1. Introduction

Spectral and intensity stability of lasers is of utmost importance for applications that utilize the coherence properties of light, such as heterodyne detection and high resolution spectroscopy. Frequency and intensity of electrical field in a laser cavity experience continuous fluctuations due to the variation of the surrounding experimental conditions. Feedback mechanisms are widely used to stabilize single mode lasers using atomic or molecular absorption lines as frequency reference [1]. A different scenario is required in the case of multimode lasers. They require additional stabilization mechanisms to lock the different operating modes. Depending on the phase relation among the modes, frequency combs and laser pulse trains can be generated in stabilized multimode systems [2].

Quantum cascade lasers (QCL) are unipolar sources based on intersubband transitions in the conduction bands [3]. Due to limited amplitude-phase coupling, QCLs exceed standard semiconductor laser stability. Laser linewidth of few kilohertz have been reported for QCLs [4–7].

In addition QCLs feature strong resonant nonlinearities due to a large optical matrix element between the upper and the lower state of the laser transition. Strong second order optical nonlinearities between laser states have also been demonstrated thanks to symmetry breaking of the confining potential in the growth direction [8–10].

In this work we present stabilization and control of the frequency separation of the modes of a QCL via two, cascaded, second order nonlinear processes in the optical cavity. In the first process two adjacent longitudinal modes beat to generate microwave field oscillating at the round trip frequency. In the second one the generated field interacts with the exiting modes, creating sidebands and locking the spectrum. This last second order nonlinear process can be either triggered by the optical field itself or by an external direct modulation of the laser bias. An extremely stable microwave signal can modulate the optical field and its frequency can control the separation of the optical modes within a certain detuning (frequency injection).

In recent years, third order resonant nonlinearity in QCLs has been demonstrated to be a powerful mechanism to generate stable frequency comb in the mid infrared region [11–13]. Degenerate and non-degenerate four-wave mixing processes generate a proliferation of the optical modes locking the optical spectrum, if laser waveguide has a limited group velocity dispersion.

The stabilization mechanism presented in this work, in contrast to standard four-wave mixing, has the advantage of being naturally adapted for external direct modulation of the laser bias. In this scheme the frequency separation of the laser modes can be locked to a stable external source for more control of the spectrum.

By engineering a laser cavity with limited losses for the microwave beating field and increased overlap between the optical and microwave mode, third order nonlinear processes based on cascaded nonlinearities cannot only be observed but they also become dominant.

In this work we present the realization of a microstrip-QCL, where mid-infrared QCL buried heterostructure is embedded in a microwave line, as in standard double metal THz QCL.

Experimental characterization of the inter-mode stability is performed by measurements of the spectrum at the round-trip frequency of electrical signal generated by the incident output optical power on an ultrafast detector, the beat note signal. Reduction of the full width half maximum of the beat note of a factor >10 with respect to the same buried heterostructure without microstrip is demonstrated reaching values < 100 kHz. We present beat note spectra for the entire dynamical range narrowing with the operating bias.

Stabilization of inter-mode spacing is further investigated by direct modulation of the laser at the round-trip frequency. Sidebands of the free running modes are created and used to lock the laser modes separation to the external modulation source. Finally, a comparative study between the microstrip-QCL [14] and the same QCL without the microstrip line is carried out. In the following we will refer to the reference buried heterostructure QCL without microstrip line simply as QCL.

We present a net improvement of the frequency response to direct modulation of the microstrip-QCL respect to the QCL, with low pass cutoff frequency > 14 GHz limited by the experimental set-up. For the microstrip-QCL we demonstrate injection locking of the laser mode separation to an external modulation source and tuning of the laser inter-mode spacing by more than 1 MHz from the free-running spacing.

2. Stabilization of the laser inter-mode separation via cascaded nonlinearity

Third order nonlinear processes are widely used to generate frequency combs, where the spectrum of the optical beam is an ultra-stable series of evenly spaced modes [15] with constant phase difference. In QCL stabilization of the spacing among longitudinal laser modes has been demonstrated via nonlinear processes in the active material itself [11,12]. By considering 4 laser consecutive modes ω_n, ω_{n + 1}, ω_{n + 2}, ω_{n + 3} interacting in an active medium through an effective χ³, four wave mixing can take place following for instance the scheme ω_n + ω_{n + 3} = ω_{n + 1} + ω_{n + 2}.

QCLs have extremely high nonlinear coefficient associated with intersubband transitions with measured values of χ³ of 0.9 10⁻¹⁵ m²V⁻² per period [16].

In this work we experimentally investigate an equivalent third order nonlinear process generated by the cascade of two second order nonlinear mechanisms [17,18]. Two incoming waves at frequency ω_n and ω_{n + 1} interact with the nonlinear material via a first χ^{(2) DFG} process, generating an intermediate field at ω_i = ω_n – ω_{n + 1,} the frequency difference between two adjacent longitudinal modes. This field interacts successively with a third wave at ω_m to generate ω_{m ± 1} = ω_m ± ω_i. Both processes can be considered to be generated by a resonant nonlinear process between the upper and lower states of the laser transition.

To estimate the value of the χ^(2)DFG of the first interaction and of the χ^(2)SFG for the second process, we use the formalism of [19], where nonlinear processes in asymmetric quantum wells are studied. The expression of χ^(2)DFG of the first interaction is ${\text{χ}}^{\left(2\right)DFG}=\frac{2N{q}^3\delta z^2{T}_1{T}_2}{{\epsilon }_0{\hslash }^2}$ and the second nonlinear interaction is ${\text{χ}}^{\left(2\right)SFG}=\frac{2N{q}^3\delta z^2{T}_2{}^2}{{\epsilon }_0{\hslash }^2}$ where N is the volume population inversion, δ is the difference between the centroid of the electron distributions in the upper and lower lasing state, q is the electron charge, z is the matrix element of the dipole moment operator, T₁ the relaxation time, T₂ the dephasing time, ε₀ the vacuum permittivity and ℏ the Planck constant.

We would point out that an essential requirement for second order nonlinear interaction is the asymmetry of the confining potential. This gives rise to a shift, along the z axis, between the centroids of the upper and lower state wave functions, thus producing a non-negligible value of δ. This value can be very large in a quantum cascade laser active region, higher than in asymmetric quantum well, with estimated value of few nanometers.

Assuming as N = 1.9 10¹⁶ cm⁻³ the average population inversion per period, T₁ = 0.29 10⁻¹² s and T₂ = 0.15 10⁻¹² s and δ = 10⁻¹⁰ m, we obtain ${\text{χ}}^{\left(2\right)DFG}$ = 1.9 10⁻⁶ mV⁻¹ and ${\text{χ}}^{\left(2\right)SFG}$ = 1.0 10⁻⁶ mV⁻¹. With the same formalism the value of the third order non linearity can be calculated at resonance for the same active region parameters obtaining the value ${\text{χ}}^{\left(3\right)}=\frac{4N{q}^4{z}^4{T}_1{T}_2{}^2}{3{\epsilon }_0{\hslash }^3}=$ 8.0 10⁻¹⁴ m²V⁻². By comparing ${\text{χ}}^{\left(3\right)}$ with the product of the two ${\text{χ}}^{\left(2\right)}$ nonlinear coefficients, ${\text{χ}}^{\left(2\right)DFG} {\text{χ}}^{\left(2\right)SFG}=$ 1.9 10⁻¹² m²V⁻², we estimate that the cascade of nonlinear processes can be a significant higher nonlinearity than the third order ${\text{χ}}^{\left(3\right)}$. However, a theoretical prediction on generated fields should take into account also the different overlap integral among the modes, the cavity losses at the different involved frequencies and the degeneracy of the process under consideration. A rigorous theoretical treatment of this problem goes beyond the scope of the present paper and will be reported in a future publication. While a quantitative theoretical prediction is beyond the scope of this work, it is clear from previous discussion that larger overlap integrals and lower losses will enhance the second order processes relative to the third order processes.

In our work we have designed a laser that favors the cascaded nonlinear process. This has been achieved by inserting the optical cavity of the QC laser in a microwave waveguide that reduces the losses of the microwave field generated by the beating between longitudinal modes and also increases its spatial overlap with the mid-infrared modes. In the microstrip-QCL microwave field propagates between two metal plates, along the dielectric stacks of the optical guide, enabling a good overlap of the microwave field with the active region. On the other hand the optical mode is confined in a typical buried heterostructure [3], preserving laser performances [20]. As reported in [20] by A. Calvar et al. the estimated microwave losses in the microstrip-QCL are a factor 1.5 lower than in the standard lasers and the overlap of the microwave field with the active region is 2.5 times higher.

In order to characterize the laser inter-mode stability we measure on a spectrum analyzer the electrical signal produced by the optical intensity of the laser incident on the QWIP detector [21]. In order to reduce optical feedback of the laser light in the cavity, affecting the stability of the system, different optical attenuators have been used for the two lasers. For a multi-mode Fabry-Perot cavity the laser electric field is $\text{E}={\displaystyle \sum }_k{\text{E}}_k(1+{\text{δ}}_{\text{k}}){\text{e}}^{\text{i}({\text{ω}}_{\text{k}}\text{t}+{\text{δ}}_{\text{ωk}}\text{t}+{\text{δ}}_{\text{φk}})}$ with ${\text{ω}}_k=\text{ω}+{\text{kω}}_{\text{RT}}$, where ω is the central laser frequency, ${\text{ω}}_{\text{RT}}$ is the cavity round-trip frequency and k is an integer. The frequency noise and the phase noise at ${\text{ω}}_k$ are respectively ${\text{δ}}_{\text{ωk}}$, ${\text{δ}}_{\text{φk}}$; the quantity ${\text{δ}}_{\text{k}}$ represents the noise on the field amplitude. The laser spectrum at ${\text{ω}}_{\text{RT}}$, the round trip frequency, is proportional to the Fourier transform of the time average of ${\text{E}}^2$ at same frequency ${\text{ω}}_{\text{RT}}$, ${\text{I}}_{{\text{ω}}_{\text{RT}}}={\displaystyle \sum }_k{\text{δ}}_{\text{k}}{\text{δ}}_{\text{k}+1}{\text{e}}^{\text{i}({\text{ω}}_{\text{RT}}\text{ t}+{\text{δ}}_{\text{φ}\left(\text{k}+1\right)}-{\text{δ}}_{\text{φk}}+{\text{δ}}_{\text{ω}\left(\text{k}+1\right)}\text{t}-{\text{δ}}_{\text{ωk}}\text{t})}$. From the expression of ${\text{I}}_{{\text{ω}}_{\text{RT}}}$ we can see that the width of the beat note spectrum decreases with the inter-mode frequency fluctuation (${\text{δ}}_{\text{ωk}}-{\text{δ}}_{\text{ω}(\text{k}+1)}$) reduction. The beat note spectra therefore give an insight on the relative stability among neighbor laser modes in a cavity [11, 22–24].

In Figs. 1(a) and 1(b) we compare measured beat note spectra for the microstrip-QCL and the QCL. Devices have been processed from the same epitaxial growth at the same time and they are of similar dimensions (3 mm x 8 µm). Spectra are reported for different operating bias and they are compared as a function of the mid-infrared output power (I_MIR) for the two devices. As reported in [20] by A. Calvar et al. the light current voltage characteristics of the two devices are similar. For both samples a single strong peak at the round-trip frequency of ~13.5 GHz is measured. The amplitude of the beat note peak of the standard laser decreases from −74 dBm to −76 dBm with increasing mid-infrared output power, while for the microstrip-QCL it increases from −96 dBm to −86 dBm. At the same time (See Fig. 1(c)) the width of beat note for the QCL increases from 650 kHz to 1200 kHz, similarly to what have been shown in [22] by A. Gordon et al.. On the contrary for the microstrip-QCL, in the same operating condition, the width of the beat note decreases from 200 kHz to 70 kHz.

 

Fig. 1 Measured beat note spectra for the QCL (a panel) and microstrip-QCL (b panel) for different optical output power (I_MIR). Span is 20 MHz (a) and 10 MHz (b). Resolution band width is 10 kHz. Presented spectra are results of average over 100 scans. QCLs operating temperature is 77K. Panel c: measured beat note full width half maximum as function of the optical power for the QCL (black dot) and microstrip-QCL (red triangles).

Download Full Size | PDF

3. Inter-mode stabilization via direct modulation at the round-trip frequency

Improved overlap factor between the optical and modulation modes together with reduced microwave propagating losses are favorable conditions to move toward the control of the inter-mode laser separation, which is an essential function for the development of dual comb spectroscopy [25]. Direct modulation is in fact a powerful tool to lock the laser modes separation to an external microwave source. With this method stable sidebands of the free running modes are created and by an appropriate design of the laser cavity they can mutually inject the adjacent longitudinal modes and lock them in phase [26].

In Fig. 2 we present the measured frequency response to direct modulation for the micro-strip-QCL and the QCL. The laser output is detected with an ultrafast QWIP with 65 GHz low pass cutoff and the signal at the modulation frequency is reported in Fig. 2. A flat frequency response is demonstrated for the microstrip-QCL up to ~14GHz; above this value measurements are limited by the injection circuit. For the QCL strong oscillations of the signal are measured in the low frequency range and the response starts to drop around 11.5 GHz. As expected from the improvement of the microwave figure of merit for the microstrip-QCL, defined as the ratio of the overlap of the modulation signal with active region over the losses, an enhancement of a factor ten of the modulated signal at 14 GHz is demonstrated.

 

Fig. 2 High frequency modulation frequency response measured with an ultrafast QWIP detector for the QCL (black line) and the microstrip-QCL (red line). Lasers are operating at 77 K.

Download Full Size | PDF

We will now focus on direct modulation at the round-trip frequency. The control of the laser mode separation by an external source is studied in the frame of the formalism of injection locking for coupled oscillators. Considering the optical case, injection locking model depicts a scheme where a stable master laser oscillating at a frequency ω₁ is injected in a slave laser oscillating at a free-running frequency ω₀, close to ω₁. The signal at ω₁, circulating in the cavity, can be regeneratively amplified by the gain, which is clamped for ω ₀ but not for ω₁. At a certain point the amplified intensity at ω ₁ begins to approach the value of the initial optical intensity at ω₀ and it can begin to saturate the laser gain to suppress laser oscillation at ω₀. In the case of direct modulation generated sidebands play the role of the stabilizing master laser. If the microwave intensity I_RF is injected in the cavity, sidebands of the optical mode of intensity I_SB = I_RF η_RF η_Mir are created where η_RF represents the modulation efficiency (η_RF takes into account the microwave losses in the optical cavity and the overlap of the modulation signal with the optical mode) and η_Mir is the laser slope efficiency.

The sideband frequency, separated from a laser mode by the injected RF frequency, F_rf, can be tuned by changing F_rf. In this way the externally applied modulation can control the laser mode separation and tune the separation over a given range defined as the frequency locking range (f_lock).

Within the picture of coupled oscillators and considering the sideband generated by the modulation as an effective injected control signal, the locking range can be written as [2, 26, 27]:

From Eq. (2) we see that the locking range, for a given injected modulation intensity, decreases with increasing mid-infrared power I_MIR. In Fig. 3 we present the beat note spectra measured for the standard laser when it is modulated close to the round-trip frequency, for I_RF = 100 mW and for two operating points (I_MIR = 11.5 mW, 21.5 mW). The beat note spectra are presented in color scale and the modulation frequency is reported on the y axis. Each horizontal scan corresponds to an average over 100 measurements with a resolution bandwidth of 10 kHz. Measurements have been performed by changing the injected frequency from lower to higher values.

 

Fig. 3 Measurements of the beat note spectra for the standard the QCL for different modulation frequency. The RF power is set is to 100 mW and the optical power is 11.5 mW in panel a and 21.5 mW in panel b. On the Y-axis is reported the modulation frequency and on the X-axis the frequency of the signal on the spectrum analyzer, renormalized with an offset at the round-trip frequency. In color scale is the intensity of the measured RF signal (dBm). Each single scan corresponds to an average over 100 measurements; the resolution bandwidth is 10 kHz.

Download Full Size | PDF

In Fig. 3(a) we see that as the driving frequency approaches the free running beat note, the spectra move toward this value until the intensity is mainly concentrated in a single narrow peak. Locking is reached and the spectra are controlled by the external injected signal. By further increasing the RF frequency the spectra come back to initial condition. The region where a single narrow peak is measured corresponds to the locking range. We want to point out that given the high modulation power of this measurement a strong pick-up noise component is present in the measurement at the injection frequency, hiding the laser contribution to the signal. Increasing the laser output power (Fig. 3(b)) the locking range decreases becoming smaller than the width of the beat note spectra.

In Fig. 4 we compare injection locking properties for the microstrip-QCL and the QCL. In this case the output power of the two lasers is the same (30 mW) and I_RF is only 10 mW for the microstrip-QCL while it is 100 mW for the QCL.

 

Fig. 4 Measurement of the beat note spectra for the microstrip-QCL (panel a) and the QCL (panel b) for different modulation frequency. The optical power is 30 mW for the two devices and the RF power is set is to 10 mW in panel a and to 100 mW in panel b. On the Y-axis is reported the modulation frequency and on the X-axis the frequency of the signal on the spectrum analyzer, renormalized with an offset at the round-trip frequency. In color scale is the intensity of the measured RF signal (dBm). Each single scan corresponds to an average over 100 measurements; the resolution bandwidth is 10 kHz.

Download Full Size | PDF

In Fig. 4(a) a clear locking range of more than 1 MHz is measured. Thanks to the reduced microwave power, the pick-up noise doesn’t affect the measurements and the strong narrow peak at the modulation frequency represents the signal coming from the laser. In the case of the QCL device no influence of the modulation on the beat note spectra has been observed.

We would like to mention that even if this result represents a relevant improvement with respect to the QCL, the locking range presented here it much smaller than what has been reported in [18] by C. Kolleck for double metal THz QCL where f_lock ~100 MHz has been measured. This is probably due to two main factors. The first one is the doping of the active region that for mid-infrared QCL is one order of magnitude higher than in THz QCL, causing higher losses in GHz range. The second one is the geometrical size of the microstrip that in the THz is of the order of 100 µm while in the mid- infrared it is only tens of µm wide. This imposes a very different overlap between the microwave and the optical fields.

In order to perform a systematic study on injection locking mechanism by direct modulation for the microstrip-QCL and the QCL, we repeat the same measurements presented in Figs. 3 and 4 for different optical power ${\text{I}}_{MIR}$ . Results are summarized in Fig. 5 where we report the injected microwave I_RF /$f_{\text{lock}}$ needed to measure a given locking range , called I_{RF Effective} as function of $\sqrt{\frac{{\text{I}}_{MIR}}{{\text{η}}_{MIR}}}$, called I_{MIR Effective}. As predicted by Eq. (3) a linear dependence is demonstrated. The slope of the curve represents the term $\frac{\pi {\tau }_{MIR}}{\sqrt{{\eta }_{RF }}}$ in Eq. (3); for the QCL the slope is 8.2 ± 0.4 µs and for the microstrip-QCL it is 0.65 ± 0.06 µs. Assuming comparable ${\tau }_{MIR}$ for the two devices, as demonstrated in [20] by A.Calvar et al., $\frac{\sqrt{{\eta }_{R{F}_{microstrip}-QCL}}}{\sqrt{{\eta }_{R{F}_{QCL }}}}=12\pm 1$. From this result we can conclude that for a given modulation power and laser slope efficiency the intensity of the sideband mode is 144 times higher in the microstrip-QCL than in the QCL.

 

Fig. 5 Graph of the measured IRF Effective ($\sqrt{{\text{I}}_{RF}}/f_{lock}$ where ${\text{I}}_{RF}$ is the injected microwave power needed to measure a given locking range ($f_{lock}$) as function of IMIR Effective ${\text{I}}_{MIR}/{\text{η}}_{MIR}$, where IMIR is the optical power and ${\text{η}}_{MIR}$the laser slope efficiency. Data are presented for the QCL (black dot) and microstrip-QCL (red triangles). In dash lines are the linear fits of the experimental data

Download Full Size | PDF

4. Conclusion

In this work we present the realization of a QCL integrated in a microstrip line. We study the stabilization of the inter-mode frequency separation and the direct modulation of this laser. Integrating the optical cavity with a microwave line, an enhancement of the second order nonlinearities is achieved. Nonlinear process acts as stabilization mechanism in the laser cavity for the entire dynamic range of the device. Reduced microwave losses and improved overlap of the modulation signal with the optical mode enable flat frequency response up to 14 GHz and injection locking of the microstrip-QCL with superior performances with respect to the QCL.