where e is the energy of the beam in joules, τ is the duration of the pulse in seconds, a is the illuminating area in square meters, and f is the repetition rate of the pulsed beam in hertz. increasing the illumination area thus reduces the amount of generated nonlinear photons unless the energy is increased. optical damage depends on the energy density, i.e. peak intensity per area i=e/(τa). therefore, both the area and energy can be easily increased without the risk of optical damage if the peak intensity per area is kept low, and yet a gain in the amount of generated nonlinear photons can be obtained because of the quadratic dependence. for example, increasing both the area and energy 1000 fold, leaves the peak intensity unchanged but increases the generated nonlinear photons by 1000 fold. this 1000 extra photons are indeed generated over a larger area. in imaging this means that the extra 1000 photons are spread over the image, which at first might not seem an advantage over multiphoton scanning microscopy. the advantage however becomes evident when the size of the image and the scanning time are considered. the amount of nonlinear photons per image frame per second generated by a wide-field multiphoton microscope compared to a scanning multiphoton microscope is given by