Figure 8.5: Frequency modulation response of 30nm spacer laser (Chip 1, bar 5, Slot 1, device 7) for dierent bias currents. 0.1mA current modulation. Measured with Optilab BPR-20-M balanced photodetector and HP 8722C RF network analyzer, using MZI with FSR = 7.06GHz
pushed to higher frequencies with pump current, as expected from classic laser dynamics theory.
4. Resonance frequency damping - All spacer lasers have damped relaxation reso- nance. This is expected due to the high-Q and TPA that suppresses intensity peaks.
Figure 8.6: Frequency modulation response of 100nm spacer laser (Chip 1, bar 1, Slot 2, device 19) for dierent bias currents. 35nA current modulation. Measured with Optilab BPR-20-M balanced photodetector MZI with FSR = 1.56GHz. Low frequencies (<500 MHz) were measured using Agilent 4395A network analyzer and high frequencies (>500MHz) using HP 8722C analyzer. Curves from the two analyzers are plotted together without any additional post-processing (stitching)
Figure 8.7: Frequency modulation response of 150nm spacer laser (Chip 1, bar 1, Slot 2, device 19) for dierent bias currents. 0.1mA current modulation. Measured with Optilab BPR-20-M balanced photodetector and Agilent 4395A network analyzer, using MZI with FSR = 1.56GHz
Figure 8.8: Frequency modulation response of dierent spacer lasers for pump current oset of 50mA.
It shows a response of few hundred MHz/mA of plasma dispersion eect, and a resonance peak at few GHz. The relaxation follows a zero of the transfer function as predicted from the (classic) rate equation analysis. This behavior is very typical of III-V lasers, as expected from thin spacer designs.
2. In all lasers, the very low frequency (<1MHz) is dominated by 1/f-like response that is attributed to thermal frequency drift.
3. The thicker spacer designs have a qualitatively dierent response: they feature dips of the frequency response curve, prior to the relaxation frequency. This is in very good qualitative agreement with the nonlinear rate equation analysis, which attributes this feature to Si free-carrier-dispersion.
4. The thick 150nm spacer in Figure 8.7 demonstrates a wide range of bias currents with a close to threshold measurement curve. This gure shows a much lower frequency response than at the higher bias currents. This is consistent with the nonlinear rate equation analysis that attributed the rise in the response to the accumulation of free-carriers in Si.
Figure 8.8 compares the three spacer design for a given oset from threshold. As
was also evident from the intensity modulation response, the resonance frequency shifts to extremely low frequencies when the mode is pushed further into silicon. It also demonstrates the dierent dynamics of the 30nm spacer, which is closer to the conventional design, compared to the thicker 100nm and 150nm spacer designs.
Despite the very good qualitative agreement with the theoretical analysis, there are still some characteristics that are not explained by the nonlinear model:
1. The position of the frequency response dip - Theory predicts that this unique dip will move to slightly higher frequencies with spacer thickness. In the exper- imental results, the dip is in fact pushed to lower frequencies.
2. The relative frequency response magnitude between the 100nm and 150nm spac- ers - The nonlinear model predicts that the thicker spacer will have a stronger frequency response at the low frequencies than the thinner spacer. In the experi- mental results of Figure 8.8 the 100nm spacer attens at value of∼700MHz/mA while the 150nm spacer does so at 350MHz/mA
It is worth noting that at the root of these discrepancies lies the assumption that dierent spacer laser are identical in all parameters but spacer thicknesses. This is very unlikely. The two lasers were fabricated separately; they have dierent threshold currents, and hence operate at dierent temperatures. Heat management is less ecient for the thick spacer design, due to the thermally isolating oxide layer. They may also have dierent internal eciencies, due to fabrication variations in the ion- implant. For all of these reasons, it is dicult to compare absolute values in dierent spacer lasers. However, the unique characteristics that were outlined in the theoretical analysis in Chapter 7 are observed in these lasers, and the trends within a specic spacer design are in agreement with the theory.
In this chapter I presented experimental results demonstrating the modulation response behavior of our lasers. The relaxation resonance was found to be at frequen- cies as low as∼100MHz, validating the theoretical predictions. Predictions regarding the role of free-carriers in Si were also validated: a zero of the intensity modulation
transfer function and a unique dip of the frequency modulation response were both empirically observed. The location of the transfer function's zero was used to esti- mate that the eective lifetime of carriers in Si is ∼30ns. This is the rst time this lifetime has been measured in the context of hybrid Si/III-V lasers.
Chapter 9
Noise performance - Theoretical analysis
Chapter 5 considered the noise due to the spontaneous emission process. The anal- ysis there was based on the Schawlow-Townes formula, which was derived by using Fermi's golden rule to calculate the rate of spontaneous decay and the coupling of this spontaneous radiation to the lasing mode. The resulting frequency noise spectra from the S-T formula is white. Further enhancement of the noise was considered by taking into account the coupling between the imaginary and real parts of the refractive in- dex through Henry's alpha parameter. The modied Schawlow-Townes formula was derived, but was still considered white noise.
As was seen in Chapter 7, the dynamic response of the laser cannot be simplied by making it a constant, especially at high frequencies, where resonance eects appear.
Furthermore, though the modied Schawlow-Townes formula withstood the test of time and was proven useful, our high-Q hybrid lasers have a new component which is absent from conventional lasers: the free-carriers in silicon. Fluctuations in the number of free-carriers in Si are bound to add excess noise to the system, and must be considered in the context of narrow-linewidth Si/III-V lasers. This chapter will use a dierent method to carry out the analysis, the Langevin noise source approach, to reconsider laser frequency noise in the presence of uctuations of carriers, photons, and temperature. Experimental results are compared to the theoretical predictions in Chapter 10.
9.1 Methodology - Langevin noise sources
The use of Langevin noise terms in the context of lasers has many analogies to the original use of Paul Langevin (1872-1946) in the context of Brownian motion of parti- cles. The same results that Albert Einstein derived by using advanced mathematical tools were derived by Langevin by using a simple, yet very dierent, technique. In his own words, Langevin describes his work innitely more simple than Einstein's [54].
Despite the simplication, his method is considered even more general than Einstein's [54]. Langevin introduced a stochastic force that pushes the Brownian particle (in the velocity space), a concept that since then was generalized into a very useful class of methods in the study of continuous random processes.
Laser noise formulas were derived using the advanced mathematical tools of quan- tum mechanics. The inclusion of quantum-mechanical operators in that analysis makes this approach very cumbersome, and many simplications must be made to obtain analytical results. The treatment of McCumber [63] in a paper from 1965 suggested the interpretation of quantum uctuations of a system that is described by rate equations in terms of ctitious Langevin noise sources. This approach was later justied both in the context of classical self-sustained oscillators [53] and for laser oscillators [51]. The same approach was further justied and used to analyze quantum noise of semiconductor lasers [67, 28, 130, 117]. Carrier uctuations and temperature diusion were also included to successfully predict the noise spectrum of semiconductor lasers [50]. This approach provides a strong, yet simple, mathematical tool to estimate laser noise spectrum and was found to agree well with experimental results [28].
The strength of the Langevin noise approach is its simplicity: one can simply add a stochastic driving force term to the rate equations. These somewhat ctitious forces are engineered to recreate noise predicted from master equations or other fundamental approaches. In the context of recombination and generation of photons and carriers, the Langevin noise force that recreates quantum mechanical results is simply shot-noise [63]. Quantum uctuations can be described using delta-function
impulses with integrated intensity of one (in the total number description). This implies that one can infer the magnitude of the noise-term just by looking at the form of the dynamic equations. A Langevin noise source suitable for diusion processes can also be derived. In the next section, Langevin noise terms are added to the dierential rate equations, and their statistical properties are discussed.