To determine how to double the number of photons in a beam of laser light with a wavelength of 500 nm, let's carefully evaluate each option:
### Option 1:
Choose a light-emitting material with an excited state at twice the energy gap from the next lowest state, so that two photons are emitted in a single transition rather than one.
This option suggests altering the material properties such that it emits two photons for each transition. While theoretically intriguing, practically, most laser materials and transitions are not designed to operate this way. For most common laser systems, this is not a feasible or straightforward solution.
### Option 2:
Double the intensity of the beam, which you might do by doubling the amount of light-emitting substance in the laser (for instance, ruby).
Intensity is directly proportional to the number of photons being emitted. By increasing the amount of the light-emitting substance, say by doubling it, you are effectively doubling the number of photon-emitting interactions. This results in a direct doubling of the number of photons in the beam. This is a practical and effective solution to increase the number of photons in the laser beam.
### Option 3:
Double the wavelength of the beam, which you might do by doubling the number of times you pump the light-emitting material to reach excited states.
Doubling the wavelength of the light does not increase the number of photons; instead, it changes the energy of the photons. The relationship between wavelength and energy is [tex]\( E = \frac{hc}{\lambda} \)[/tex], where [tex]\( E \)[/tex] is energy, [tex]\( h \)[/tex] is Planck's constant, [tex]\( c \)[/tex] is the speed of light, and [tex]\( \lambda \)[/tex] is the wavelength. Thus, this approach fundamentally changes the character of the light rather than increasing the number of photons.
### Option 4:
Double the frequency of the beam, which means photons are being emitted twice as fast by the light-emitting material.
Doubling the frequency corresponds to halving the wavelength because frequency ([tex]\( f \)[/tex]) and wavelength ([tex]\( \lambda \)[/tex]) are inversely related by the speed of light ([tex]\( c = \lambda f \)[/tex]). However, emitting photons "twice as fast" does not necessarily double the number of photons unless power or energy input is also doubled accordingly, which is implied indirectly but not explicitly stated. This method involves changing the nature of the emission process and is not as straightforward as simply increasing the intensity.
Conclusion:
By analyzing and comparing practical feasibility and direct impact on the number of photons emitted in the laser beam, Doubling the intensity of the beam (Option 2), for instance by doubling the amount of the light-emitting substance, is the most straightforward and effective solution. This directly affects the number of photons and is a practical method used in laser systems. Therefore, the correct choice is option 2.
