La razón renacentista como sueño de totalidad

This section briefly discusses the shape of the solutions to the model. Figure 2.12 shows the simulated internal photon flux density distribution modeled with the parameters L = 0.6 cm,

ˆ

ξsp = 0.0005× 1024 photons (cm3 s)−1, ¯g = 1.384 (cmA)−1, αw = 4 cm−1, φs = 85.4×

1024 photons (cm2 s)−1. The blue curves represent the photons traveling to the right, φ1(z),

the green curves represent the photons traveling to the left, φ2(z), while the red curve is the

sum of both, φ1(z) + φ2(z).

Figure 2.12 Simulated internal photon flux density distribution modeled with the parameters L =

0.6 cm, ˆξsp = 0.0005× 1024 photons (cm3 s)−1, ¯g = 1.384 (cmA)−1, αw = 4 cm−1, φs = 85.4×

1024 photons (cm2 s)−1. Blue curve: φ₁(z); Green curve: φ₂(z); Red curve: φ₁(z) + φ₂(z). Left plot:

Uncoated, symmetric resonator with R₁ = R₂ = 28% at I = 5A. Center plot: Coated, asymmetric

resonator with R₁ = 28% and R₂ = 1% at I = 5A. Right plot: Coated, asymmetric resonator with

R₁= 28% and R₂= 1% at I = 7 A.

The left plot shows the uncoated, symmetric resonator with R1 = R2 = 28% at I = 5 A. The

green and blue branches are clearly curved upward in their respective directions of propagation and have similarities to exponential functions. The red line has a minimum in the center of the resonator, and since it is this combined photon density that saturates the gain, the gain has a maximum at the center. Note how at the facets, at z = 0 and z = 0.6 cm, due to the reflectivity of 0.28, the reflected photon flux has a value of 0.28 times the outgoing, which are

the boundary conditions of the problem. In these areas, where the amplitude of the outgoing wave has a greater amplitude than the returning wave, the combined wave (red) nearly has

the form of an outward traveling wave. Only towards the center, where φ1 = φ2, does the

wave take the shape of a standing wave.

The center plot shows the case where the right facet is AR-coated and has a residual

reflectivity of R2 = 1% and is driven at the same current. Thus, at z = 0.6 cm, the green

curve is a hundred times smaller than the blue curve and the resonator is not symmetric. It is interesting that although most light is lost to the right, the steady state demands that the internal photon density is highest at this end. Note also that the absolute value of the photon flux is three orders of magnitude smaller than for the left plot. This is because due to the increased losses from the right facet, the lasing threshold is not reached yet and what can be seen here is only amplified spontaneous emission. The right plot shows the same coated laser as the central plot, but driven above threshold at I = 7 A to reach approximately the same magnitude of photon flux as the case plotted on the left. Note how the blue and green curves in the right plot have a much greater curvature than the equivalent curves in the central plot. This is due to the fact that due to stronger pumping there is much greater gain (exponential character), while the curves in the central plot are more strongly dominated by spontaneous emission (linear character).

Figure 2.13 Simulated internal photon flux density distribution, φ₂(z), for the uncoated, symmetric

resonator with R₁= R₂= 28%, modeled with the parameters L = 0.6 cm, ˆξsp= 0.0005× 1024photons

(cm3s)−1, ¯g = 1.384 (cmA)−1, αw= 4 cm−1, φs= 85.4×1024photons (cm2s)−1. Left plot (upwards):

I = (1, 2, 3) A. Center plot (upwards): I = (5, 6, 7)A. Right plot (upwards): I = (10, 11, 12, 13, 14)A. The left-traveling photons were not plotted, since in a symmetric resonator they are just the mirror images of the plotted curves.

Figure 2.13 shows the internal photon flux density distribution of the same uncoated laser as in Fig. 2.12 a), modeled with the same parameters, but driven at various currents. The

0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 IA Φ _out H 10^24 phot cm^2 s L

Figure 2.14 P-I curve associated with the plotted photon densities of Fig. 2.13. Dots mark the

currents at which the photon densities are plotted in Fig. 2.13. For better readability, the power is plotted in units of photon flux density, but this can be converted to actual power with the use of Eq. (2.63).

photons traveling to the left and the sum ϕ1(z) + ϕ2(z) are omitted, since for a symmetric

resonator ϕ2(z) = ϕ1(L − z), and the green and red curves are redundant. The left plot

shows current values of 1, 2, and 3 A (lowest curve first). These are all below the lasing threshold as can be seen in Fig. 2.14, which shows the corresponding simulated P-I curve, where the current values of the various plots in Fig. 2.13are marked (the first three dots that lie below the threshold kink are the values of the curves in the left plot and so on). For better comparability with Fig. 2.13, the power plotted Fig 2.14 is in units of photon flux density, but this can be converted to actual power with the use of Eq. (2.63). The curve at 3 A is very nearly linear. This is because at 3 A the gain γ almost exactly equals the distributed loss αw. This is the so-called transparent resonator case, where there is neither gain nor loss

and the only light propagating is the unobstructed (but accumulating) spontaneous emission. The two lower-lying curves are bent downwards in the direction of propagation, meaning loss per unit distance is greater than gain, thus the accumulating spontaneous emission is partially scattered. The curves in the central plot are taken at 5, 6, and 7 A, which all lie above threshold. It can be seen that the absolute photon densities are 4-5 orders of magnitude higher than in the left plot and that towards higher pump currents the curves get more curved, i.e. more exponential in character. The right plot shows pump currents of 10, 11, 12, 13, and 14 A, which is approximately three times the threshold current. Here gain saturation start to play a role, since the sum of the photon flux densities reach up to 200 scale units, while the saturation photon flux is 85. Thus the curves overall lie closer together since stronger pumping is partially compensated by stronger saturation.

2.5.6 Application of the model to determine the residual facet re-