Power-dependent PL: Lorentzian fits + allometric (power-law) fits

Data: input.csv has 1024 energy points (1.196 to 1.606 eV) and one merged spectrum per power. The powers are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90 and 100 uW (15 spectra).

Method
- Despiking: points more than 8*MAD+50 counts away from a 7-point median filter were replaced by the median value. This changed 2 points at 45 uW, 1 at 60 uW and 1 at 100 uW (cosmic rays near 1.605 eV).
- Each spectrum was fitted over the full range with y = y0 + sum_i (2*A_i/pi) * w_i / (4*(x-xc_i)^2 + w_i^2), using three Lorentzians:
  P1 = main peak near 1.33 eV; P2 = weaker peak near 1.535 eV; P3 = a rising edge above 1.6 eV, which lies mostly outside the window and is included only to absorb that edge.
- Peak intensity = Lorentzian peak height H = 2A/(pi*w), measured above the baseline y0. The fit R^2 is 0.92 at 5 uW and 0.97 to 0.994 at all other powers.
- Allometric fit: y = a * x^b (Origin Allometric1), with x = power in uW.

Fit results per power (H in counts, xc in eV)
P(uW)  H1      xc1      w1      H2     xc2
5      122.7   1.3180   0.0692  79.1   1.5363
10     226.6   1.3231   0.0707  109.4  1.5307
15     319.3   1.3252   0.0701  153.9  1.5322
20     409.8   1.3269   0.0675  199.6  1.5334
25     500.8   1.3281   0.0663  244.8  1.5346
30     593.9   1.3291   0.0667  281.4  1.5363
35     669.2   1.3295   0.0658  299.1  1.5352
40     792.7   1.3307   0.0646  354.0  1.5373
45     853.1   1.3310   0.0644  372.3  1.5359
50     947.8   1.3319   0.0647  417.9  1.5357
60     1131.8  1.3324   0.0636  489.8  1.5358
70     1321.5  1.3336   0.0628  544.2  1.5372
80     1501.9  1.3338   0.0621  606.6  1.5397
90     1723.3  1.3344   0.0613  659.0  1.5392
100    1989.3  1.3351   0.0602  715.9  1.5391

Allometric fits y = a*x^b
Plot 1, peak intensity vs power:
  P1 (main, ~1.33 eV): a = 20.1 +/- 1.6,  b = 0.990 +/- 0.019  (linear, b close to 1)
  P2 (~1.535 eV):      a = 18.7 +/- 0.8,  b = 0.793 +/- 0.010  (sublinear)
  (For reference, the integrated Lorentzian area A gives b = 0.911 +/- 0.014 for P1 and 0.80 +/- 0.03 for P2.)
Plot 2, peak position vs power:
  P1: a = 1.31028 +/- 0.00046 eV, b = 0.00412 +/- 0.00010. The peak blueshifts from 1.3180 to 1.3351 eV (about 17 meV) between 5 and 100 uW.
  P2: a = 1.5285 +/- 0.0021 eV,  b = 0.0014 +/- 0.0004. The position is nearly constant, at about 1.531 to 1.540 eV.

Robustness: I also tried a fit window cut at 1.59 eV, 2 or 3 peaks, and a baseline fixed at zero. P1 intensity gave b between 0.988 and 0.996 in every variant.

Plot files: intensity_vs_power.svg, position_P1_vs_power.svg and position_P2_vs_power.svg. Fit table: fit_results.csv.

The main-peak intensity exponent is reported below.

RESULT: b = 0.990 ± 0.019
