Theory of Image Formation in Confocal Microscopy




For a long time I have been trying to find a simple mathematical description for the three-dimensional (3-D) image in a microscope or confocal microscope. Eventually, together with my colleagues, Dr. Wieland Weise, Dr. Oleg Lobkis and Prof. Siegried Boseck, we developed the theory of three dimensional image formation of strong scatterers in scanning acoustic microscopy (SAM) and scanning optical microscopy ( Zinin, Weise, Lobkis, Boseck. Wave Motion. 1997. 25. 212-235 ). We formulated the theory of the image formation in reflection and transmission microscopes using the angular spectrum approach (P. Zinin and W. Weise, "Theory and applications of acoustic microscopy", in T. Kundu ed., Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, CRC Press, Boca Raton, chapter 11, 654-724 (2003). ). The theory showed that the image of a closed object could be represented as a two fold two-dimensional Fourier transform of a far field scattering amplitude of the object. Contrast in the reflection microscope is mainly due to scattering from the boundary of the object and can be described as a thin layer whose medium surface is in coincidence with the boundary of the objects. In the transmission microscope the contrast is connected with the contour of the object. The proposed theory provides the opportunity to investigate the imaging process for objects having an arbitrary shape. As you can see from the images of the spheres provided below, the theory works very well in acoustic microscopy. Later, we showed that the same theory can be used for image interpretation in optical scanning microscopy. Images of a simple spherical particle look surprisingly different from what we expected without knowing the theory.


Reflection SAM


Theory: x-z scan of 3-D image of steel sphere

Model of the reflection and transmission acoustic microscopes

Experiment: x-z scan of 3-D image of steel sphere.


Transmission SAM





Theory: x-z scan of 3-D image of steel sphere.

Theory: x-z scan of 3-D image of liquid drop

Theory: x-z scan of 3-D image of plexiglass sphere.


Important conclusion from the theory we developed is that size of the spherical particle can be determined only from image taken by a transmission microscope. The size of the image of the spherical particle in reflection microscope is less than the real size of the particle and is equal to a sin(α), where a is the radius of the particle, and α is semiaperure angle of the lens. Our theory has found several direct application in practical microscopy: surface imaging and in developing Emulated Transmission Confocal Raman Microscopy. 




Subsurface Imaging: The angular spectrum approach was used to develop the theory of subsurface imaging in acoustic microscopy. It take into account reflection and transmission of the sound beam at the liquid-solid interface. The theory is of importance for understanding acoustical images of the internal microstructure of the non-transparent solids (Lobkis, Kundu,  Zinin. Wave Motion. 1995. 21 183-201).


SAM image made by OXSAM of a standard epoxy layer on aluminum at 300 MHz: (a) Z = 0; (b) Z = - 20Ám. Subsurface image clearly identifies voids at the epoxy/aluminum interface.




Emulated Transmission Confocal Raman Microscopy: A single cell or clusters of cells on glass substrate can be clearly seen in optical microscopes operating in transmission configuration. However, the glass substrate emits strong Raman signal as compared to that of a single cell, and this hampers the Raman signal coming from the sample. Replacing the glass substrate by other non-transparent materials introduces some degree of difficulty in optical imaging of the cell in the reflection mode. In order to obtain high-contrast optical images and to measure Raman spectra of cells attached to a substrate using the reflection confocal Raman microscope we developed an "emulated transmission confocal Raman microscopy" where the cells are attached to a reflective mirror (Zinin, Misra, Kamemoto, Yu, Sharma. J. Opt. Soc. Am.. 24, 2779, 2007)..