Global coupling of QCLs: inclusion of dynamics

A. Gavrielides; T. C. Newell

doi:10.1364/OE.28.007746

1. Introduction

For many mid-infrared quantum cascade lasers (QCLs) technological applications such as remote chemical sensing we require high output powers and very good beam quality. There are several ways to increase the power by scaling the individual devices to obtain higher powers but these methods lead to multi-mode operation and reduction of the brightness. The most preferred approach is to integrate several individual narrow-ridge lasers on a single chip and phase lock them by a specific coupling scheme. Phase locking in an in-phase mode will provide high power and excellent beam quality of the outcoupled coherent light. There are several method to achieve coupling of individual QCLs which have also been attempted successfully with other of types of lasers, such as using evanescent coupling [1], leaking wave coupling of the arrays [2], and Y-coupling [3] as an effective mutual global coupling of arrays, which have been attempted successfully with other type of lasers such as $CO_2$, diode laser, fiber lasers etc.

Talbot imaging refers to a lensless imaging in which an image of periodic array of objects is re-imaged after the propagation of the light of a specific distance that is related to the periodicity of the array. The Talbot effect is therefore a near field diffraction effect and it was first observed by Talbot [4] in 1836 in images of periodic gratings and x-ray crystallography. Subsequently Rayleigh [5] provided an analysis and explanation of the effect. This effect offers an interesting way to globally couple an array and it has been used successfully to couple an array of diode lasers grown in the same chip [6]. To affect the feedback/coupling a flat mirror was inserted at a fractional distance of the Talbot distance $Z_T=2 {d^2} / {\lambda }$, at which the pattern repeats in a specific manner. d is the center-to-center spacing of the QCLs and $\lambda$ the wavelength. The analysis in [6] is predicated on several assumptions two of which are of most importance: (i) interelement coupling arises through diffraction in the Talbot cavity only (i.e., the waveguides of the array are not evanescently coupled), and (ii) there is no residual reflection from the interface between the waveguide array and the diffraction region.

In the next few sections we would like to analyze the equations that may describe the lasing characteristics of the Talbot coupled QCLs system by first analyzing the small signal gain or threshold gain of the system in section II. Section III we will derive the system’s nonlinear equations as ODEs in time to understand the mode selection and the saturation characteristics of the coupled QCLs. In section IV we adopt the QCL global coupled model and show its characteristics in the small signal gain approximation. Full numerical results of three coupled QCLs in an Talbot integrated cavity are presented in Section V. In particular, the predictions of the supermodes and lasing characteristics are based on the parameters of the coupled system and are compared with specific experiments presented in the literature. Finally, in section VI we will present our conclusions.

One of the basic approaches to determine the characteristics of the coupled QCL system is to examine the threshold of the system and determine which of the supermodes is most likely to lase based on the criterion of having the lowest threshold. In general this may not be the case if the system is above threshold and in particular in the strongly saturated regime. In addition it is not possible to determine the dynamics of the system and the stability of the various supermodes unless we have a theory that can predict the development of the modes in time. Such a system of differential equations will be derived and analyzed for the Talbot coupling of the case of several lasers and indeed for any other possible global coupling geometry. The analysis is also very important from the point of view of knowing the efficiency of extraction which actually depends on the particular supermodes that are lasing and on their stability.

2. Small signal gain analysis

In this section we would like to point out that it is important to distinguish two cases of laser coupling and distinguish them based on two of its properties. Focusing on Fig. 1 we assume that the N lasers are independent from each other without the feedback of the external Talbot cavity as the reflectivity at either or both facets of the laser are finite and non zero. Indeed this case will violate the second assumption of [6]. The threshold gain $e^{g_0L}={r_0^{-2}}$ and is related to the reflectivity of the array at the two ends of each of the lasers where the reflectivity is equal to the interface of semiconductor material and and air and it is of the order of $r_0\sim 50\%$. Therefore, the lasers may be above threshold and operating in the saturated regime depending on the degree of pumping even without the feedback from the external Talbot cavity. This particular case is formulated by a model in which the lasers operate independently and are coupled by the perturbation of the external feedback. In this case a general model in which in addition to the external perturbation also includes possible delayed global feedback has been analyzed in [7]. In particular, for the case of Talbot coupling a model specific for QCLs was published in [8] and subsequently in [9,10]. A generic model that may include QCLs for the purposes of discussion for semiconductor lasers reads:

(1)$$\frac{dE_n}{dt}=(1+i\alpha)N_nE_n+\eta\sum_{m=1}^NR_{nm}E_m(t-\tau)$$

(2)$$\frac{dN_n}{dt}=P-N_n-(1+2N_n)|E_n|^2$$

Fig. 1. Schematic of the linear Talbot cavity for coupling of n independent lasers. (Fig. 1 in [6])

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where $\eta$ is the degree of coupling and $R_{nm}$ is the Talbot matrix obtained by determining the amount of radiation from field of $m^{th}$ laser is injected in the $n^{th}$ laser. Its form is specific and can be obtained by assuming various field distributions for the individual $m^{th}$ laser. It is clear that if the coupling is turned off, that is set $\eta =0$ then the lasers operate independently each having intensity $I_n=P$ and random phases relative to each other. Similarly at threshold with the Talbot coupling active linearizing Eqs. (1) and (2) we find that $N_n\sim P_{th}$:

(3)$$[\frac{\eta e^{i\omega\tau}}{(1+i\alpha)P_{th}-i\omega}\bar{R}+\bar{I}]\mathcal{E}=0$$

The threshold condition is given by

(4)$$P^\nu_{th}=\eta| \lambda_\nu|\qquad \omega_\nu=-\pi-\phi_\nu$$

where we assumed that $\omega$ in the denominator is small and also that for convenience $\alpha =0$. $\lambda$ are the eigenvalues of the Talbot coupling matrix R.

On the other hand if we examine the experimental realizations of Talbot coupling in most of the recent experiments [11,12] we find that the array of lasers one of its facets is AR coated. With no Talbot feedback, i.e. $\mathcal {R}=0$, the array will be below threshold and non lasing. Similarly and more importantly in [13–17] the Talbot cavity is an integral and monolithic part of the laser array. Indeed, there is no discontinuity in index of refraction between the laser array and the Talbot cavity and therefore no feedback into the lasers unless $\mathcal {R}\neq 0$. Therefore, we assume as a first condition for the model that for $\mathcal {R}=0$, then the fields $E_n=0$ for all n. As a second condition and requirement for the model is to satisfy the linearized threshold condition derived and used in [13–18] that have shown that is in good agreement with the extensive experiments reported in these references. The condition is:

(5)$$[r_0re^{i\sigma L}\mathcal{R}-\mathcal{I}]\mathcal{E}=1$$

where $\sigma L=\frac {n_g}{c}\omega -ig_{th}$, and the enhancement factor was set to zero for convenience but easily recovered when required. $r_0$ is the reflectivity of the laser facet and $r$ is the reflectivity of the outcoupled element. Using the eigenvalues of the Talbot reflectivity matrix $\mathcal {R}$ defined as $\lambda _\nu$ a complex number, we find for the threshold value and the shift of the frequency of a given supermode $\nu$:

(6)$$g^\nu_{th}L-\ln(\frac{1}{r_0r})=\ln(\frac{1}{|\lambda_\nu|})\qquad \frac{n_g}{c}\omega_\nu=-\phi_\nu.$$

It is clear that the two conditions, one derived from the usual models Eq. (4) and the condition derived using the usual linearized threshold conditions Eq. (6) do not agree at all and it appears that a new model is required.

3. Model formulation: QCLs

In order to obtain a description of the compound cavity we will use Fig. 2 as a generic geometry. In particular the ring geometry will allow us to integrate the PDE equations for the fields and for the inversion over z in the (z,t) plane [19]. We will exclude hole burning that may occur in certain cases but for the QCL case they will be neglected due the assumption of fast carrier diffusion.

Fig. 2. Abstraction of a ring compound cavity with n gain segment components. The location of certain planes are denoted by $z_1$ and $z_2$ and are related to $\zeta _n=\frac {z_n}{v}$. The red box indicates the location of the gain or frequency filter.

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We will start with the diode laser equations since they are simplest and as the inversion or carriers equations are straightforward to modify as are required for QCLs. In this case since all the gain sections are of equal length by the growth, processing and cleaving process, this assumption allows a simple treatment of the problem. Then at a later publications we may investigate unequal paths or length in the individual ridges in order to tackle the more difficult problem of QCLs detuning or in particular for fibers in which the lengths of the gain sections of the fibers are definitively unequal.

For QCLs we will use the simplest set of equations for amplifiers Eqs. (33) and (34) as written in Appendix A. After transformation of the coordinates (t,z) to ($\tau ,\zeta$) with $\tau =t-\frac {z}{v},\quad \zeta =\frac {z}{v}$, we find:

(7)$$\frac{\partial E_{n}}{\partial\zeta}=\frac{v}{2L}(1+i\alpha)Z_nE_{n}$$

(8)$$\gamma_e^{-1}\frac{\partial Z_n}{\partial\tau}=P-Z_n-Z_n|E_{n}|^2$$

The subscript of the field and the inversion $n$ denotes the particular QCL gain segment as indicated in Fig. 2. We will set the origin of the round trip calculation at $z_1=0$ or more appropriate at $\zeta _1=0$. The gain segments are located at $\zeta _2^+\leq \zeta \leq \zeta _1^-$ the $\pm$ indicate that the fields are the individual QCLs $E_n$ associated with the gain segments. On the other hand $\zeta _2^-$ and $\zeta _1^+$ are associated with the Talbot fields.

Eq. (7) then can be integrated in this interval of $\zeta$ to obtain for the fields

(9)$$E_n(\tau,\zeta_1^-)=r_0e^{\frac{(1+i\alpha)}{2}G_n}E_n(\tau,\zeta_2^+)$$

where $r_0$ the reflectivity of the end facet of the QCL segments and

(10)$$G_n(\tau)=\frac{v}{L}\int_{\zeta_2}^{\zeta_1}Z_n(\tau,\zeta)d\zeta.$$

To obtain the inversion equation we integrate Eq. (8) over $\zeta$ and apply the definition of Eq. (10) to derive

(11)$$\gamma_e^{-1}\frac{d G_n}{d\tau}=P-G_n-[I_{n}(\tau,\zeta_1^-)-I_{n}(\tau,\zeta_2^+)]$$

where the intensities are the square of the fields at the two locations. In order to complete a round trip of the ring we now must integrate the sourceless field equation for the section $\zeta _2^-\geq \zeta \geq \zeta _1^+$ to connect the fields at $\zeta _1$ with the fields at the same location after one roundtrip.

Beginning with the fields of the individual segments at $\zeta _1^-$ we now obtain the fields in the Talbot cavity from

(12)$$E_{n}^T(\tau,\zeta_1^+)=\sum_{m=1}^Nv_{nm}E_{m}(\tau,\zeta_1^-)$$

where $v_{nm}$ are the expansion coefficients as can be calculated either by expansion into free space modes or cavity modes for the section of the Talbot cavity [6,14]. The superscript on the field indicates a Talbot cavity mode.

For the purposes of calculating the Talbot reflection we assume that the fields have an aperture function parameterization, that is the total field in the gain sections is defined as in [13,14]:

\mathcal{E}_n(\tau,\xi,x)=E_n(\tau,\xi)U_n(x)\qquad U_n(x)=\sqrt{\frac{2}{a}}\cos(\pi\frac{x-nd}{a})\qquad nd-\frac{a}{2}\leq x\leq nd+\frac{a}{2}

with n=1,2$\cdots N$ the number of gain segments in the array and $d$ the period of the array and $a$ the width of the gain segment, while the Talbot cavity modes are defined

\mathcal{E}_m^T(\tau,\xi,x)=E_n(\tau,\xi)\Psi_m(x)\qquad \Psi_m(x)=\sqrt{\frac{2}{W}}\sin(m\pi\frac{x}{W})\qquad 0\leq x\leq W

where W is the width of the Talbot cavity and m=1,2$\cdots N_T$, the number of Talbot cavity modes. The expansion coefficients $u_{nm}$ in Eq. (12) are the overlap integral of the two transverse functions. To this end we integrate

(13)$$\frac{\partial E_{n}}{\partial\zeta}=0$$

Propagation to $\zeta _2^-$ after reflection at an possible outcoupler located halfway in the Talbot cavity gives

(14)$$E_{n}^T(\tau,\zeta_2^-)=rE_{n}^T(\tau,\zeta_1^+)$$

while the phase propagation through the cavity will be included in the Talbot reflection coefficients. Finally we project the Talbot modes into the gain ridge lasers modes in order to propagate the laser modes in the individual gain sections.

(15)$$E_{k}(\tau,\zeta_2^+)=r\sum_{n=1}^NR_{km}E_{m}(\tau,\zeta_1^+)$$

where

(16)$$R_{km}=\sum_{n=1}^{N_T}u_{nk}v_{nm}e^{i2\beta_nL_T}$$

where $\beta _n$ is the propagation wave vector of the Talbot modes and $2L_T$ is the round trip length of the Talbot cavity. We use Eq. (9) and (15) to complete a round trip in the ring to obtain:

(17)$$E_k(\tau,\zeta_1^-+S)=rr_0e^{\frac{(1+i\alpha)}{2}G_n}\sum_{m=1}^NE_m(\tau,\zeta_1^+)$$

where $S=2\frac {L+L_T}{v}$. However, we would like also to include the filter located between $\zeta _1^-$ and $\zeta _1^+$ that represents the bandwidth of the gain line shape. In frequency space we would use a Lorentzian as

(18)$$E_k(\omega,\zeta_1^-+S)=\mathcal{L}(\omega)E_k(\omega,\zeta_1^+)$$

We will select from the fields at the various locations the field $E_n(\tau ,\zeta _1^+)\equiv A_n(\tau )$ as the fundamental field that represents the dynamics of the resonator. Then we use the boundary condition $E_n(\tau ,\zeta _1^-+S)=E_n(\tau +S,\zeta _1^-)=A_n(\tau +S)$ to relate $A(\tau )$ to its the advanced field. On the other hand in the time domain Eq. (18) as it express Eq. (17) with a gain filter is given by the convolution:

(19)$$A_k(\tau+S)=\int_{-\infty}^\tau L(\tau-x)P_k(x)dx$$

where $\mathcal {L}(\tau )=L(\tau )u(\tau )$ where $u(\tau )$ is the step function defined as $u(x)=0$ for $x\leq 0$ and $u(x)=1$ for $x\geq 0$. The Lorentzian in time is given by

(20)$$L(\tau)=\gamma e^{(i\Omega-\gamma)\tau}$$

where $\gamma$ and $\Omega$ is the bandwidth and the center frequency of the gain lineshape. The function $P_k(\tau )$ is the function on the RHS of Eq. (17). The Fourier transform of Eq. (19) gives:

(21)$$[\gamma-i(\Omega-\omega)]A_k(\omega)=\gamma e^{-i\omega S}\mathcal{F}\{P_k(\tau)\}|_\omega$$

Taking the inverse Fourier transform of Eq. (21) we obtain the field equation that describes the compound cavity of QCLs

(22)$$\gamma^{-1}\frac{dA_k}{d\tau}=-A_k+rr_0e^{\frac{(1+i\alpha)}{2}G_k(\tau-S)}e^{-i\Omega S}\sum_{m=1}^NR_{km}A_m(\tau-S)$$

The factor $e^{-i\Omega S}$ arises from the phase shift associated with the center line of the gain lineshape and it has the same meaning as it appears in the Lang-Kobayashi laser model with external delay. For our convenience we can set the center line frequency as the origin of the frequency and will neglect it on future treatments. The equation that now determines the inversion is obtained by substituting Eq. (15) in Eq. (11) to close the system of equations:

(23)$$\gamma_e^{-1}\frac{dG_k}{d\tau}=P-G_k(\tau)-(r_0^2e^{G_k(\tau)}-1)r^2|\sum_{m=1}^NR_{km}A_m(\tau)|^2$$

As a final transformation to eliminate the dimensional time by defining $s=\gamma \tau$ and $T=\frac {\gamma }{\gamma _e}$. Similarly the delay is scaled by $\theta =\gamma S$. We also rescale all the fields by $r$ that appears in the inversion to eliminate it since it is superfluous. However, its appearance in the field equation is very important since its value determine the number of frequencies lines under the gain curve as well as the threshold gain, as we will see in the later sections. Then we have:

(24)$$\frac{dA_k}{ds}=-A_k+rr_0e^{\frac{(1+i\alpha)}{2}G_k(s-\theta)}\sum_{m=1}^NR_{km}A_m(s-\theta)$$

(25)$$T\frac{dG_k}{ds}=P-G_k(s)-(r_0^2e^{G_k(s)}-1)|\sum_{m=1}^NR_{km}A_m(s)|^2$$

These equations will be analyzed and numerically integrated in section 5 to determine the steady states (supermodes) and their stability.

4. Model

In order to construct a useful model for the averaged field and inversion in the cavity we assume that the formation of hole burning is weak and the diffusion of the carriers over the spatial wavelength of the field are washed out. Under this assumption we can neglect the interaction of the forward and backward fields and assume that the lasers are acting as unidirectional ring lasers. Naturally the loading of the gain by the two field is not included as should be included in the two directional case but we may assume that in the strong saturating regime the effect will be negligible. Here we will adopt Eqs. (24) and (25) for the case of QCLs grown on the same chip and having equal ridge lengths. The steady state equations with $E_n=A_ne^{i\omega s}$ where $A_n$=complex, are:

(26)$$i\omega E_n=-A_n+rr_0e^{\frac{(1+i\alpha)}{2}G_n}e^{i\omega\theta}\sum_{m=1}^NR_{nm}A_m$$

(27)$$|\sum_{m=1}^NR_{nm}A_m|^2=\frac{P-G_n}{r_0^2e^{G_n}-1}$$

In the case of threshold $A_n\rightarrow 0$ we have $P_{th}=G_n$ where $P_n$ indicates a threshold associated with a supermode. Equation (28) then becomes:

(28)$$i\omega A_n=-A_n+rr_0e^{\frac{1}{2}(P_{th}+i(\alpha P_{th}-\omega\theta))}\sum_{m=1}^NR_{nm}A_m=0$$

Defining the vector $\mathcal {E}=A_n$ and $\sigma =\frac {1}{2}(-iP_{th}+(\alpha P_{th}-\omega \theta ))$ we can recast Eq. (28) as a matrix equation

(29)$$(rr_0\frac{e^{i\sigma}}{i\omega+1}\bar{R}-\bar{I})\mathcal{E}=0.$$

In the limit of $\omega \rightarrow 0$ and with $\alpha =0$, it is the identical equation derived in [13] and the same form as Eq. (6). It is an eigenvalue equation for the eigenvalues of $\sigma$ with eigenvectors $\mathcal {E}$ of the supermodes. Also for our simulations later we can identify the threshold for our problem as $P_{th}=2[\ln (\frac {1}{rr_0})+\ln (\frac {1}{|\lambda _\nu |})]$, including the factor of 2 as it appear in the definition of $P_{th}$ in the line above Eq. (29).

5. QCL numerical results

We will concentrate and use Eqs. (24) and (25) as the representative equations to determine the dynamics of Talbot coupled QCLs. As a useful first look we will examine and analyze the experimental results offered in [20]. A list of all of the most important parameters that are needed to determine the constants appearing in Eqs. (24) and (25) are in Table 1. The Talbot distance for the system of lasers is given by $Z_T$=$2d^2n_g$/ $\lambda$ with $\lambda$ the free space wavelength of the QCLs, and it is determined to be $Z_T=842\mu m$.

Table 1. Compound laser parameters.

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It is also important from the integrated laser structure to locate the three lasers symmetrically over the width of the Talbot cavity by defining the centering variable $c$=$(W-(N+1)d)$/2. Finally the delay that appears in the dynamic equations is obtained from

(30)$$\theta=\gamma\frac{2(L+L_T)}{c}n_g$$

Based on the values on Table 1 we find $\theta =2.2$ for a Talbot cavity length $L_T=105.3\mu m$ corresponding to $Z_T/8$. Since $L\gg L_T$ we expect that the delay will not effect the selection of the supermode but rather effect only the dynamics. The rest of the parameters required in the solution of the dynamical equations are $T=0.1$ and $P\geq P_{th}$ and we will for the purpose of this early discussion take the linewidth enhancement factor $\alpha =0$.

Following the discussion of the previous section we find that for $L_T=105.3\mu m$ the eigenvalues of $R_{nm}$ are

(31)$$|\lambda^{th}_\nu|= \left[{\begin{array}{c} 0.657\\ 0.54\\ 0.44 \end{array}}\right]$$

which eigenvalues correspond to:

(32)$$\mathcal{E}^{th}_\nu= \left[{\begin{array}{ccc} 0.507 & 0.707 & -0.5\\ 0.705 & 0.0 & 0.71\\ 0.507 & -0.707 & -0.5 \end{array}}\right]$$

where the first column is the symmetric mode with the largest eigenvalue, the second the mixed mode and the third the antisymmetric mode. Based on the eigenvalue of the symmetric mode we find that the threshold gain $P_{th}=3.39$ and is the smallest of the other two supermodes. Therefore we would expect that the symmetric mode will be the dominant mode for a Talbot cavity length $L_T=Z_T/8$. The imaginary parts of the eigenvalues and of the eigenvectors are of $O(10^{-3})$ and have not been included for transparency.

From the numerical calculations we indeed find that Fig. 3(a) shows the far field of the dominant mode which is symmetric for a Talbot cavity length of $Z_T/8$. The amplitudes are almost equal $[0.505\quad 0.617\quad 0.505]$ and constant in time. The frequencies are locked to $\omega =-0.388$ and all the phases are equal. In particular we verified numerically that the threshold of the laser is given by $P_{th}=3.39$ as was computed analytically above. Similarly in Fig. 3(b) the numerical results confirm that at a Talbot cavity length of $L_T=Z_T/4$ the antisymmetric mode is the dominant mode. The amplitudes are of the same order [0.641 -0.761 0.641] and constant in time as in the symmetric case. The frequency shift is $\omega =-0.524$ at which all modes are locked. We have performed several other computations by exploring the dependence on the pumping constant P. For larger values of $4\leq P\leq 20$ we find that there is no change to the solutions that we presented so far either for the symmetric or the antisymmetric case.

Fig. 3. (a) Far field of the symmetric mode for $L_T=Z_T/8$, and (b) antisymmetric mode for $L_T=Z_T/4$. The other parameters are: P=8.0 and $\alpha =0$ for both cases.

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Another avenue of numerical exploration is the enhancement parameter $\alpha$ effects on the supermodes if it is non zero. We expect that QCLs have the enhancement parameter in the range of 0-2 and it is of interest in the dynamics of the laser. For $\alpha =1$ we find that there is no change to the dynamics of the laser for the symmetric mode at $L_T=Z_T/8$ and P=4. However, as the parameter P is increased we find that the laser dynamics bifurcate into an oscillatory behavior. In Fig. 4(a) we show a time series of the amplitude $B_n$ and the phase difference of the symmetric mode at $P=8$. The field is express as $A_n=B_ne^{{\phi }_n}$ as usual. The amplitudes of the supermode oscillates at a slow frequency with a period of $T=150$ for the lasers 1 and 3 while the amplitude 2 has a period of twice that i.e $T=300$. The similar situation is exhibited by the phase difference $\phi _n(s)-\phi _n(s-\theta )$. The effect of this oscillation can be seen in the far field pattern of the laser Fig. 4(b). The peak intensity has been reduced by about $30\%$ while a Gaussian noise background can be easily discerned. Such effects were indeed clearly seen in the experiments of [20] but more studies are necessary to identify them and explain them fully. For the antisymmetric mode that is dominant for a cavity length of $L_T=Z_T/4$ we find no bifurcations into an oscillatory state for $\alpha =1$ and for $P=4-16$.

Fig. 4. (a) Time series of field amplitude $B_n(s)$ and phase difference $\phi _n(s)-\phi _n(s-\theta )$ of the symmetric mode for $L_T=Z_T/8$, and (b) average far field of the symmetric mode for $L_T=Z_T/8$. The color order of the lines are $B_1$=(red), $B_2$=(blue) and $B_3$=(green) and similarly for the phase differences. The other parameters are: P=8.0 and $\alpha =1$.

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Finally we have found a bifurcation for $P=8$, $\alpha =2$ and Talbot cavity length equal to $L_T=Z_T/4$ that supports the antisymmetric supermode. In Fig. 5(a) is the time series of the three amplitudes of the antisymmetric mode all locked together at the optical frequency and also pulsating with a period of 2.2 that corresponds to the delay $\theta$. Similarly the phases are locked simultaneously to the optical frequency and oscillating or pulsating at the delay period. The far field of the antisymmetric mode is not effected by the oscillation since all fields are locked and it is identical to that shown in Fig. 3(b). More detailed calculations and bifurcation diagrams as a function of P and $L_T$ for various realistic values of the enhancement factor are required.

Fig. 5. (a) Time series of field amplitude $B_n(s)$ and (b) phase difference $\phi (s)_n-\phi _n(s-\theta )$ of the antisymmetric mode for $L_T=Z_T/4$. The color order of the lines are $B_1$=(red), $B_2$=(blue) and $B_3$=(green) and similarly for the phase differences. The other parameters are: P=8.0 and $\alpha =2$.

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6. Conclusions

We have constructed a physical simple laser model for an Talbot integrated cavity that couples N-sections of QCLs ridges assuming that the ridges have equal lengths. The model reduces in the small signal gain approximation and matches the calculation of the threshold gains of the possible supermodes available by the N-ridge structure. We concentrated to the simplest of several experimental publications [13–17] to the three QCL integrated Talbot cavity of [20]. The full model computations for the two cases of the Talbot cavity length of $L_T=Z_T/8$ and for $L_T=Z_T/4$ produces a stable steady state symmetric supermode and a stable steady state antisymmetric supermode respectively over a large range of pumping P above threshold and with $a=0$ in agreement with the experiments. In addition we examined a restricted range of values of enhancement factor $a\neq 0$ and discovered a time pulsating regime bifurcation unto an unlocked regime for the case of the symmetric supermode. The average far field pattern of the supermode retains its symmetric nature, however it appears that it now sits on an noise understructure. Similarly for the case of the antisymmetric mode there exits a regime of pulsating intensity and phase while the modes remain locked. For completeness we would like to include the full set of QCLs equations for an integrated cavity taking care of the carrier dynamics properly. They are listed in Appendix B.

There several other schemes that use modified array structures that radiate and couple through Talbot as described in [21] and in [16] in order to suppress antisymmetric or higher order modes. The model proposed in this paper will be fairly usful to also analyze such systems as long as the reflectivity of the modified Talbot structures is calculated properly. This is a interesting avenue for further investigations. For completeness we would like to include here the full QCLs equations for an integrated Talbot cavity taking care of the carrier dynamics correctly. We use the set of equations in Appendix A,

Appendix A

For QCLs the equations that we use are the simplest diode amplifier equations:

(33)$$\frac{1}{v}\frac{\partial E_s}{\partial t}+\frac{\partial E_s}{\partial z}=\frac{1}{2L}(1+i\alpha)ZE_s$$

(34)$$\gamma_e^{-1}\frac{\partial Z}{\partial t}=P-Z-Z|E_s|^2$$

where here $\gamma _e$ is the carrier’s decay rate and the pumping is $P=\frac {G_NL}{v_g}(\frac {J}{q\gamma _e}-N_0)$ and the field is normalized to as $\sqrt {\frac {G_N}{\gamma _e}}E\rightarrow E$. The quantity $G_N$ is given by $G_N=N_p\Gamma v\sigma _{32}$ where $\sigma _{32}$ is the emission cross-section of the transition and $N_p$ the number of stages.

A more expanded set of equation that include all three levels are the prototype equation in [22]. We will display them here as they apply for a section of an amplifier.

(35)$$\frac{1}{v}\frac{\partial E}{\partial t}+\frac{\partial E}{\partial z}=\frac{1}{2}(1+i\alpha)N_pg(N_3-N_2)E$$

(36)$$\frac{\partial N_3}{\partial t}=\frac{I_{in}}{q}-\frac{N_3}{\tau_{32}}-\frac{N_3}{\tau_{31}}-g(N_3-N_2)|E|^2$$

(37)$$\frac{\partial N_2}{\partial t}=\frac{N_3}{\tau_{32}}-\frac{N_2}{\tau_{21}}+g(N_3-N_2)|E|^2$$

We define $Z=N_pgL(N_3-N_2)$ and $V=N_pgLN_2(\frac {\tau _{32}}{\tau _{21}}-1)$. in addition we rescale the field as $\sqrt {g\tau _{32}}E\rightarrow$E. The final equations are:

(38)$$\frac{1}{v}\frac{\partial E}{\partial t}+\frac{\partial E}{\partial z}=\frac{1}{2L}(1+i\alpha)ZE$$

(39)$$\frac{\partial Z}{\partial t}=s_1[P-\gamma_4Z-2\gamma_3V-2Z|E|^2]$$

(40)$$\frac{\partial V}{\partial t}=s_2[Z-V-Z|E|^2]$$

where $s_1=\frac {1}{\tau _{32}}$ and $s_2=\frac {1}{\tau _{32}}(\frac {\tau _{32}}{\tau _{21}}-1)$ the unnormalized decay rates of $Z$ and $V$, and the normalized rates $\gamma _3=(\frac {\tau _{32}}{\tau _{21}}-1-\frac {\tau _{32}}{\tau _{31}})(\frac {\tau _{32}}{\tau _{31}}-1)^{-1}$, and $\gamma _4=(2+\frac {\tau _{32}}{\tau _{31}})$ as defined also in [22]. Also here for the amplifier $P=N_pgL\tau _{32}\frac {I_{in}}{q}$.

Appendix B

To include the proper carrier dynamics of QCLs from [22] to an array coupled by a Talbot cavity we use the set of equations in Appendix A, Eqs. (38)–(40) and following the procedure outlined in Section 3, we obtain:

(41)$$\frac{dA_k}{ds}=-A_k+rr_0e^{\frac{(1+i\alpha)}{2}G_k(s-\theta)}\sum_{m=1}^NR_{km}A_m(s-\theta)$$

(42)$$T_1\frac{dG_k}{ds}=P-\gamma_4G_k-2\gamma_3K_k-2(r_0^2e^{G_k(s)}-1)|\sum_{m=1}^NR_{km}A_m(s)|^2$$

(43)$$T_2\frac{dK_k}{ds}=G_k-K_k+(r_0^2e^{G_k(s)}-1)|\sum_{m=1}^NR_{km}A_m(s)|^2$$

where $T_1=\gamma /s_1$, and $T_2=\gamma /s_2$, where $\gamma$ is the band width of the spectral gain of the QCLs.

Disclosures

The authors declare no conflicts of interest.

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Number of lasers N	3
Length of lasers L	2 mm
Index of refraction $n_{g}$	3.1
Ridge width a	12 $μ m$
Period d	25 $μ m$
Wavelength	4.6 $μ m$
Gain bandwidth $γ^{- 1}$	50 GHz
Carrier’s decay rate $γ_{e}$	2 ps
Reflectivity $r_{0}$	0.51
Talbot cavity width W	120 $μ m$

Global coupling of QCLs: inclusion of dynamics

Abstract

1. Introduction

2. Small signal gain analysis

3. Model formulation: QCLs

4. Model

5. QCL numerical results

6. Conclusions

Appendix A

Appendix B

Disclosures

References

Cited By

Figures (5)

Tables (1)

Equations (45)

Optics Express