Chapter 13: Optical Amplifiers ============================== Source: Hunsperger, sixth edition (2009), Problems 13.1--13.8, printed pages 273--274. Problem 13.1 — Three semiconductor amplifier architectures ---------------------------------------------------------- **Working.** A traveling-wave amplifier suppresses facet reflections so the signal is amplified predominantly in one transit. A Fabry–Perot amplifier retains reflecting facets and reinforces resonant signals through repeated passes, while remaining below threshold. An injection-locked amplifier is a laser biased above threshold whose oscillation locks to a suitably matched injected optical signal. **Answer.** Traveling-wave, Fabry–Perot, and injection-locked amplifiers. They differ in feedback strength and whether the unseeded device oscillates. **Check.** All obtain added optical energy from the electrical supply. An above-threshold injection-locked device is not described by the same unsaturated linear formula as a below-threshold amplifier. Problem 13.2 — Select EDFA pump wavelengths ------------------------------------------- **Working.** Pumping must overlap an erbium absorption band that feeds the metastable upper laser manifold. The useful 980 nm band has finite width, so 982 nm can also excite it; the 1480 nm band directly pumps the upper laser manifold. **Answer.** 980 nm, 982 nm, and 1480 nm are the intended useful choices. The listed 855 and 1350 nm values are not the conventional ground-state pump bands for the illustrated silica EDFA. **Check.** Pump absorption depends on host, temperature, and the actual absorption spectrum. A transition band is not a delta function, so rejecting 982 nm merely because it differs from 980 nm is incorrect. Problem 13.3 — Signal gain with ASE separated --------------------------------------------- **Working.** Amplified spontaneous emission is added noise, not amplified signal. Using 900 mW signal output and 1 mW signal input, .. math:: G_{\rm sig}=900,\qquad G_{\rm sig,dB}=10\log_{10}(900)=29.5424\,{\rm dB}. **Answer.** Signal gain is approximately 29.54 dB. **Check.** Adding the 50 mW ASE would give total output power 950 mW but would overstate useful signal gain. If “stimulated power” denotes only newly generated power, including the transmitted 1 mW changes the result slightly to :math:`10\log_{10}(901)`. Problem 13.4 — Third-harmonic modelocking ----------------------------------------- **Working.** A linear two-mirror cavity of length 2 m has :math:`T_{\rm rt}=2n_gL/c`. With negligible dispersion, :math:`n_g\simeq1.75`, so .. math:: f_{\rm rt}=\frac{c}{2(1.75)(2)}=42.8275\,{\rm MHz},\qquad f_{\rm lock}=3f_{\rm rt}=128.482\,{\rm MHz}. **Answer.** Approximately 128.5 MHz. **Check.** A ring cavity would have a different round-trip length. The optical carrier wavelength does not set the repetition rate once optical path length is specified. Problem 13.5 — Amplifier wavelength ranges ------------------------------------------ **Working.** Gain bandwidth depends on the active transition or Raman shift, not just the presence of a waveguide. .. list-table:: Ranges in the context of the sixth edition :header-rows: 1 * - Amplifier - Interpretation * - SOA - Different semiconductor compositions span roughly 0.8--1.6 micrometres; one device covers only its material gain band. * - EDFA - Roughly 1.52--1.60 micrometres across C/L-band implementations; strongest conventional gain is near 1.55 micrometres. * - Raman - Pump-selectable Stokes gain, with silica's main peak near :math:`\nu_s=\nu_p-13.2\,{\rm THz}`. * - TDFA - The book discusses S-band systems around 1.48--1.52 micrometres and a pumping scheme with a broader 1.43--1.50 micrometre response. **Answer.** Give a range together with host, pumping scheme, and device family; Raman amplification has no single universal absolute band. **Check.** Stokes Raman gain is at lower frequency and longer wavelength than the pump. A 1500 nm pump cannot generate a 1480 nm Stokes signal; that numerical passage in the chapter is inconsistent. Problem 13.6 — Check the proposed Fabry–Perot amplifier first ------------------------------------------------------------- **Working.** The net single-pass power gain is .. math:: G_s=e^{(\Gamma g-\alpha)L} =e^{[(0.9)(50)-4](0.035)}=4.19965. For equal facet reflectivity 0.33, the round-trip power factor is :math:`R^2G_s^2`, and :math:`RG_s=1.38588>1`. **Answer.** (a) The specified unsaturated device is already above lasing threshold. There is no stable small-signal Fabry–Perot amplifier gain at that operating point without saturation or a changed bias. (b) With negligible facet reflection and the assumed same material gain, the traveling-wave gain is 4.19965, or about 6.23 dB. **Check.** Algebraically evaluating :math:`(1-R)^2G_s/(1-RG_s)^2` above threshold produces a number outside the formula's physical domain. Carrier density may also change when feedback is removed at fixed injection current. Problem 13.7 — Stimulated photon generation rate ------------------------------------------------ **Working.** Divide stimulated optical power by one photon's energy: .. math:: \dot N_\gamma=\frac{P\lambda_0}{hc} =\frac{(0.850)(1536\times10^{-9})}{hc} =6.57254\times10^{18}\,{\rm s}^{-1}. **Answer.** Approximately :math:`6.57\times10^{18}` photons/s. **Check.** Use watts rather than milliwatts. This is a generation rate, not the stored number of photons in the amplifier. Problem 13.8 — A second above-threshold amplifier specification --------------------------------------------------------------- **Working.** With :math:`G_s=15` and :math:`R=0.65`, the round-trip power multiplier is .. math:: R^2G_s^2=(0.65)^2(15)^2=95.0625. **Answer.** The unsaturated specification is far above threshold, so it does not define a stable linear Fabry–Perot amplifier. Stable operation would require :math:`G_s<1/R=1.53846`. **Check.** The squared denominator in the formal resonant-gain expression does not make the above-threshold solution physical. Gain clamping or injection-locking would require additional information.