Wave theory of V z

A simplified picture of the transducer, lens surface, and reflecting object is shown in Fig. 7.4. The waves radiated by the transducer are refracted by the lens so as to form a spherical wavefront centred on the focal point of the lens. Each point on this wavefront can be described by its angular coordinates from the focus let these be 6 for the zenithal angle (i.e. the angle to the lens axis, again taken to be normal to the specimen surface) and for the azimuthal angle. Thus the spherical wave emerging from the lens can be described by 

Since L (9, 0) and L2(9, 0) depend only on the geometry and material of the lens, it is convenient to combine them in a single pupil function, which may be defined as 

Most imaging lenses are designed to have axial symmetry, so that P is independent of 0. Anisotropic specimens will be considered in Chapter 11, but if the specimen is isotropic then the 0 integration may be absorbed in P, so that the expression for V simplifies to 

If the specimen is moved away from the focal position, then this will cause a phase shift that depends on 6. If the wavenumber in the coupling fluid is k = 2n/Xo, then the z component of the wavevector is kz = k cos 6. Defocusing the specimen by an amount z causes a phase delay of 2zkz, or 2zk cos 0 (the factor of two arises because both the incident wave and the reflected wave suffer a change in path length). Expressing this phase delay as the complex exponential of a phase angle, the response of the microscope with a defocus z is 

This is the basic expression for the response of the microscope to a uniform isotropic specimen at a defocus z. It should be permanently imprinted on the mind A V(z) curve for glass calculated using this expression is shown in Fig. 7.5. The primary difficulty in achieving exact agreement with measured curves lies in knowing the exact pupil function of the lens in the experiment. 

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