# Effects of external optical feedback on semiconductor lasers

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

We have considered the emission characteristics of a semiconductor laser operating in isolation.

In practice, however, a small portion of the emitted light is inevitably fed back into the laser cavity owing to parasitic reflections that may occur at a surface outside the cavity. In optical communication systems, such unintentional reflections may occur at the near end or the far end of the fiber link.

It is often observed that even a relatively small amount of external optical feedback can significantly affect the performance of a semiconductor laser. Considerable attention has therefore been given to investigating the effects of external optical feedback on the static, dynamic, spectral, noise, and modulation characteristics of the semiconductor laser.

In this tutorial, we briefly describe these effects using the single-mode rate equations modified to account for the weak external feedback.

## Modified Rate Equations

For a theoretical analysis of the optical feedback effects on laser performance, the rate equations of the rate equations for semiconductor lasers tutorial need to be modified.

In the case of strong feedback, both the laser cavity and the external cavity (formed by the laser facet and the external reflecting surface) should be considered. This case is discussed in a later tutorial in relation to coupled-cavity single-frequency lasers.

When the feedback is relatively weak, it is possible to consider the rate equations for the laser cavity alone. However, the field equation (6-2-9) [refer to the rate equations for semiconductor lasers tutorial] should be modified by adding a term that includes the contribution of the feedback. The situation is similar to the case of external injection.

Using Equations (6-2-8), (6-2-9), and (6-2-17) [refer to the rate equations for semiconductor lasers tutorial], we obtain

\[\tag{6-7-1}\begin{align}\frac{\text{d}E}{\text{d}t}=\text{i}&(\omega_0-\omega_\text{th})E(t)+\frac{1}{2}(G-\gamma)(1-\text{i}\beta_\text{c})E(t)\\&+\kappa{E}(t-\tau)\exp(\text{i}\omega_0\tau)\end{align}\]

where the last term accounts for the reflection feedback and is generally obtained using the concept of an effective facet reflectivity.

In Equation (6-7-1), \(\omega_0\) is the mode frequency in the presence of feedback whereas \(\omega_\text{th}\) is the mode frequency of the solitary laser operating near threshold.

A simple way to obtain an expression for \(\kappa\) is to consider the field at the laser facet facing the external cavity. During each round trip, the field immediately after reflection is

\[\tag{6-7-2}E'(t)=(R_\text{m})^{1/2}E(t)+(1-R_\text{m})(f_\text{ext})^{1/2}E(t-\tau)\exp(\text{i}\omega_0\tau)\]

where \(R_\text{m}\) is the facet reflectivity, \(f_\text{ext}\) is the fraction of output power reflected back into the laser cavity, \(\tau\) is the external-cavity round-trip time, and \(\theta=\omega_0\tau\) is the phase shift that occurs during a single round trip.

Multiple reflections in the external cavity are neglected since \(f_\text{ext}\ll1\) for the case of weak feedback.

Equation (6-7-2) shows that during each round trip inside the laser cavity, a small fraction \((1-R_\text{m})(f_\text{ext}/R_\text{m})^{1/2}\) of the intracavity field reenters the laser cavity after some delay and phase shift.

Since this happens at a rate of \(\tau_\text{L}^{-1}\), the coefficient \(\kappa\) is

\[\tag{6-7-3}\kappa=\frac{(1-R_\text{m})}{\tau_\text{L}}\left(\frac{f_\text{ext}}{R_\text{m}}\right)^{1/2}\]

where \(\tau_\text{L}\) is the laser-cavity round-trip time.

The feedback fraction \(f_\text{ext}\) in Equation (6-7-3) should include all coupling, absorption, and diffraction losses that may occur during the round trip in the external cavity.

In the case of reflections occurring at the far end of a fiber of length \(L_\text{f}\), the feedback fraction is given by

\[\tag{6-7-4}f_\text{ext}=\eta_\text{c}^2\exp(-2\alpha_\text{f}L_\text{f})R_\text{f}\]

where \(\eta_\text{c}\) is the power-coupling efficiency, \(\alpha_\text{f}\) is the absorption coefficient, and \(R_\text{f}\) is the reflectivity of the fiber-air interface.

Equation (6-7-1) can be used to obtain the rate equations for the amplitude and phase of the intracavity field. The carrier rate equations (6-2-20) [refer to the rate equations for semiconductor lasers tutorial] remains unchanged.

In terms of the photon and carrier populations \(P\) and \(N\) and the optical phase \(\phi\), the modified rate equations are

\[\tag{6-7-5}\begin{align}\dot{P}=&(G-\gamma)P(t)+R_\text{sp}\\&+2\kappa[P(t)P(t-\tau)]^{1/2}\cos[\omega_0\tau+\phi(t)-\phi(t-\tau)]\end{align}\]

\[\tag{6-7-6}\dot{N}=I/q-\gamma_\text{e}N(t)-GP(t)\]

\[\tag{6-7-7}\begin{align}\dot{\phi}=&-(\omega_0-\omega_\text{th})+\frac{1}{2}\beta_\text{c}(G-\gamma)\\&-\kappa\left[\frac{P(t-\tau)}{P(t)}\right]^{1/2}\sin[\omega_0\tau+\phi(t)-\phi(t-\tau)]\end{align}\]

where, as in the rate equations for semiconductor lasers tutorial, the spontaneous-emission contribution has been included through \(R_\text{sp}\).

The effect of reflection feedback is included through the parameter \(\kappa\) given by Equation (6-7-3).

When \(\kappa=0\), Equations (6-7-5) to (6-7-7) reduce to the single-mode rate equations of a solitary laser considered in previous tutorials.

The difference-differential form of the modified rate equations implies that the external-reflection feedback would affect significantly the dynamic behavior of a semiconductor laser.

Before discussing the feedback-induced features, let us consider what amount of minimum external feedback is necessary in order to influence the device behavior in a significant way.

A rough estimate can be obtained by comparing the various terms in Equation (6-7-5) and requiring that the contribution of the feedback term proportional to \(\kappa\) exceed that of spontaneous emission \(R_\text{sp}\).

In the steady state, this requirement implies that \(\kappa{P}\gg{R}_\text{sp}\), or after using Equation (6-7-3),

\[\tag{6-7-8}f_\text{ext}\gg(R_\text{sp}\tau_\text{L}/P)^2\]

where we have set \(R_\text{m}/(1-R_\text{m})^2\) equal to \(1\) in this order-of-magnitude calculation after using \(R_\text{m}\approx0.32\).

The condition (6-7-8) shows the important device parameters that determine the sensitivity of a semiconductor laser to external parasitic reflections: the smaller the right-hand side, the more sensitive is the laser to external reflections.

In particular,

- Index-guided lasers are more sensitive than gain-guided lasers because of a smaller \(R_\text{sp}\).
- Short-cavity lasers are more sensitive because of a smaller \(\tau_\text{L}\).
- Reflection sensitivity increases with an increase in the output power.

The condition (6-7-8) can be written in other equivalent forms by noting that \(\tau_\text{L}^{-1}=\Delta\nu\) (the longitudinal-mode spacing) and from Equation (6-5-52) [refer to the noise characteristics of semiconductor lasers tutorial] \(R_\text{sp}/P\sim\Delta{f}=\tau_\text{c}^{-1}\), where \(\Delta{f}\) is the solitary-laser CW line width and \(\tau_\text{c}\) is the corresponding coherence time.

We then obtain

\[\tag{6-7-9}f_\text{ext}\gg\left(\frac{\tau_\text{L}}{\tau_\text{c}}\right)^2\qquad\text{or}\qquad{f_\text{ext}}\gg\left(\frac{\Delta{f}}{\Delta\nu}\right)^2\]

These inequalities show that single-frequency lasers, oscillating predominantly in a single longitudinal mode, are more sensitive to external reflections because of their longer coherence time, or equivalently because of their smaller spectral width.

## Steady-State Behavior

The steady-state behavior of a semiconductor laser can be readily analyzed using Equations (6-7-5) to (6-7-7) after assuming that \(P\), \(N\), and \(\phi\) are time-independent.

For simplicity, we neglect \(R_\text{sp}\), which mainly affects the threshold sharpness as discussed in the steady-state characteristics of semiconductor lasers tutorial.

In the absence of reflection feedback (\(\kappa=0\)), the solution is

\[\tag{6-7-10}\bar{G}=\gamma\qquad\bar{\omega}_0=\omega_\text{th}\qquad\bar{P}=(I-I_\text{th})/q\gamma\]

where the threshold current \(I_\text{th}=q\gamma_\text{e}N\).

If \(\Delta{G}\) and \(\Delta\omega\) represent the gain and frequency shifts introduced by the feedback, they are given by

\[\tag{6-7-11}\Delta{G}=G-\bar{G}=-2\kappa\cos(\omega_0\tau)\]

\[\tag{6-7-12}\Delta{\omega}=\omega_0-\omega_\text{th}=\kappa[\sin(\omega_0\tau)+\beta_\text{c}\cos(\omega_0\tau)]\]

The corresponding output power can be obtained using Equation (6-7-6) and is given by

\[\tag{6-7-13}P=(I-I_\text{th})/[q(\gamma+\Delta{G})]\]

The output power may increase or decrease with feedback depending on whether \(\Delta{G}\) is negative or positive.

For given values of \(\kappa\) and \(\tau\), Equations (6-7-11) to (6-7-13) can be used to obtain the steady-state characteristics. The device behavior, however, depends on whether the frequency equation (6-7-12) allows single or multiple solutions.

It can be readily verified that a single solution is obtained when the feedback parameter

\[\tag{6-7-14}C=\kappa\tau(1+\beta_\text{c}^2)^{1/2}\lt1\]

Using Equations (6-7-3) and (6-7-14) we note that the value of \(C\) depends on both the feedback fraction \(f_\text{ext}\) and the external-cavity optical length \(L_\text{ext}\) (\(\tau=2L_\text{ext}/c\)).

Let us first consider the case of a short external cavity such that \(C\lt1\).

The frequency shift \(\Delta\omega\) can be obtained by solving Equation (6-7-12). Equations (6-7-11) and (6-7-13) then yield the gain change \(\Delta{G}\) and the corresponding mode power \(P\).

Both the output power \(P\) and the frequency shift \(\Delta\omega\) can however be changed by changing \(\tau\) through variations in the external cavity length. The situation can be visualized by treating the round-trip phase shift (\(\theta=\omega_0\tau\)) occurring in the external cavity as a continuous parameter. Its parametric variation results in an ellipse in the \(\Delta\omega-\Delta{G}\) plane, as shown in Figure 6-25.

As \(\theta\) is varied by changing \(\tau\), both the mode frequency and the mode power change accordingly. The periodic variation of \(P\) and \(\omega_0\) with the external-cavity length has been observed experimentally.

The relative power change depends on the feedback fraction. If we use Equations (6-7-3) and (6-7-11), it is easy to see that \(|\Delta{G}/G|_\text{max}\approx(f_\text{ext})^{1/2}\); i.e., a 1% feedback can change the mode gain and the power each by 10%. This result again emphasizes the sensitivity of semiconductor lasers to external reflections.

The effect of external reflections on the light-current (\(L-I\)) characteristics is to introduce periodic ripples. Lang and Kobayashi have discussed the \(L-I\) curves and the current-induced spectral shifts in the presence of feedback.

When the feedback is relatively strong, a hysteresis is observed in both the oscillation frequency and the \(L-I\) curve. These experimental results can be explained by taking into account the thermal variation of the refractive index. As the current is increased, the temperature rise under CW operation shifts the longitudinal-mode frequency of the laser cavity, which affects the stability of the lasing mode.

When the feedback conditions are such that the parameter \(C\) defined by Equation (6-7-14) exceeds unity, Equation (6-7-12) can be satisfied for multiple values of \(\omega_0\) while \(\tau\) is kept fixed.

The number of allowed stable solutions increases as \(C\) increases. The frequency difference between the two neighboring solutions is approximately given by \(\tau^{-1}\), or the longitudinal-mode spacing of the external cavity.

These multiple solutions are referred to as the ** external-cavity modes** (XCM), which appear in the vicinity of the

**(LCM), the dominant longitudinal mode of the solitary laser.**

*laser-cavity mode*Figure 6-26 shows schematically the longitudinal-mode spectra of semiconductor laser with and without feedback.

With reference to Figure 6-25, the multiple solutions of Equations (6-7-11) and (6-7-12) are discrete points on the lower limb of the ellipse. The solution closest to the bottom of the ellipse corresponds to the lowest threshold mode, which becomes the dominant mode in Figure 6-26. Other solutions correspond to XCMs that have smaller amplitudes because of the mode selectivity provided by the gain margin (different \(\Delta{G}\) values in Figure 6-25).

Feedback-induced XCMs have been observed and studied in detail. Their number and amplitudes depend on the external-cavity length \(L_\text{ext}\) as well as on the feedback fraction \(f_\text{ext}\).

They are readily observed for relatively long external cavities (\(L_\text{ext}\ge1\text{ m}\)) since the feedback parameter parameter \(C\gg1\) in that case, even for weak feedback (\(f_\text{ext}\le10^{-3}\)).

In particular, the feedback from the far end of an optical fiber can generate a large number of XCMs in the power spectrum of a semiconductor laser. Both the parameter \(C\) and the feedback fraction \(f_\text{ext}\) are shown in Figure 6-27 as a function of the fiber length for three values of the fiber loss after using Equations (6-7-4) and (6-7-14).

The feedback parameter \(C\gg1\) for fiber lengths in a wide range of 1 m to 100 km. Clearly, far-end reflections can generate additional XCMs under typical operating conditions of an optical communication system. Among other effects, the presence of far-end reflections increases laser-intensity noise and can lead to an increase in the bit error rate.

## Dynamic Behavior

The previous section discussed the steady-state solutions of the modified rate equations (6-7-5) to (6-7-7). However, a stationary solution may not be dynamically stable; i.e., a small deviation from the steady state may grow indefinitely.

As in the relaxation oscillations part of the transient response of semiconductor lasers tutorial, the dynamic stability is examined through small-signal analysis of the modified rate equations.

In contrast to the relaxation-oscillation analysis presented in the transient response of semiconductor lasers tutorial, the optical phase \(\phi\) now plays critical role since the external feedback provides a reference phase (similar to the case of injection locking).

In the small-signal analysis, the variables \(P\), \(N\), and \(\phi\) are perturbed by a small amount around their steady-state values and Equations (6-7-5) to (6-7-7) are linearized in terms of small perturbations \(\delta{P}\), \(\delta{N}\), and \(\delta\phi\).

The resulting set of homogenous linear equations can be readily solved after assuming that perturbations vary in time as \(\exp(zt)\). The procedure is similar to that of the relaxation oscillations part of the transient response of semiconductor lasers tutorial except for the complications arising because the rate equations take the difference-differential form in the presence of reflection feedback.

The decay or growth rate \(z\) of perturbations is obtained by the transcendental equation

\[\tag{6-7-15}\begin{align}(z+\Gamma_\text{N})[(z+\kappa_\text{c}\xi)(z+\kappa_\text{c}\xi+\Gamma_\text{P})+&\kappa_\text{s}^2\xi^2]\\&+GG_\text{N}P[z+(\kappa_\text{c}-\beta_\text{c}\kappa_\text{s})\xi]=0\end{align}\]

where

\[\tag{6-7-16}\kappa_\text{c}=\kappa\cos(\omega_0\tau)\qquad\kappa_\text{s}=\kappa\sin(\omega_0\tau)\]

\[\tag{6-7-17}\xi(z)=1-\exp(-z\tau)\]

and other parameters have been defined in the relaxation oscillation part of the transient response of semiconductor lasers tutorial.

A steady-state solution represents a dynamically stable state of the system if all roots of Equation (6-7-15) have a negative real part.

In general, the roots are obtained numerically for a given set of parameters.

However, in the case of weak feedback such that \(\kappa\tau\ll1\), the solutions of Equation (6-7-15) satisfy \(|\tau{z}|\ll1\), and one can replace \(\xi(z)\) with \(\tau{z}\) in Equation (6-7-15). The resulting polynomial has three roots: \(z=0\) and \(z=-\Gamma_\text{R}\pm\text{i}\Omega_\text{R}\), where \(\Gamma_\text{R}\) and \(\Omega_\text{R}\) represent, respectively, the decay rate and the frequency of the relaxation oscillations given by

\[\tag{6-7-18}\Gamma_\text{R}=\frac{1}{2}\left[\Gamma_\text{N}+\frac{(1+\kappa_\text{c}\tau)}{(1+\kappa_\text{c}\tau)^2+\kappa_\text{s}^2\tau^2}\Gamma_\text{P}\right]\]

\[\tag{6-7-19}\Omega_\text{R}\approx\bar{\Omega}_\text{R}\left[\frac{1+(\kappa_\text{c}-\beta_\text{c}\kappa_\text{s})\tau}{(1+\kappa_\text{c}\tau)^2+\kappa_\text{s}^2\tau^2}\right]^{1/2}\]

where \(\bar{\Omega}_\text{R}=(GG_\text{N}P)^{1/2}\) and is the relaxation-oscillation frequency of the solitary laser.

Equations (6-7-18) and (6-7-19) show that the external feedback affects both the frequency and the decay rate of relaxation oscillations.

Under certain feedback conditions, \(\Omega_\text{R}^2\) can become negative, making \(z\) real and positive. The CW state is then unstable and the laser is likely to exhibit self-pulsing.

Using Equations (6-7-16) and (6-7-19), a necessary condition for stability is

\[\tag{6-7-20}1+C\cos(\omega_0\tau+\tan^{-1}\beta_\text{c})\gt0\]

where the feedback parameter \(C\) is given by Equation (6-7-14).

When \(C\gt1\), the inequality in Equation (6-7-20) is violated for some phases \(\theta\), making the corresponding steady-state solution unstable.

For large values of \(C\), however, it is necessary to use Equation (6-7-15) for the linear stability analysis. The device behavior is then sensitive to the exact value of \(\tau\), and small changes can enhance or damp relaxation oscillations.

In general, for a given external-cavity length, the steady state becomes unstable for a range of feedback levels \(f_\text{ext}\) and phases \(\theta\). The semiconductor laser would then exhibit feedback-induced pulsations.

By contrast, under certain conditions the external feedback can stabilize the output of a self-pulsing semiconductor laser.

Recently, a new kind of reflection-induced instability has been observed wherein the semiconductor laser exhibits pulsations with a period of about \(5-10\tau\). Furthermore, during each pulsation, ripples of width \(\tau\) are superimposed on the pulse.

It is found to be necessary to go beyond the preceding linear stability analysis in order to explain the pulsation behavior. The physical mechanism behind this instability can be understood by referring to Figure 6-25.

In the absence of external reflections, the operating point is at the origin (\(\Delta\omega=\Delta{G}=0\)).

In the presence of reflections, the steady state corresponds to operation at the bottom of the ellipse. The approach to the steady state requires traveling along the lower portion of the ellipse.

However, as the steady-state point is approached, an instability due to carrier-induced phase fluctuations shifts the operating point from the bottom to the top of the ellipse, resulting in a drop of the output power. The whole process then repeats itself.

The frequency chirp associated with self-pulsations manifests as a substantial broadening of the CW line width that has been observed experimentally.

## Noise Characteristics

The external feedback affects the intensity noise as well as phase noise.

The intensity-noise spectrum at high frequencies is found to consist of a large number of equally spaced peaks whose frequency separation coincides exactly with the external-cavity mode spacing \(\tau^{-1}\). Temkin et al. have observed an additional low-frequency peak that is associated with the instability described in the previous section.

The high-frequency noise peaks can be explained using the small-signal analysis of the modified rate equations after adding the Langevin noise sources and following the procedure of intensity noise part of the noise characteristics of semiconductor lasers tutorial.

Explanation of the low-frequency noise requires consideration of relatively large fluctuations.

The effect of external reflections on the phase noise has attracted attention, since it can lead to considerable narrowing or broadening of the line width of a single longitudinal mode depending on the relative phase of the returned light.

The problem has been analyzed theoretically using the modified rate equations. Following an analysis similar to that given in the phase noise and line width part of the noise characteristics of semiconductor lasers tutorial, the line width in the presence of external feedback is approximately given by

\[\tag{6-7-21}(\Delta{f})_\text{ref}\approx\frac{\Delta{f}}{[1+C\cos(\omega_0\tau+\phi_\text{R})]^2}\]

where \(\phi_\text{R}=\tan^{-1}(\beta_\text{c})\) and \(\Delta{f}\) is the solitary-laser line width given by Equation (6-5-52) in the noise characteristics of semiconductor lasers tutorial.

Equation (6-7-21) shows that, depending upon the feedback phase \(\omega_0\tau\), the laser line width can broaden or narrow.

The maximum narrowing occurs when \(\omega_0\tau+\phi_\text{R}=2m\pi\), and the reduction factor in that case is \((1+C)^2\). Clearly the line width can be reduced considerably for large values of \(C\) if the external-cavity length is fine-tuned to adjust the phase \(\omega_0\tau\). Reductions by more than three orders of magnitude have been observed.

As discussed earlier, when \(C\gt1\) there are multiple stable steady states over which the laser can be made to oscillate by changing the external-cavity length. In the vicinity of mode hops, the line width can vary by a large amount with a small change in \(\tau\).

For \(C\lt1\), only one mode oscillates for all values of the phase. Considerable line broadening can occur in this case in a certain range of phase values, as seen clearly by an inspection of Equation (6-7-21).

The effect of external feedback on the phase noise at high frequencies has also been discussed. The phase noise is considerably enhanced in the vicinity of the relaxation-oscillation frequency \(\Omega_\text{R}\).

This results in enhanced side peaks when \(\omega=\omega_0\pm\Omega_\text{R}\) in the optical spectrum of the semiconductor laser. Figure 6-28 shows the experimentally observed power spectrum when a small fraction (\(f_\text{ext}\approx10^{-3}\)) of the power is reflected back from the far end of a 7.5-km-long fiber.

The side peaks are displaced from the center-line by 4.88 GHz, which corresponds to the relaxation-oscillation frequency of the laser. These side peaks are present even for the case of a solitary laser, but their amplitude is usually below 1% of the central peak.

Physically one can interpret the growth of the sides peaks in the optical spectrum (see Figure 6-28) as unclamping of the relaxation oscillations such that the laser power is self-modulated at the relaxation-oscillation frequency. In other words, feedback destabilizes the CW state and makes the laser output periodic.

With a further increase in the feedback, the amplitude of the side peaks (AM sidebands) grows, and the optical spectrum becomes broader. When the external feedback exceeds a critical level, the optical spectrum becomes extremely broad with a line width in the range of a few gigahertz.

This extreme broadening of the laser line width is referred to as coherence collapse, since it implies that the semiconductor laser has a much shorter coherence time than the value expected for the solitary laser in the absence of the feedback.

The phenomenon of coherence collapse has attracted considerable attention since it represents an example of feedback-induced chaos in semiconductor lasers resulting from destabilization of the CW or self-pulsing state.

The presence of spontaneous-emission fluctuations makes it difficult to distinguish between the chaotic and stochastic dynamics, but the evidence of chaos has been seen in several experiments. Numerical solutions of the rate equations [Equations (6-7-5) to (6-7–7)] clearly predict the onset of feedback-induced chaos when he feedback exceeds a critical level.

Considerable experimental and theoretical work has been carried out to understand the chaotic dynamics in semiconductor lasers.

From a practical standpoint, the coherence-collapse region should be avoided in the design of coherent communication systems requiring narrow-line-width semiconductor lasers.

The effect of external feedback on the RIN has also been studied. In the low-feedback regime in which the steady state remains stable, an analytic expression for RIN can be obtained. In general, RIN can increase considerably with feedback.

Similarly to the case of laser line width, RIN is sensitive to the feedback phase [the parameter \(\omega_0\tau\) in Equation (6-7-21)] which can cause variations in RIN as much as 10-20 dB. In fact, RIN becomes minimum under the same conditions in which the line width is minimum. In practice, however, the feedback phase is not easy to control, and RIN often increases significantly with feedback.

The feedback-induced enhancement of RIN is of considerable concern in the design of optical data-storage systems where the unintended feedback from the optical disk can severely degrade the system performance. A technique, known as high-frequency injection, solves the problem by modulating the laser current at frequencies much higher than the data rate.

However, RIN depends on many parameters such as modulation frequency, modulation depth, and round-trip time in the external cavity. In general, RIN decreases with an increase in the modulation current, and the decrease is larger for longer external-cavity lengths. It can increase abruptly above a certain modulation depth whose exact value depends, among other things, on the modulation frequency and the external-cavity length.

The use of the high-frequency injection technique requires optimization of several operating parameters which generally vary from laser to laser.

It is clear that parasitic reflections in an optical communication system can degrade system performance. The problem of parasitic reflections becomes severe for coherent systems requiring semiconductor lasers with a stable amplitude and phase.

The use of isolators between the laser and the fiber then become a necessity. The amount of required isolation is governed by Equations (6-7-4) and (6-7-9), and may exceed 50 dB (\(f_\text{ext}\le10^{-5}\)).

The next tutorial introduces ** DFB (distributed feedback) semiconductor lasers**.