Change of Refractive Index

The change in the refractive index, group index, and effective index can be extracted from the amplified spontaneous emission spectra. By considering adjacent longitudinal mode, FP mode spacing _FP at single current bias is [20],

(4.9) From the waveguide theory, the group velocity v can be expressed as _g

(4.10) Since both n and _e  are function of . The group index n can be expressed as _g

(4.11) From Eq. (4.9) and Eq. (4.11), we can get a compact expression for group index

(4.12) According to Eq. (4.1), we define ₁ and ₂ as two adjacent peak wavelength in ASE spectrum. The corresponding refractive indices are ₁ ^{1 1}

e 2

The change of the refractive index can be expressed as

(4.13) where    ₂ and ₁  m m₂m₁. Since the refractive indices are function of wavelengths and injection currents, the change of the refractive index can also be expressed as,

e e

Thus, manipulating of Eq. (4.13) and Eq. (4.14) gives

(4.15) By substituting Eq. (4.9) into Eq. (4.15) and after some mathematical manipulation, we can obtain the induced change in the refractive index.

(4.16) where _p _p( )I₂ _p( )I₁ is the wavelength shift of FP peaks due to an increase in current injection. The equation is very useful to study the change in the refractive index from wavelength shifts of a single Fabry-Perot mode due to the change of injection current from I₁ to I₂.

4.2.3 Linewidth Enhancement Factor

Semiconductor lasers exhibit a strong variation of refractive index and net modal gain when injected carrier concentration changes. The parameter describing this dependency is called linewidth enhancement factor _e [23]. The value of _e is greatly important for many application of semiconductor lasers, such as laser linewidth, chirp, and response to optical feedback. Large values of _e can result in a detrimental effect on a laser and can produce significant antiguidance, selffocusing, filamentation in broad-area emitters, and chirp under modulation. The increasing chirp will produce significant dispersion, which severely influences to high-speed signal processing in optical communications. As a result, minimization of _e becomes a big issue for semiconductor lasers. For strained InGaAs single-quantum-well (QW) lasers which operate near 980 nm, a typical value of _e has been shown to be 2 or higher at

carrier concentrations near threshold [24]. At the optical communication wavelengths of 1.3 and 1.55m, lasers exhibit usually significantly high value of _e unless modulation doping [25] or multi-quantum-wells are employed in the laser device structure [26]. From the Kramers-Kronig relation, a symmetric gain profile will yield a zero _e at the peak gain since the refractive index remains unchanged. Instead of QWs, the density of states of an ideal quantum dot (QD) has a series of delta-function at the quantized energy levels, indicating that ideal quantum dot lasers are expected to have reduced linewidth enhancement factor, which yields important impacts to optical communication area. A substantial reduction in _e should be realized by using quantum dot lasers [27,28].

The linewidth enhancement factor _e is defined as [29]

(4.17) where n and ^' n are real and imaginary parts of refractive index, respectively. The complex ^'' propagation constant in the cavity is given by

(4.18) Substituting Eq. (4.14) into Eq. (4.13) yields the linewidth enhancement factor

(4.19) where n^e

N



 is the change in the refractive index due to the injected carrier changes, and g N



 is the change in the gain due to the injected carrier changes.

_e  n^'

4.3 Net Modal Gain, Refractive Index Change, Linewidth Enhancement Factor, and Group Index of The QD

Plasmonic Laser Under CW Mode Bias

This part presents the experimental data of the net modal gain, refractive index change, group index, and linewidth enhancement factor of the QD plasmonic laser under continuous wave (CW) mode bias. Those data are extracted from the amplified spontaneous emission spectra of the QD Fabry-Perot plasmonic laser. The emission spectrum falls on near-IR region (1200nm~1350nm).

4.3.1 Net Modal Gain

The QD Fabry-Perot laser is operated in CW mode bias to obtain the DC characteristics of the device. The reflectivities of each facet is estimated to be 0.299 by using the refractive index of GaAs (n_GaAs=3.413 [30]). The cavity length is 1150m with error less than 0.5%. The ASE spectrum is obtained from incremental current injection of 2mA until the threshold condition of the laser is reached. The clear Fabry-Perot spectrum is evident when the emission intensity is strong enough compared to noise level. As the CW current increases, the gain increases to the amount offsetting the material losses and mirror loss. Above the threshold operation, the maximum modal gain is located at 1287 nm and the power density of the lasing modes are much higher compared with other longitudinal modes since stimulated emission has occurred. Further increase in the current injection will produce the lasing action of this device.

1250 1260 1270 1280 1290 1300 1310 1320 -20

-15 -10 -5 0 5 10

15 16 mA

18 mA 20 mA 22 mA

Net modal gain (cm

)

Wavelength (nm)

Fig 4.2 Current-dependent net modal gain extracted from the ASE spectra under CW current bias at 293 K.

Fig. 4.2 shows the current-dependent net modal gain of the QD laser under CW operation at 293K. The net modal gain is extracted from ASE spectra by using the Hakki-Paoli method. As the injection current increases, the net modal gain keeps growing until the laser reaches threshold condition. After lasing action takes place, the net modal gain is pinned and equals to the mirror loss, which is about 10.5 cm^1. The plateau of the gain curves on the long wavelength side gives the intrinsic loss of the cavity medium, which is about 8 cm^1. The threshold current of the QD

plasmonic laser is about 22.5 mA, corresponding to an carrier density of 0.978 kA cm , which / ² is comparably low as typical QD lasers.

It should be pointed out that the extraction of the net modal gain has its own difficulty on the longer or shorter wavelength side. The ASE spectrum covers certain wavelength range, and drops rapidly on both side. As the emission intensity is not large enough, insufficient signal-to-noise ratio (SNR) makes it difficult to obtain precise spectra and net modal gain.