In fluorescence microscopy, an important tool in biology, a laser beam is absorbed by target moleculesin a sample. These molecules are then imaged by a microscope as they emit longer-wavelengthphotons in quantum jumps back to lower energy levels, a process known as fluorescence. A variation on this technique is two-photon excitation. If two photons are absorbed simultaneously, theirenergies add. Consequently, a molecule that is normally excited by a photon of energy Ephoton canbe excited by the simultaneous absorption of two photons having half as much energy. For thisprocess to be useful, the sample must be irradiated at the very high intensity of at least 1032 photons/m2· s.This is achieved by concentrating the laser power into a very short pulse (100 fs pulse length) andthen focusing the laser beam to a small spot. The laser is fired at the rate of 108 pulses each second. Suppose a biologist wants to use two-photon excitation to excite a molecule that in normalfluorescence microscopy would be excited by a laser with a wavelength of 420 nm. If she focusesthe laser beam to a 2.0-µm-diameter spot, what minimum energy must each pulse have?

In fluorescence microscopy, an important tool in biology, a laser beam is absorbed by target moleculesin a sample. These molecules are then imaged by a microscope as they emit longer-wavelengthphotons in quantum jumps back to lower energy levels, a process known as fluorescence. A variation on this technique is two-photon excitation. If two photons are absorbed simultaneously, theirenergies add. Consequently, a molecule that is normally excited by a photon of energy Ephoton canbe excited by the simultaneous absorption of two photons having half as much energy. For thisprocess to be useful, the sample must be irradiated at the very high intensity of at least 1032 photons/m2· s.This is achieved by concentrating the laser power into a very short pulse (100 fs pulse length) andthen focusing the laser beam to a small spot. The laser is fired at the rate of 108 pulses each second. Suppose a biologist wants to use two-photon excitation to excite a molecule that in normalfluorescence microscopy would be excited by a laser with a wavelength of 420 nm. If she focusesthe laser beam to a 2.0-µm-diameter spot, what minimum energy must each pulse have?