Single longitudinal mode laser diodes

Laser is a light source that emits photons described by a “near coherent” optical field. By “coherent optical field,” one means the monochromatic electromagnetic field at optical frequency (ω 0) with constant amplitude (A) and phase (θ). Therefore, a coherent optical field can be expressed as A cos (ω 0 t + θ), with phase θ determined by a reference starting time. Any laser built as an oscillator must take spontaneously emitted photons as its initial driven seed, since a laser doesn’t have any coherent light as its input. Knowing the fact that the spontaneous emission is a random process, one therefore cannot expect that the laser will give an ideal coherent optical field output. Rather, the laser emits photons described by the ideal coherent optical field, driven by spontaneously emitted photons, for a certain amount of time τ 0 until the emerging of another group of spontaneously emitted photons. Consequently, its output field will experience a sudden change on its amplitude and phase at time τ 0 and this sequence keeps repeating indefinitely. As such, one can express the laser output as a “quasi-monochromatic” electromagnetic field at optical frequency ω 0 as [A + δ a (t)] cos [ω 0 t + θ (t)], with δ a (t) as a random process with zero mean and θ (t) also a random process with uniform distribution between 0 and 2φ. Once a laser is operated under a bias beyond its threshold, A > > | δ a (t) | is satisfied. Hence the only nonideality of a laser output from a coherent optical field lies in 110its random phase. The difference becomes more apparent if one observes the output optical spectrum of a laser. Actually, as the Fourier transform of an ideal coherent optical field, its frequency-domain spectrum is a delta function that appears at ω 0 with an amplitude of | A |. The quasi-monochromatic field, however, has its averaged spectrum in the shape of a “broadened” delta function still peaked at ω 0, but with an amplitude in 1/2 φ | A | τ 0 and a width of Δ ω = 2 φ/τ 0. This is because the averaged duration between two consecutive abrupt phase changes is τ 0, which means the quasi-monochromatic field can be described as a series of truncated ideal coherent fields known as a wave train, with τ 0 as the truncation window or the averaged length of the wave train. The Fourier transform of a single truncated coherent field piece corresponds to the convolution between a delta function and a sampling function in the form of sin (ω τ 0/2)/(ω τ 0/2) as the Fourier transforms of the ideal coherent field and the flat window function (i.e., 1 for t inside τ 0 and 0 elsewhere), respectively, which gives the result as the aforementioned broadened delta function. Since distinct truncated coherent field pieces in the wave train differ by a time shift only, their Fourier transforms differ just by a phase. Hence, the frequency-domain amplitude spectra of the quasi-monochromatic field as a wave train composed of all these pieces are overlapped as a single peak as shown in Figure 30.1.