Linewidth Enhancement Factor and Amplification of Angle-Modulated Optical Signals in Injection-Locked Quantum Cascade Lasers

1 Introduction

Quantum cascade lasers (QCLs) [1–9] are intersubband lasers. In normal interband p-n junction laser an electron makes a downward transition from the conduction band to valence band resulting in an emission of a photon. The emission wavelength of diode lasers, calculated from the difference in energy of the two levels, in general, falls in the visible, near and mid-infrared (IR) region of the spectrum. In contradistinction to normal diode laser, an electron in a QCL falls down a staircase-like potential in the conduction band and generates a series of photons of same energy in a cascade. There can be several staircases in the conduction band of the QCL corresponding to several subbands. The active region of the QCL consists of a multi-quantum well structure in general, and in each potential well there are subband energy levels. The cascaded nature of emission of a series of photons by a single electron gives rise to relatively high power of the QCL. Another point of difference of QCL with normal diode laser is that the uppermost subband electron relaxation time in the three-level system of QCL is much shorter [1] (typical value of 4.3 ps) compared with that of normal interband diode laser which is in the nanosecond region. In the lower-middle subband to which the electron makes a radiative transition from the uppermost level, the electron relaxation time is further shorter (~0.6 ps) so that the electrons are removed very fast from this level. The electrons are injected from the injector into the active region of the QCL through resonant tunnelling in a very short time (~0.2 ps). These relaxation times are much shorter than the cavity round trip time (~50 ps) [10] in QCL.

Injection locking [11, 12] is basically a nonlinear phenomenon. It has been well investigated in the radiofrequency and microwave oscillators [11, 12]. QCL is an oscillator operating in the mid-IR and far-IR regions. The modal gain of the QCL decreases with the increase in laser intensity which is known as gain saturation. Since intensity is proportional to the square of the electric field amplitude, the gain is a nonlinear function of the laser electric field amplitude. It is the gain nonlinearity which is responsible for injection locking in QCL. This is analogous to the mode-locking phenomenon [13] where the loss modulation takes place in a saturable absorber. A rate equation model [14] of the injection-locked QCL has been developed where the role of gain nonlinearity has been ignored. We have developed a transmission line model of the injection-locked QCL which takes into account the laser intensity-dependent gain, viz. gain nonlinearity of the active region of the device.

In injection locking, a free-running slave QCL is injected by another QCL called as the master QCL whose lasing frequency is close to the free-running frequency of the slave QCL. Depending upon the injection IR power level relative to that of the free-running slave QCL and the detuning of the master QCL from the slave QCL, the slave QCL begins to oscillate at the master QCL frequency instead of its own frequency and its phase is locked. The slave QCL is said to be injection locked to the master QCL.

The injection-locked QCLs have many applications in the field of optical communication. They can act as high-gain amplifiers [15] of angle-modulated mid-IR signals and can have high-intensity modulation bandwidths [16]. There are atmospheric windows in 3–5 μm and 8–14 μm wavelength ranges. Free-space optical communication with high speed and wide bandwidth is possible in these wavelength regions. Light detection and ranging (LIDAR) which is an optical analog of microwave/millimetre wave radar can be designed to operate in these window wavelengths.

The linewidth enhancement factor (LEF) [17–24] which arises from phase amplitude coupling of a QCL resulting from carrier density variation in the active region plays an important role in the injection locking [25–28] process of QCLs and possesses considerable influence in the amplification process of angle modulated IR signals by injection-locked QCLs. In this paper, we investigate analytically the role of LEF on the injection-locked QCL amplifiers using transmission line model [25] developed by the authors. To the best of our knowledge no analysis, except the one describing resonant amplification [29] in QCL, has appeared in literature yet on the role of LEF on the injection-locked QCL amplifiers. It is seen from the present analysis that with a given injection power the bandwidth of the injection-locked QCL amplifier favours higher value of LEF. Second, for a given value of LEF, a higher IR injection power leads to a considerable increase in QCL amplifier bandwidth. These are some important outcomes of the analysis.

2 Analysis

A transmission line model [25] for a Fabry–Perot (FP) QCL has been developed for analysing the injection-locking phenomenon in QCL. The injection-locked QCL acts as a good amplifier of mid-IR optical angle-modulated signals [15, 27]. A schematic block diagram of injection-locked QCL amplifier of angle-modulated IR signal is shown in Figure 1.

Figure 1:

A schematic block diagram of injection-locked QCL amplifier of angle-modulated IR signals. IR, infrared; PM, phase modulator.

The active region of the QCL is assumed to have a length l with cavity loss αc and mirror loss αm. The refractive index of this active region is taken as n. An idea of modal gain saturation of the QCL can be obtained from the theory presented in [17, 26, 29]. The intensity-dependent modal gain of the QCL can be expressed as

where I is the laser intensity within the cavity, Is is the saturation intensity and g0 is the unsaturated value of gain. Since I∝E2, the IR field-dependent gain under weak saturation condition can be approximated as

Here, E is the IR electric field amplitude and Es is the saturation value of the IR electric field amplitude of the slave QCL.

The input impedance of the equivalent transmission line is given by

where μ0 and ε0 are the absolute permeability and absolute permittivity of free space (air).

In complex representation, the angle-modulated lightwave is assumed to have a complex amplitude

while the complex amplitude of the output lightwave from the injection-locked QCL is written as

Here, θi is an arbitrary phase angle and ξt represents input angle modulation, while θ0t stands for angle modulation of the output lightwave from the injection-locked QCL.

The input–output phase error,

where ψt=θi−θ0t is the phase error in the absence of angle modulation.

The electric field, E, and the magnetic field, H, of the electromagnetic wave travelling on the transmission line are related as

Again, in wave representation

and

Using eqs (8), (9) and (3) in eq. (7) and then applying the principle of harmonic balance we obtain the following phase equation of the slave QCL injected by the angle-modulated IR signal:

where δ=ω−ω0ω0 is the normalized detuning of the injection IR signal of radian frequency ω from the cavity resonance radian frequency ω0. Q is the external quality factor of the cavity, y=(|E|2/|Ef|2) with Ef being the free-running electric field of the slave QCL. C1, C2 and C3 are the laser constants expressed as C1=n−1nαml, C2=n−1nlg0|Ef|2|Es|2, C3=n−1nlg0|Ef|4|Es|4. Taking, |Ef|2/|Es|2=0.8 we get the values of QCL constants C1, C2, C3 as C1=4.02, C2=9.86, C3=7.89.

The LEF (α) results from the coupling between gain and refractive index of the active region under carrier density variation. Mathematically, LEF is defined as

where λ0 is the vacuum wavelength of the QCL, n stands for refractive index, N implies carrier density in the active region and G=A0N1−N2 stands for carrier-density-dependent linear gain of the QCL. N1 and N2 are carrier densities in the uppermost and middle-lower lasing subbands of a three-level system. A0 is the differential gain in m3/s. This LEF (α) plays a dominant role in the amplification process of angle-modulated optical signals in an injection-locked QCL.

As derived in [17], the LEF produces a change in cavity resonance frequency given by

where ω0 is the cavity resonance frequency in radian, Δω0 is the change in ω0 produced by the LEF, α, Pi is the injection lightwave power, Ps is the free-running output power of the QCL and ζ is a parameter expressed as [17]

Here, Γ is the optical confinement factor, A0 is the differential gain in m³/s, A0′ is the modified gain in m², S is the photon density, τ1 is the uppermost subband electron relaxation time. c is the vacuum velocity of light, h is the Plank constant, ν is the frequency of IR light, V is the volume of active region, n is the refractive index of the active region and nˉ is the group refractive index.

The detuning parameter δ=ω−ω0ω0 should be replaced by δ−Δω0ω0 in the phase equation [15, 17, 27] of the injected QCL. Considering the effect of LEF, the phase equation of the injected QCL [17] is given by

Let the angle modulation be sinusoidal in nature. Then,

where ωm is the radian frequency of modulation and δm is an arbitrary phase angle.

The output of the locked QCL will also be angle modulated. We assume solution of ψt as

where ψ0 is the dc phase error in absence of angle modulation and δ0 is an arbitrary phase angle. We consider low level injection, i.e. Pi<<Ps.

Now, dψtdt=−dθ0tdt.

Substituting, eqs (15) and (16) in eq. (14) and applying the principle of harmonic balance, we get

where

Squaring and adding the equations of eq. (17) and simplifying, we get

Here we have assumed the angle modulation amplitude to be small so that J0ψm≈1, J0ξm≈1, J1ψm=ψm2 and J1ξm=ξm2.

Equating the dc terms from both sides of eq. (14) and simplifying we get

From eq. (24), we write

where

At ω=ω0, ψ0=ν. Since (Pi/Ps)<<1 and cosψ0<1, the numerator of eq. (23) is less than the denominator. So, in general, ψm<ξm. When the modulation frequency ωm→0, the factor k=2Qωmω0→0. In that case, ψm=ξm, i.e. the angle modulation amplitude remains the same in the low modulation frequency region. As the angle modulation frequency ωm increases, the ratio of output to input angle modulation amplitude decreases. The variation of the ratio ψm/ξm with the angle modulation frequency, fm, using LEF, α, as a parameter is shown in Figure 2. Similar variation using the relative optical injection power level, Pi/Ps, as a parameter for α=1 is shown in Figure 3. The amplifier bandwidth increases with increase in the injection power level, Pi/Ps.

Figure 2:

Frequency response of the injection-locked QCL amplifier of angle-modulated optical signals using LEF α as a parameter.

Figure 3:

Frequency response of the injection-locked QCL amplifier of angle-modulated optical signals using injection power ratio (Pi/Ps) as a parameter.

The values of QCL parameters used in numerical calculation are shown in Table 1.

The calculated variation of the input–output dc phase error ψ0 as a function of the detuning of the injection signal carrier frequency from the free-running frequency of the slave QCL using LEF, α, as a parameter is shown in Figure 4. For α=0, the phase error curve passes through the origin. The maximum detuning on either side of the zero detuning point is different for different values of α, since the upper-side and lower-side lockbands (LBs) are functions ofα.

From eq. (23), we see that if α=0 then

where γ and η are given by eqs (20) and (21), respectively.

Equation (27) is similar with that in [15] with α=0.

Figure 4:

Calculated variation of dc input–output phase error with the detuning of the injection signal carrier from the free-running slave QCL frequency using LEF α as a parameter.

3 Lockband estimation

Let us ignore the angle modulation of the input lightwave in this case. Then, ξt=0 and ψt=θi−θ0t. Now, from the steady-state form of the phase eq. (14), we can write

where

and

The locking band lying on the upper side of the free-running slave QCL frequency (USLB) is obtained from the condition, ψ=(π/2).

Then, USLB=ω+−ω0=Rsinπ2−μ

where

Similarly, the lower-side lockband (LSLB) is calculated as

Here, ω+ and ω− stand for injection frequencies corresponding to upper and lower extremities of the LB, respectively.

Total LB

If the LEF, α=0, then

The ratio of total LBs of the locked slave QCL with α≠0 and with α=0 is given by

The calculated variations of USLB, LSLB and |LBtotal with Pi/Ps are shown in Figure 5. The LEF, α, is thus responsible for locking asymmetry [30–32] in QCL. The LB asymmetry expressed by the ratio LSLB/USLB is plotted in Figure 6 as a function of LEF, α. The asymmetry increases rapidly with the increase in the value of α.

Figure 5:

Calculated variation of upper-side, lower-side and total lockband with the square root of the normalized injection power (Pi/Ps).

Figure 6:

Calculated variation of the ratio of lower-side to upper-side lockband sections as a function of LEF α.

LB asymmetry [30, 32] introduced by the LEF α is present in both QCL and interband semiconductor lasers. But, there is a difference regarding the magnitude of this LB asymmetry between QCLs and the interband laser diodes. This difference is introduced by the factor ζ which depends upon the upper most lasing level carrier relaxation time τ1 and also upon the lasing wavelength λ0. The magnitude of the factor Γτ1A0S in the denominator of the expression for ζ given in eq. (13) is of the order of unity in QCL but this factor is much greater than unity in interband diode lasers. Effectively ζ becomes nearly independent of τ1 (~ns) in diode lasers. However, ζ is a function of τ1 (~ps) in QCL. All other parameters remaining the same, the magnitude of ζ is greater in interband diode lasers compared with the same in QCL owing to different values of τ1 alone. So, the different values of τ1 in QCL and diode laser tend to reduce the locking asymmetry in QCL compared with the diode laser. Again, ζ∝λ02. For mid-IR QCL, the lasing wavelength λ0 is considerably greater than the same for diode lasers operating in the visible or near IR region. So, this wavelength dependence of ζ tends to enlarge the locking asymmetry in QCL compared with that for diode laser. The overall enlargement or contraction in LB asymmetry of the QCL will be determined by these two opposite effects, viz. which one surmounts the other.

When compared with the existing rate equation model [14] of the injection-locked QCL, it is seen that it does not take into account the gain nonlinearity of the QCL which is responsible for injection locking. Our transmission line model [25, 26], on the other hand, takes the gain nonlinearity of the QCL into consideration and develops the required nonlinear analysis for injection locking of QCL.

4 Conclusion

We have made a nonlinear analysis of the effect of LEF on the amplification of angle-modulated optical signals in injection-locked slave QCL. The higher the value of LEF the more is the tendency of preservation of angle modulation index, the limiting value of output angle modulation depth being equal to the input angle modulation index. For a given angle modulation frequency, the higher the value of LEF the higher is the output angle modulation index. The LEF also modifies the value of input-output dc phase error and at the same time introduces asymmetry in the locking characteristics of the slave QCL. The LSLB is larger than the upper-side LB. However, since the value of LEF is small in QCL compared with the normal interband semiconductor laser, the locking asymmetry tends to be less pronounced in QCL due to the smallness of LEF alone.

The smaller value of upper most subband electron relaxation time in QCL compared with that in the interband diode laser tends to reduce the LB asymmetry in QCL. On the other hand, longer lasing wavelength of mid-IR QCLs tends to increase the LB asymmetry in QCL in comparison with the interband semiconductor lasers.