In this problem you will investigate the parameters involved in a single-molecule fluorescence experiment. Specifically, the incident photon power needed to see a single molecule with a reasonable signal-to-noise ratio will be determined.

a. Rhodamine dye molecules are typically employed in such experiments because their fluorescence quantum yields are large. What is the fluorescence quantum yield for Rhodamine B (a specific rhodamine dye) where k _f = 1 × 10⁹ s^–1 and k _ic = 1 × 10⁸ s^–1? You can ignore intersystem crossing and quenching in deriving this answer.

b. If care is taken in selecting the collection optics and detector for the experiment, a detection efficiency of 10% can be readily achieved. Furthermore, detector dark noise usually limits these experiments, and dark noise on the order of 10 counts s–1 is typical. If we require a signal-to-noise ratio of 10:1, then we will need to detect 100 counts s^–1. Given the detection efficiency, a total emission rate of 1000 fluorescence photons s^–1 is required. Using the fluorescence quantum yield and a molar extinction coefficient for Rhodamine B of ~40,000M^–1 cm^–1, what is the intensity of light needed in this experiment in terms of photons cm^–2 s^–1?

c. The smallest-diameter focused spot one can obtain in a microscope using conventional refractive optics is one-half the wavelength of incident light. Studies of Rhodamine B generally employ 532-nm light such that the focused-spot diameter is ~270 nm. Using this diameter, what incident power in watts is required for this experiment ? Don’t be surprised if this value is relatively modest.