaviour of the phase under different imaging conditions for objects of different sizes, shapes and thicknesses. These theoretical predictions have been compared to experimental phase results to test the validity of the optical transfer function theory and have also been used to construct guidelines for the imaging of thick objects and the interpretation of phase images.

Thick ObjectsThe depth of field of an imaging system is related to its ability to image points separated along the optical axis and depends on the numerical aperture of the objective lens:

Objects whose thickness is less than the depth of field of the system will be perfectly focused by the system, however images of objects whose thickness extends outside the depth of field will contain out of focus information.

If we assume the Born Approximation, thick objects can be treated as being the sum of many thin objects, each of which can be perfectly focused by the imaging system at some point along the optical axis.

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NA-1λ D

Phase Images of Thick ObjectsTheoretical phase predictions were made for objects of different thicknesses, T, and diameters, W, with circular cross sections perpendicular to the optical axis. These predictions show the imaging conditions under which accurate phase values are obtained for objects with different eccentricity, . In general, the results show that accurate phase values can be obtained for objects with eccentricities less than or equal to one.

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ReferencesE.D.Barone-Nugent, A.Barty and K.A.Nugent, Quantitative Phase Amplitude Microscopy I: Optical Microscopy , J.Microscopy, 206, 194-203 (2002).Barty A, Nugent K.A., Roberts A. and Paganin D., Quantitative Phase Microscopy, Optics Letters, 23, 817-819 (1998) Streibl, N., Three-Dimensional Imaging by a Microscope, J.Opt.Soc.Am., A2, 121-127 (1985)

ResultsThe theoretical phase predictions for a sphere were compared to the experimental phase values from a rat red blood cell. These results can be seen below. Under normal imaging conditions the agreement between the theory and the experiment is very good implying that the optical transfer functions can be used to predict phase images of thick objects. While this example focuses on a simple object, the method can easily be used to deal with more complex objects.

Experimental ObjectsTo obtain experimental phase values under different imaging conditions simple experimental objects were needed with little internal structure and a regular shape to ensure that changes in phase values could be ascribed to changes in the imaging conditions not to properties of the experimental object itself. These conditions were met by rat red blood cells which have a constant refractive index throughout and are roughly spherical in certain conditions.

These cells have diameters, and consequently thicknesses, ranging from 5-8μm. When imaged with a X40, NA=0.7 objective these cells extended over 7-10 depths of field making them thick objects and providing many different imaging conditions to view the cells under.

Phase Image of Rat Red Blood Cell

Phase Profile of Rat Red Blood Cell

Amplitude Transfer Function

Phase Transfer Function

Optical Transfer Functions - Theoretical ModelThe three dimensional optical transfer functions can be used to simulate an imaging system and to simulate intensity images of objects. The exact shape of the transfer functions depends on the position of the image plane along the optical axis and the condenser setting.

The phase and amplitude transfer functions decouple from each other making it easy to simulate intensity images at different points along the optical axis of the imaging system. These intensity images can then be used to predict the phase images of different objects. This model has been shown to work for thin objects.

The phase image of a thick object can be predicted by treating each thin slice of the object separately. The phase image of each thin slice can be calculated and then the overall phase image can be found by adding together the phase from each slice.

ConclusionsThe Optical Transfer Function formalism is a good way to describe the imaging of three dimensional objects and can be used to find phase values of thick objects under different imaging conditions. This theory predicts phase images of objects which accurately reflect the optical properties of the object if the ellipticity of the object is less than or equal to one even when the sample is thicker than the depth of field of the imaging system.