Temporal focusing allows for optically sectioned wide-field microscopy. The optical sectioning arises because this method takes a pulsed input beam, stretches the pulses by diffracting off a grating, and focuses the stretched pulses such that only at the focal plane are the pulses re-compressed. This approach generates nonlinear optical processes at the focal plane and results in depth discrimination. Prior theoretical models of temporal focusing processes approximate the contributions of the different spectral components by their mean. This is valid for longer pulses that have narrower spectral bandwidth but results in a systematic deviation when broad spectrum, femtosecond pulses are used. Further, prior model takes the paraxial approximation but since these pulses are focused with high numerical aperture (NA) objectives, the effects of the vectorial nature of light should be considered. In this paper we present a paraxial and a vector theory of temporal focusing that takes into account the finite spread of the spectrum. Using paraxial theory we arrive at an analytical solution to the electric field at the focus for temporally focused wide-field two-photon (TF2p) microscopy as well as in the case of a spectrally chirped input beam. We find that using paraxial theory while accounting for the broad spectral spread gives results almost twice vector theory. Experiment results agree with predictions of the vector theory giving an axial full-width half maximum (FWHM) of andrespectively as long as spectral spread is taken into account. Using our system parameters, the optical sectioning of the TF2p microscope is found to be . The optical transfer function (OTF) of a TF2p microscope is also derived and is found to pass a significantly more limited band of axial frequencies than a point scanning two-photon (2p) microscope or a single photon (1p) confocal microscope.
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