# Noise Characteristics of Semiconductor Lasers

This is a continuation from the previous tutorial - ** transient response of semiconductor lasers**.

In the preceding deterministic description, laser power and frequency were assumed to remain constant in time once the steady state had been reached. In reality, however, laser output exhibits intensity as well as phase fluctuations.

At the most fundamental level, the origin of these fluctuations lies in the quantum nature of the lasing process itself. A proper description therefore requires a quantum-mechanical formulation of the rate equations.

In general, intensity noise reaches its peak in the vicinity of the laser threshold and then decreases rapidly with an increased in the drive current. The intensity-noise spectrum shows a peak near the relaxation-oscillation frequency as a consequence of the laser's intrinsic resonance.

Phase fluctuations produce spectral broadening of each longitudinal mode and are responsible for the observed line width.

The noise characteristics of semiconductor lasers have been studied extensively both theoretically and experimentally.

## Langevin Formulation

Within the semiclassical treatment, fluctuations arising from the spontaneous-emission process and the carrier-generation-recombination process are incorporated by adding a Langevin noise source to the single-mode rate equations (6-2-14), (6-2-20), and (6-2-18) [refer to the rate equations of semiconductor lasers tutorial]; these then become

\[\tag{6-5-1}\dot{P}=(G-\gamma)P+R_\text{sp}+F_\text{P}(t)\]

\[\tag{6-5-2}\dot{N}=I/q-\gamma_\text{e}N-GP+F_\text{N}(t)\]

\[\tag{6-5-3}\dot{\phi}=-(\omega_0-\omega_\text{th})+\frac{1}{2}\beta_\text{c}(G-\gamma)+F_\phi(t)\]

Physically \(F_\text{P}\) and \(F_\phi\) arise from spontaneous emission, while \(F_\text{N}\) has its origin in the discrete nature of the carrier generation and recombination processes (shot noise).

In the presence of Langevin noise sources, \(P\), \(N\), and \(\phi\) become random and their dynamics are governed by the stochastic rate equations (6-5-1) to (6-5-3).

The problem simplifies considerably if we make the Markovian assumption; i.e., we assume that the correlation time of the noise sources is much shorter than the relaxation times \(\gamma^{-1}\) and \(\gamma_\text{e}^{-1}\) (system has no memory).

Under the Markovian assumption, the Langevin forces satisfy the general relations

\[\tag{6-5-4}\langle{F}_i(t)\rangle=0\]

\[\tag{6-5-5}\langle{F_i(t)}F_j(t')\rangle=2D_{ij}\delta(t-t')\]

where angle brackets denote ensemble average and \(D_{ij}\) is the diffusion coefficient associated with the corresponding noise source.

In a rigorous approach, \(D_{ij}\)'s are obtained by evaluating the second moments of the dynamic variables with the help of Equations (6-5-1) to (6-5-3). They can also be obtained using physical arguments.

Their explicit expressions are

\[\tag{6-5-6}D_\text{PP}=R_\text{sp}P\qquad{D_{\phi\phi}}=R_\text{sp}/4P\qquad{D_{\text{P}\phi}}=0\]

\[\tag{6-5-7}D_\text{NN}=R_\text{sp}P+\gamma_\text{e}N\qquad{D_\text{PN}}=-R_\text{sp}P\qquad{D_{\text{N}\phi}}=0\]

Here \(N\) and \(P\) represent the steady-state ** average** values of the carrier and photon populations, respectively, and are the same as obtained previously in the steady-state characteristics of semiconductor lasers tutorial.

To obtain the noise characteristics, the steady-state values \(P\), \(N\), and \(\phi\) are perturbed by small amounts \(\delta{P}\), \(\delta{N}\), and \(\delta\phi\), respectively.

In the small-signal analysis the stochastic rate equations (6-5-1) to (6-5-3) are linearized, and we obtain

\[\tag{6-5-8}\delta\dot{P}=-\Gamma_\text{P}\delta{P}+(G_\text{N}P+\partial{R_\text{sp}}/\partial{N})\delta{N}+F_\text{P}(t)\]

\[\tag{6-5-9}\delta\dot{N}=-\Gamma_\text{N}\delta{N}-(G+G_\text{P}P)\delta{P}+F_\text{N}(t)\]

\[\tag{6-5-10}\delta\dot{\phi}=\frac{1}{2}\beta_\text{c}G_\text{N}\delta{N}+F_\phi(t)\]

where the small-signal decay rates \(\Gamma_\text{P}\) and \(\Gamma_\text{N}\) are defined by Equations (6-4-16) and (6-4-17) [refer to the transient response of semiconductor lasers tutorial].

Except for the presence of the noise terms, Equations (6-5-8) and (6-5-9) are identical to those obtained in the transient response of semiconductor lasers tutorial for the analysis of relaxation oscillations.

The phase equation (6-5-10) is also considered here to incorporate phase fluctuations.

The power dependence of gain arising from nonlinear effects such as spectral-hole burning is included through \(G_\text{P}\). However, its contribution to the phase is neglected since the index change associated with such nonlinear phenomena is expected to be much smaller than the carrier-induced index change.

The linear set of Equations (6-5-8) to (6-5-10) can be solved in the frequency domain using Fourier analysis. If the Fourier transform of an arbitrary function \(f(t)\) is defined as

\[\tag{6-5-11}\tilde{f}(\omega)=\displaystyle\int\limits_{-\infty}^{\infty}f(t)\exp(-\text{i}\omega{t})\text{d}t\]

the solutions for the Fourier components are

\[\tag{6-5-12}\delta\tilde{P}(\omega)=\frac{(\Gamma_\text{N}+\text{i}\omega)\tilde{F}_\text{P}+(G_\text{N}P+\partial{R_\text{sp}}/\partial{N})\tilde{F}_\text{N}}{(\Omega_\text{R}+\omega-\text{i}\Gamma_\text{R})(\Omega_\text{R}-\omega+\text{i}\Gamma_\text{R})}\]

\[\tag{6-5-13}\delta\tilde{N}(\omega)=\frac{(\Gamma_\text{P}+\text{i}\omega)\tilde{F}_\text{N}-(G+G_\text{P}P)\tilde{F}_\text{P}}{(\Omega_\text{R}+\omega-\text{i}\Gamma_\text{R})(\Omega_\text{R}-\omega+\text{i}\Gamma_\text{R})}\]

and

\[\tag{6-5-14}\delta\tilde{\phi}(\omega)=\frac{1}{\text{i}\omega}(\tilde{F}_\phi+\frac{1}{2}\beta_\text{c}G_\text{N}\delta\tilde{N})\]

where the decay rate \(\Gamma_\text{R}\) and the frequency \(\Omega_\text{R}\) of relaxation oscillations are given by Equations (6-4-21) and (6-4-22), respectively [refer to the transient response of semiconductor lasers tutorial].

Equations (6-5-12) to (6-5-14) clearly show that noise is enhanced in the vicinity of \(\omega=\Omega_\text{R}\) because of the intrinsic resonance exhibited by the carrier and photon populations.

These equations can be used to study how processes such as spontaneous emission and carrier recombination affect the light intensity, the optical phase, and the carrier population. In the next two sections we consider separately the intensity noise and the phase noise.

## Intensity Noise

The power emitted by a semiconductor laser fluctuates around its steady-state value. The spectrum of these fluctuations can be measured by detecting the laser output using a wide-bandwidth photodiode and a spectrum analyzer. Such a measurement yields the noise spectrum associated with the total power.

The noise spectrum of a single mode can be obtained at first selecting a single longitudinal mode using a monochromator.

The intensity noise at a given frequency \(\omega\) is characterized by the ** relative intensity noise** (RIN) defined as

\[\tag{6-5-15}\text{RIN}=S_\text{P}(\omega)/P^2\]

where the spectral density

\[\tag{6-5-16}S_\text{P}(\omega)=\displaystyle\int\limits_{-\infty}^\infty\langle\delta{P}(t+\tau)\delta{P}(t)\rangle\exp(-\text{i}\omega\tau)\text{d}\tau\]

is related to \(\delta\tilde{P}(\omega)\) simply by

\[\tag{6-5-17}S_\text{P}(\omega)=\lim_{T\to\infty}\frac{1}{T}|\delta\tilde{P}(\omega)|^2\]

The ensemble average in Equation (6-5-16) was replaced by a time average over the interval \(T\) assuming an ergodic stochastic process.

We substitute \(\delta\tilde{P}(\omega)\) from Equation (6-5-12) and make use of the relation that

\[\tag{6-5-18}\lim_{T\to\infty}\frac{1}{T}[\tilde{F}_i^*(\omega)\tilde{F}_j(\omega)]=2D_{ij}\]

which follows from Equation (6-5-5) and implies that the Langevin noise sources have a white (frequency-independent) spectrum.

Using \(D_\text{PP}\), \(D_\text{PN}\), and \(D_\text{NN}\) from Equations (6-5-6) and (6-5-7), we obtain

\[\tag{6-5-19}\text{RIN}=\frac{2R_\text{sp}[(\Gamma_\text{N}^2+\omega^2)+G_\text{N}^2P^2(1+\gamma_\text{e}N/R_\text{sp}P)-2\Gamma_\text{N}G_\text{N}P]}{P[(\Omega_\text{R}-\omega)^2+\Gamma_\text{R}^2][(\Omega_\text{R}+\omega)^2+\Gamma_\text{R}^2]}\]

Figure 6-10 shows the calculated intensity-noise spectra at several laser powers for a 1.3-μm InGaAsP laser using the parameter values shown in Table 6-1 [refer to the rate equations of semiconductor lasers tutorial].

At a given laser power, the noise is relatively low when \(\omega\ll\Omega_\text{R}\) and is enhanced significantly in the vicinity of \(\omega=\Omega_\text{R}\).

At a given frequency, RIN decreases with an increase in the laser power \(P\). To see the functional dependence of RIN on \(P\), consider the low-frequency limit when \(\omega\ll\Omega_\text{R}\).

Since \(\Gamma_\text{R}\ll\Omega_\text{R}\), the denominator in Equation (6-5-19) can be replaced by \(P\Omega_\text{R}^4\).

Further, at low power levels the terms proportional to \(P\) and \(P^2\) in the numerator make a relatively small contribution. If we neglect these terms and use Equation (6-4-23) [refer to the transient response of semiconductor lasers tutorial] to express \(\Omega_\text{R}\) in terms of \(P\), we find that RIN decreases with laser power at \(P^{-3}\).

The inverse cubic dependence of RIN on \(P\) has been observed for AlGaAs as well as InGaAsP lasers.

At higher bias levels, the neglected terms in Equation (6-5-19) become important and RIN varies more slowly approaching the \(P^{-1}\) dependence. This behavior has also been observed experimentally.

Another measure of the intensity noise is provided by the autocorrelation function \(C_\text{pp}(\tau)\) defined as

\[\tag{6-5-20}C_\text{pp}(\tau)=\langle\delta{P(t)}\delta{P(t+\tau)}\rangle/P^2\]

From Equations (6-5-15) and (6-5-16) \(C_\text{pp}(\tau)\) is related to \(\text{RIN}(\omega)\) by the Fourier-transform relation

\[\tag{6-5-21}C_\text{pp}(\tau)=\frac{1}{2\pi}\displaystyle\int\limits_{-\infty}^\infty\text{RIN}(\omega)\exp(\text{i}\omega\tau)\text{d}\omega\]

The integral in (6-5-21) can be performed by using the method of contour integration. The result is

\[\tag{6-5-22}C_\text{pp}(\tau)=\frac{R_\text{sp}\exp(-\Gamma_\text{R}\tau)}{2\Gamma_\text{R}P}\text{Re}\left(\frac{\Gamma_\text{e}^2+(\Omega_\text{R}+\text{i}\Gamma_\text{R})^2}{\Omega_\text{R}(\Omega_\text{R}+\text{i}\Gamma_\text{R})}\exp(\text{i}\Gamma_\text{R}\tau)\right)\]

where \(\text{Re}\) stands for the real part and

\[\tag{6-5-23}\Gamma_\text{e}^2=\Gamma_\text{N}^2+G_\text{N}^2P^2(1+\gamma_\text{e}N/R_\text{s}P)-2\Gamma_\text{N}G_\text{N}P\]

The intensity autocorrelation function oscillates with \(\tau\) (as a consequence of relaxation oscillations) and vanishes when \(\tau\) becomes comparable to the damping time \(\Gamma_\text{R}^{-1}\) of relaxation oscillations.

The quantity of practical interest is \(C_\text{pp}(0)\) since it is related to the variance of intensity fluctuations. Indeed, one can define the signal-to-noise ratio (SNR) of the laser light as

\[\tag{6-5-24}\text{SNR}=\frac{1}{\sqrt{C_\text{pp}(0)}}=\left(\frac{1}{2\pi}\displaystyle\int\limits_{-\infty}^\infty\text{RIN}(\omega)\text{d}\omega\right)^{-1/2}\]

By substituting \(C_\text{pp}(0)\) from Equation (6-5-22) the \(\text{SNR}\) is given by

\[\tag{6-5-25}\text{SNR}=\left(\frac{2\Gamma_\text{R}P}{R_\text{sp}}\right)^{1/2}\left(1+\frac{\Gamma_\text{e}^2}{\Omega_\text{R}^2+\Gamma_\text{R}^2}\right)^{-1/2}\]

The last factor can be approximated by \(1\) since \(\Gamma_\text{e}^2\ll\Omega_\text{R}^2+\Gamma_\text{R}^2\) in most cases of practical interest. The \(\text{SNR}\) is then given by a remarkably simple expression

\[\tag{6-5-26}\text{SNR}=\left(\frac{2\Gamma_\text{R}P}{R_\text{sp}}\right)^{1/2}\]

It shows that the SNR depends on the rate of spontaneous emission and degrades as \(R_\text{sp}^{-1/2}\) with an increase in it. It also shows that the damping rate \(\Gamma_\text{R}\) of relaxation oscillations plays a crucial role in determining the SNR.

The power dependence of SNR is not immediately obvious from Equation (6-5-26) since \(\Gamma_\text{R}\) itself depends on power. By using \(2\Gamma_\text{R}=\Gamma_\text{N}+\Gamma_\text{P}\) from Equation (6-4-21) [refer to the transient response of semiconductor lasers tutorial] and \(\Gamma_\text{N}\) and \(\Gamma_\text{P}\) from Equations (6-4-16) and (6-4-17) [refer to the transient response of semiconductor lasers tutorial], the SNR becomes

\[\tag{6-5-27}\text{SNR}=\left(1+\frac{P}{R_\text{sp}}[\gamma_\text{e}+N(\partial\gamma_\text{e}/\partial{N})+(G_\text{N}-G_\text{P})P]\right)^{1/2}\]

Near or below threshold, SNR reduces to 1, as it should, since the intensity fluctuations then correspond to those of thermal noise with an exponential distribution for the intensity.

In the above-threshold regime, SNR begins to increase, first as \(\sqrt{P}\) and then linearly with \(P\). Far above threshold, the SNR varies with \(P\) as

\[\tag{6-5-28}\text{SNR}=\left(\frac{G_\text{N}-G_\text{P}}{R_\text{sp}}\right)^{1/2}P\]

Since \(G_\text{P}\) is negative because of the gain suppression occurring at high powers [see Table 6-2 in the transient response of semiconductor lasers tutorial], the nonlinear gain plays an important role in determining the SNR of the output power. In fact, \(|G_\text{P}|\) is typically much larger than \(G_\text{N}\), and the SNR is directly proportional to \(\sqrt{|G_\text{P}|}\).

According to Equation (6-5-28), the SNR would keep on improving with an increase in the laser power. However, this feature is an artifact of the assumption made in Equation (6-4-18) [refer to the transient response of semiconductor lasers tutorial] by truncating the Taylor series. A more accurate treatment shows that the nonlinear gain eventually limits the SNR at a level of about 30 dB at high powers.

At a power level of a few milliwatts the SNR of most semiconductor lasers is about 20 dB. Thus the root-mean-square value of intensity fluctuations is close to 1% under typical operating conditions.

For a multimode laser, the qualitative behavior shown in Figure 6-10 is observed experimentally if the total power in all modes is detected. However, low-frequency noise is found to be significantly higher for individual longitudinal modes. Figure 6-11 shows this behavior for a laser with a mode-suppression ratio of more than 20.

At frequencies below 500 MHz the RIN for the dominant mode is higher by more than 30 dB compared to the total intensity noise. This increase is referred to as the ** mode-partition noise** and has attracted considerable attention. It arises when the main and the side modes fluctuate in such a way as to leave the total intensity relatively constant.

An alternative way to characterize intensity noise is through the probability distribution function of the random variable \(P\). Experimentally it is obtained by repetitively sampling the light intensity (using a fast detector) for a sufficiently short duration (less than ~ 100 ps) so that the signal represents instantaneous intensity and fluctuations do not average out.

As one may expect, the observed distribution depends on the sampling-time window and rapidly narrows with an increase in the sampling time. The evidence for mode-partition noise was also found in the recorded distribution functions for the main mode and the side modes.

Mode-partition noise can be modeled by adding the Langevin-noise terms to the multimode rate equations (6-2-25) and (6-2-26) [refer to the rate equations for semiconductor lasers tutorial] so that

\[\tag{6-5-29}\dot{P}_m=(G_m-\gamma_m)P_m+R_\text{sp}+F_m(t)\]

\[\tag{6-5-30}\dot{N}=I/q-\gamma_\text{e}N-\sum{G_mP_m}+F_\text{N}(t)\]

where the sum is over the total number modes. The Langevin forces satisfy Equations (6-5-4) and (6-5-5) with the diffusion coefficients

\[\tag{6-5-31}D_{mm}=R_\text{sp}P_m,\qquad{D_{mn}}=0,\qquad{D_{m\text{N}}}=-R_\text{sp}P_m,\qquad{D_\text{NN}}=R_\text{sp}\sum{P_m}+\gamma_\text{e}N\]

The spontaneous emission rate is assumed to be the same for all modes.

The multimode rate equations generally need to be solved numerically. However, they can be solved analytically by following the linearization procedure that assumes fluctuations from the steady state to be small.

The analytic solution is particularly simple when only two modes are considered. This case is of practical importance since many semiconductor lasers are designed to oscillate predominantly in a single longitudinal mode and can be modeled by considering a single side mode accompanied by the dominant main mode.

Fluctuations in the two modes are given by

\[\tag{6-5-32}\delta\tilde{P}_1(\omega)=[(\Gamma_2+\text{i}\omega)\tilde{F}_1-\theta_{12}P_1\tilde{F}_2+G_\text{N}P_1(\Gamma_2+\text{i}\omega-\theta_{12}P_2)\delta\tilde{N}]/D(\omega)\]

\[\tag{6-5-33}\delta\tilde{P}_2(\omega)=[(\Gamma_1+\text{i}\omega)\tilde{F}_2-\theta_{21}P_2\tilde{F}_1+G_\text{N}P_2(\Gamma_1+\text{i}\omega-\theta_{21}P_1)\delta\tilde{N}]/D(\omega)\]

where

\[\tag{6-5-34}\delta\tilde{N}(\omega)=(\tilde{F}_\text{N}-\bar{G}_1\delta\tilde{P}_1-\bar{G}_2\delta\tilde{P}_2)/(\Gamma_\text{N}+\text{i}\omega)\]

\[\tag{6-5-35}D(\omega)=(\Gamma_1+\text{i}\omega)(\Gamma_2+\text{i}\omega)-\theta_{12}\theta_{21}P_1P_2\]

\[\tag{6-5-36}\Gamma_1=R_\text{sp}/P_1+\beta_{11}P_1,\qquad\Gamma_2=R_\text{sp}/P_2+\beta_{22}P_2\]

The effects of nonlinear gain, included in the single-mode case through a single parameter \(G_\text{p}\), are now included through four parameters introduced by expanding \(G_1\) and \(G_2\) in Equation (6-5-29) as

\[\tag{6-5-37}G_1=\bar{G}_1+G_\text{N}\delta{N}-\beta_{11}\delta{P_1}-\theta_{12}\delta{P_2}\]

\[\tag{6-5-38}G_2=\bar{G}_2+G_\text{N}\delta{N}-\beta_{22}\delta{P_2}-\theta_{21}\delta{P_1}\]

The parameters \(\beta_{11}\) and \(\beta_{22}\) govern self-saturation responsible for a self-induced decrease of the modal gain. The parameters \(\theta_{12}\) and \(\theta_{21}\) govern cross-saturation reflecting the fact that the gain of a particular mode is also affected by the intensity of other modes.

The RIN for the main mode, side mode, and the total intensity can be obtained by following the same procedure outline before for the single-mode case.

Figure 6-12 shows the RIN spectra for the main mode for several values of \(S/P\), where \(P\) and \(S\) are average photon populations for the main and side modes.

In all cases the RIN for the total intensity \(P+S\) nearly coincides with the curve shown for \(S/P=10^{-5}\) in Figure 6-12.

For a 50-dB MSR (mode-suppression ratio) the side mode is too weak to affect the main-mode RIN. However, the main-mode RIN is considerably enhanced at frequencies below 1 GHz when the MSR is 10 and 20 dB. The enhancement can exceed 40 dB at low frequencies (< 100 MHz), in qualitative agreement with the experimental data shown in Figure 6-11.

The physical origin of such an enhancement is not immediately obvious. Considerable insight is obtained by solving the multimode rate equations (6-5-29) and (6-5-30) numerically.

Numerical simulations show that even though fluctuations in \(P(t)\) are relatively small most of the time, occasionally the side-mode intensity becomes comparable to that of the main mode. Figure 6-13 shows such an event.

These events last for a relatively small time (a few nanoseconds). However, the main-mode intensity drops considerably during such events, resulting in large fluctuations in \(P(t)\) from its average value. Since such events occur on a relatively long time scale (> 10 ns), it is only the low-frequency RIN that is enhanced.

The origin of such drop-off events is related to the gain saturation and to the fact that the same electron population provides gains for both the modes.

The side mode exhibits larger fluctuations because of its lower average intensity. An unusually large random increase in the side-mode intensity induces a simultaneous decrease in the main-mode intensity, lasting a few nanoseconds because of relaxation oscillations.

## Phase Noise and Line Width

Quantum fluctuations associated with the lasing process affect both the intensity and the phase of the optical field. A change in phase leads to a frequency shift \(\delta\omega_\text{L}=\delta\dot{\phi}\).

The frequency or phase noise is of interest in evaluating the performance of coherent optical communication systems. The line width of a single longitudinal mode under CW operation is a manifestation of phase fluctuations occurring inside the laser. The solution (6-5-14) can be used to obtain the frequency-noise spectrum as well as the line width.

Two mechanisms contribute to phase fluctuations.

In Equation (6-5-14), the first term (\(\tilde{F}_\phi\)) is due to spontaneous emission. Each spontaneously emitted photon changes the optical phase by a random amount.

The second term (\(\frac{1}{2}\beta_\text{c}G_\text{N}\delta\tilde{N}\)) shows that fluctuations in the carrier populations also lead to a phase change. Physically, this is so because a change in \(N\) affects not only the optical gain but also the refractive index (or the optical path length) and consequently the phase.

The parameter \(\beta_\text{c}\) provides a proportionality between the gain and index changes [see Equation (6-2-17) in the rate equations for semiconductor lasers tutorial]. It is sometimes referred to as the line-width enhancement factor since it was found that the second term in Equation (6-5-14) increases the line width by a factor of \(1+\beta_\text{c}^2\).

The spectral density of the frequency noise is defined similar to the intensity-noise case [see Equations (6-5-16) and (6-6-17)] and is

\[\tag{6-5-39}S_{\dot{\phi}}(\omega)=\langle|\omega\delta\tilde{\phi}(\omega)|^2\rangle\]

where \(\delta\tilde{\phi}(\omega)\) is given by Equation (6-5-14).

The general expression for \(S_{\dot{\phi}}(\omega)\) is fairly complicated. It can however be considerably simplified by noting that the main contribution to \(\delta\tilde{N}\) comes from the term proportional to \(\tilde{F}_\text{P}\) in Equation (6-5-13).

Using Equations (6-5-6) and (6-5-18), an approximate expression for the frequency-noise spectrum is

\[\tag{6-5-40}S_{\dot{\phi}}(\omega)\approx\frac{R_\text{sp}}{2P}\left(1+\frac{\beta_\text{c}^2\Omega_\text{R}^4}{[(\Omega_\text{R}^2-\omega^2)^2+(2\omega\Gamma_\text{R})^2]}\right)\]

where we have used Equation (6-4-23) [refer to the transient response of semiconductor lasers tutorial].

Equation (6-5-40) shows that the frequency-noise spectrum is relatively flat when \(\omega\ll\Omega_\text{R}\) and peaks at the relaxation-oscillation frequency \(\Omega_\text{R}\). The qualitative behavior is similar to the intensity-noise spectrum shown in Figure 6-10.

In both cases, even though the Langevin noise is white (frequency-independent), noise in the vicinity of \(\Omega_\text{R}\) is selectively enhanced by the intrinsic resonance manifested by a semiconductor laser. This enhancement has been observed experimentally.

The line width of a single longitudinal mode can be obtained by recording the optical power spectrum using a high-resolution FP interferometer. Mathematically, the power spectrum is obtained by taking the Fourier transform of the field autocorrelation and is given by [see Equation (6-5-16)]

\[\tag{6-5-41}S_E(\omega)=\displaystyle\int\limits_{-\infty}^\infty\langle{E}^*(t+\tau)E(t)\rangle\exp(-\text{i}\omega\tau)\text{d}\tau\]

where \(E(t)\) is the optical field such that

\[\tag{6-5-42}E(t)=(P+\delta{P})^{1/2}\exp[-\text{i}(\omega_0t+\phi+\delta\phi)]\]

\(P\) and \(\phi\) are the steady-state values, and \(\omega_0\) is the frequency of the single longitudinal mode under consideration.

The power spectrum \(S_E(\omega)\) can be calculated using the small-signal solution given by Equations (6-5-12) to (6-5-14). It consists of a central peak at the mode frequency \(\omega_0\) followed by much weaker satellite peaks at \(\omega_0\pm{n\Omega_\text{R}}\) (\(n\) being an integer).

These side peaks arise from relaxation oscillations and have been experimentally observed. Furthermore, a small symmetry in the amplitudes of the side peaks was observed and is interpreted to be due to a correlation between intensity and phase fluctuations.

To simplify the analysis, we neglect intensity fluctuations and set \(\delta{P}\) equal to \(0\) in Equation (6-5-42). The only effect of this simplification is that side-peak asymmetry is excluded from the theoretical treatment. Using Equations (6-5-41) and (6-5-42), we then obtain

\[\tag{6-5-43}S_E(\omega)=P\displaystyle\int\limits_{-\infty}^\infty\langle\exp(\text{i}\Delta_\tau\phi)\rangle\exp[-\text{i}(\omega-\omega_0)\tau]\text{d}\tau\]

where

\[\tag{6-5-44}\Delta_\tau\phi=\delta\phi(t+\tau)-\delta\phi(t)\]

is the difference in phase fluctuations at times \(t\) and \(t+\tau\).

Assuming that \(\Delta_\tau\phi\) has a Gaussian distribution, the average in Equation (6-5-43) is easily performed to obtain

\[\tag{6-5-45}\langle\exp(\text{i}\Delta_\tau\phi)\rangle=\exp[-\frac{1}{2}\langle(\Delta_\tau\phi)^2\rangle]\]

To evaluate the variance \(\langle(\Delta_\tau\phi)^2\rangle\) associated with the stationary random process \(\Delta_\tau\phi\), we note by taking the Fourier transform of Equation (6-5-44) that

\[\tag{6-5-46}\Delta_\tau\phi=\frac{1}{2\pi}\displaystyle\int\limits_{-\infty}^\infty\delta\tilde{\phi}(\omega)[\exp(\text{i}\omega\tau)-1]\exp(\text{i}\omega{t})\text{d}\omega\]

Since cross-correlation between the frequency components \(\delta\tilde{\phi}(\omega)\) vanishes, it can be easily shown that

\[\tag{6-5-47}\langle(\Delta_\tau\phi)^2\rangle=\frac{1}{\pi}\displaystyle\int\limits_{-\infty}^\infty\langle|\delta\tilde{\phi}(\omega)|^2\rangle(1-\cos\omega\tau)\text{d}\omega\]

If we use \(\delta\tilde{\phi}(\omega)\) from Equation (6-5-14) and evaluate the average using Equation (6-5-17), the integral can be done by the method of contour integration. The result is

\[\tag{6-5-48}\langle(\Delta_\tau\phi)^2\rangle=\frac{R_\text{sp}}{2P}\left((1+\beta_\text{c}^2b)\tau+\frac{\beta_\text{c}^2b}{2\Gamma_\text{R}\cos\delta}[\cos(3\delta)-\exp(-\Gamma_\text{R}\tau)\cos(\Omega_\text{R}\tau-3\delta)]\right)\]

where

\[\tag{6-5-49}b=\Omega_\text{R}/(\Omega_\text{R}^2+\Gamma_\text{R}^2)^{1/2}\qquad\text{and}\qquad\delta=\tan^{-1}(\Gamma_\text{R}/\Omega_\text{R})\]

The optical spectrum is obtained by substituting Equations (6-5-45) and (6-5-48) in Equation (6-5-43) and evaluating the Fourier transform numerically.

Figure 6-14 shows an example of the calculated spectrum for a semiconductor laser by using parameter values from Table 6-2 [refer to the transient response of semiconductor lasers tutorial] together with \(\beta_\text{c}=5\). Only half of the spectrum is shown since the spectrum is symmetric around \(\omega=\omega_0\).

It consists of a dominant central peak located at \(\omega_0\) and multiple satellite peaks located at \(\omega=\omega_0\pm{m}\Omega_\text{R}\), where \(m\) is an integer. The amplitude of the satellite peaks decreases with the frequency spacing.

In practice, often only the first pair of satellite peaks located at \(\omega=\omega_0\pm\Omega_\text{R}\) is visible experimentally. Even the amplitude of this pair of satellite peaks is much smaller compared with the central peak (typically 1% or less at a power level of 1 mW). As the laser power increases, the central peak narrows and the amplitude of satellite peaks decreases even more. These spectral features have been observed experimentally.

The physical origin of the satellite peaks is related to relaxation oscillations which are responsible for the term proportional to \(b\) in Equation (6-5-48). As a result, the location and the amplitude of satellite peaks is strongly affected by the frequency and damping rate of relaxation oscillations, which are in turn affected by the laser power. At a power level of a few milliwatts the spectrum is dominated by the central peak whose width determines the line width of the laser mode.

An approximate expression for the power spectrum can be obtained by using Equation (6-5-39) in (6-5-47) to yield

\[\tag{6-5-50}\langle(\Delta_\tau\phi)^2\rangle=\frac{1}{\pi}\displaystyle\int\limits_{-\infty}^\infty{S_{\dot{\phi}}(\omega)}\frac{(1-\cos\omega\tau)}{\omega^2}\text{d}\omega\]

where the frequency-noise spectral density \(S_{\dot{\phi}}(\omega)\) is given by Equation (6-5-40).

The origin of the side peaks in the power spectrum is a direct consequence of the relaxation-oscillation-induced enhancement of \(S_{\dot{\phi}}(\omega)\) at \(\omega=\Omega_\text{R}\). If we neglect this enhancement by assuming that \(S_{\dot{\phi}}(\omega)\) is constant and replace it by its value at \(\omega=0\), the integration is readily performed. The result is

\[\tag{6-5-51}\langle(\Delta_\tau\phi)^2\rangle=\tau{S_{\dot{\phi}}(0)}\]

Using Equations (6-5-43), (6-5-45), and (6-5-51), the power spectrum is found to be a simple Lorentzian centered at \(\omega=\omega_0\) and having an FWHM of \(\Delta\omega=S_{\dot{\phi}}(0)\).

Using \(\Delta\omega=2\pi\Delta{f}\), the line width \(\Delta{f}\) is given by

\[\tag{6-5-52}\begin{align}\Delta{f}&=\frac{1}{2\pi}S_{\dot{\phi}}(0)\\&=\frac{R_\text{sp}(1+\beta_\text{c}^2)}{4\pi{P}}\\&=(1+\beta_\text{c}^2)\Delta{f_0}\end{align}\]

where we have used Equation (6-5-40) and the fact that \(\Gamma_\text{R}\ll\Omega_\text{R}\).

This expression shows that the line width of a semiconductor laser is enhanced by a factor of \(1+\beta_\text{c}^2\) compared with its value \(\Delta{f_0}\) expected on the basis of the modified Schawlow-Townes formula.

Henry was the first to point out the significance of this enhancement factor in an attempt to reconcile the experimental results with theory. The enhancement factor was present in the earlier work on laser line width, but had been thought to be unimportant since \(\beta_\text{c}\ll1\) for lasers other than semiconductor lasers.

Physically the two terms in the last line of Equation (6-5-52) can be interpreted to arise from instantaneous and delayed phase fluctuations resulting from each spontaneous-emission event.

The contribution \(\Delta{f_0}\) is due to instantaneous phase change governed by the Langevin-noise source \(F_\phi(t)\) in Equation (6-5-10). the contribution \(\beta_\text{c}^2\Delta{f_0}\) is due to the \(\delta{N}\) term in Equation (6-5-10).

Its origin lies in the following sequence of events. Each spontaneously emitted photon changes the laser power, which changes the gain (or equivalently the carrier population); this in turn affects the refractive index and consequently the optical phase. The resulting delayed phase fluctuation is affected by relaxation oscillations and leads to satellite peaks at multiples of \(\Omega_\text{R}\) as well as a broadening of the central peak by \(1+\beta_\text{c}^2\).

Equation (6-5-52) shows that the line width \(\Delta{f}\) depends only on \(\beta_\text{c}\), \(R_\text{sp}\), and \(P\). If we choose \(\beta_\text{c}\) equal to \(5\) and use values of \(R_\text{sp}\) and \(P\) from Table 6-2 [refer to the transient response of semiconductor lasers tutorial], we find that \(\Delta{f}\approx70\text{ MHz}\) at a power level of 1 mW.

This provides an order-of-magnitude estimate since both \(\beta_\text{c}\) and \(R_\text{sp}\) are likely to vary from device to device. As the laser power increases, the line width decreases as \(P^{-1}\) in accordance with Equation (6-5-52). Such an inverse dependence is observed experimentally at low power levels (<10 mW) for most semiconductor lasers.

However, often the line width is found to saturate to a value in the range \(1-10\text{ MHz}\) at a power level above 10 mW. Figure 6-15 shows such a line-width-saturation behavior for several 1.55-μm DFB lasers.

It also shows that the line width can be considerably reduced by using a multiquantum-well (MQW) design for the DFB laser. The reduction is due to a smaller value of the parameter \(\beta_\text{c}\) realized by such a design.

The line width can also be reduced by increasing the cavity length \(L\), since \(R_\text{sp}\) decreases and \(P\) increases at a given output power as \(L\) is increases. Although not obvious from Equation (6-5-52), \(\Delta{f}\) can be shown to vary as \(L^{-2}\) when the length dependence of \(R_\text{sp}\) and \(P\) is incorporated.

As seen in Figure 6-15, \(\Delta{f}\) is reduced by about a factor of \(4\) when the cavity length is doubled. The 800-μm-long MQW DFB laser is found to exhibit a line width as small as \(270\text{ kHz}\) at a power output of 13.5 mW.

It should be stressed, however, that the line width of most semiconductor lasers is typically in the range of \(10-100\text{ MHz}\).

Figure 6-15 shows that as the laser power increase, the line width not only saturates but begins to rebroaden. Several mechanisms have been invoked to explain such a behavior; a few among them are side-mode interaction, \(1/f\)-noise, spatial-hole burning, nonlinear gain, and current noise.

The influence of weak side modes on the line width is of particular importance from the practical standpoint. It can be studied by considering the multimode equations (6-5-29) and (6-5-30) together with the phase equation (6-5-10).

In the specific case of a single mode it is possible to calculate the frequency-noise spectrum and the line shape by using Equations (6-5-14), (6-5-32), and (6-5-33). It turns out that cross-saturation, governed by the parameters \(\theta_{12}\) and \(\theta_{21}\) in Equations (6-5-37) and (6-5-38), plays an important role and can lead to considerable spectral broadening even when the MSR exceeds 20 dB.

The issue of laser line width is of considerable importance for lightwave systems. A narrow-line-width optical source is required for coherent communication systems.

Considerable effort has been directed toward developing semiconductor lasers that not only operate in a single longitudinal mode but whose wavelength can also be tuned over a considerable range while maintaining a narrow line width (~ 1 MHz or less).

The next tutorial covers the topic of ** modulation response of semiconductor lasers**.