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Abstract
We introduce a one-dimensional analytical model to determine the steady-state thermal behavior of a sampled grating distributed Bragg reflector laser coupled to a thermoelectric element cooler. The governing equations are a system of ordinary differential equations that can be solved analytically. This method can easily be used when the laser is integrated with more functional elements, e. g., semiconductor optical amplifiers and amplitude modulators. The results are verified by the numerical finite element method and experimental data reported by other researchers. A comparison between the results of the analytical and numerical methods reveals the high consistency between them.
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1. Introduction
Widely tunable semiconductor lasers are important parts of high-capacity coherent optical telecommunication. These lasers are the key elements in wavelength division multiplexing (WDM) [1], optical packet switching (OPS) [2], and wavelength routing in access networks [3]. The miniature structure and high performance of these lasers (e. g., narrow linewidth and high side mode suppression ratio (SMSR) [4]) make them attractive choices in the communication field. Furthermore, these lasers can be flexibly integrated with some functional elements such as electro-absorption modulators (EAM) [5] and semiconductor optical amplifiers (SOA) [6] on-chip.
One of the most successful wide-tunable lasers is sampled grating distributed Bragg reflector (SG-DBR) [7], which can cover wavelength ranges over some ten nanometers quasi-continuously by using Vernier effect [8]. SG-DBR lasers are composed of four sections: two sampled distributed Bragg reflectors as the front and rear wavelength-dependent mirrors, an active section that provides optical gain, and a phase section that helps to laser fine-tuning. The phase, wavelength, and output power can be manipulated separately by injecting individual currents in these sections.
However, a primary concern of these lasers is the temperature rise due to the electric currents. This temperature arising produces mode-hoping leading to cross-talking and lowering the useful life of the laser. One of the most common methods of managing temperature in semiconductor lasers is a thermoelectric element cooler (TEC), which works based on the Peltier effect [9]. Since these small solid-state coolers do not have any moving parts, they do not make any noise and are typically stable. They can pump heat with high precision and are cost-efficient, miniaturized, and simply pluggable into any laser module. These features make them very promising especially in situations where regular refrigerators can not be used, e.g., electronic cooling.
TECs consist of a number of thermocouples (n-and p-type semiconductors) connected thermally in parallel and electrically in series to transport heat from one side to another. Thermoelements are sandwiched between two ceramics with high thermal conductivity. TECs have recently attracted considerable attention, e.g., [9,10], basic information about thermoelectric coolers and generators (TEGs) is provided. In this respect, physical insight [11–13] can also be helpful. Bell has provided proven and potential applications of the TEC and TEG [14]. Some researchers have presented an analytical approach for the TEC and TEG’s steady-state and transient thermal behavior [15–18]. In [17], the temperature distribution of the TEG has been compared considering different boundary conditions. Some researchers have used numerical calculations such as the finite element method (FEM) using COMSOL multiphysics and ANSYS softwares [19,20], coupled-mode theory [21], and computational fluid dynamics (CFD) [22].
Additionally, several considerable articles have been published on the optimization of the TEC [23,24], advanced thermoelectric material [25], flexible TECs [26], and nanostructure TECs [27]. All of these researches confirm the importance of thermoelectric devices.
The present study aims to investigate the thermal behavior of an SG-DBR laser combined with a TEC in a laser module. Temperature rise is among the most important issues in complex-structure semiconductor lasers. Temperature drastically affects the emission wavelength. However temperature control is rather complicated due to some control injection currents and the simultaneous effect of currents on several parameters of the laser (e.g., refractive index and cavity length). Hence, these lasers need to be precisely stabilized [28].
In this paper, we provide an analytical model to investigate the steady-state thermal behavior of an SG-DBR laser when coupled with a TEC. The proposed mathematical analysis is relatively precise and leads to closed-form relations, which is important in the integrated tunable semiconductor lasers industry. Furthermore, the model can easily extend to the situations in which different functional parts are integrated with the laser (e. g., SOA and EAM). To the best of our knowledge, most studies have typically examined the thermal behavior of the TEC-SG-DBR laser module numerically [29] or under the assumption that the cold temperature or injection heat flux into the TEC is constant.
The remainder of this paper is organized as follows: in Section 2, we present the theoretical analysis of the steady-state solution of the SG-DBR laser-TEC system. We calculate the temperature distribution of the laser and TEC in the optimum situation. Section 3 is devoted to the investigation of the analytical results. Finally, the conclusions are summarized in Section 4.
2. Theoretical model
As shown in Figs. (1) and (2), the structure of the laser module consists of two main parts: laser and TEC. The laser structure is made of a laser waveguide and chip. The Laser waveguide has at least four sections: front and rear distributed Brag reflectors, active and phase sections ( all are driven by different electric currents). Other functional sections (e. g., SOA and EAM) can be integrated too. Each section contains three layers: p-doped, n-doped and intrinsic. The laser chip is only heated under the avoidable leakage electric currents from the waveguide, but submounts are heated or cooled by the heat transfer process. Due to the laser chip’s distributed heating, a one dimensional distributed heat conduction model for the laser chip is presented in this paper. TEC operates according to the Peltier effect. By applying electric current to the TEC, heat can transfer from one ceramic into another in a direction that depends on the polarity of the thermo-element. It transfers in the same direction as the electric current in the p-type, and in the opposite direction in the n-type element.
Fig. 1. Schematic structure of the SG-DBR laser module; in this module, an SG-DBR laser and a TEC module are packaged. SG-DBR: sampled grating distributed Bragg reflector; TEC: thermoelectric element cooler.
Fig. 2. Schematic structure of an SG-DBR tunable laser; each section has a separate injection electric current. For simplicity, we assume the structure has three layers: intrinsic, n-type and p-type layers. SG-DBR: sampled grating distributed Bragg reflector.
The boundary condition is the heat conduction between semiconductor laser and TEC elements. We reduce this problem to a two-point boundary value problem assuming that the hot side temperature of the TEC is known and the cold side is unknown. TEC distributed heat transfer analysis of the two-point value boundary is based on the analytical approach proposed by Huang et al. [15] and Lam et al. [16]. We applied the heat conduction lumped model for coupling the distributed heat model for combined laser and TEC.
In the following, we first investigate the steady-state solution of the heat transfer equation in an SG-DBR laser without the TEC and submounts, followed by presenting a theoretical model for heat transfer in SG-DBR laser-TEC combined system. We first decouple the coupled equations into uncoupled ones which already know the solutions. Then, we solve the complicated SG-DBR laser-TEC combined system thermal behavior by replacing existing parameters with new ones. The new parameters show the effect of coupling between two systems.
2.1 Heat conduction in SG-DBR laser
To find the temperature distribution in an SG-DBR laser, we start with the conduction heat equation:
where $k_l$ is the thermal conductivity and $\alpha _t= {\textstyle k_l}/{\textstyle \rho _l c_l}$ is the thermal diffusivity. In this paper, both parameters are assumed to be constant. $Q$ is the heat generation rate per unit volume. $\rho _l$ is the density, and $c_l$ is the specific heat. Currents injected in a narrow bridge (Fig. (2), generate the volume energy in every layer and section as follows [30]: where $V_{j-a}$ and $V_{j-p}$ are active and passive junction voltages; $J$ is the electric current density; $J_{th}$ is the threshold electric current density; $\eta _i$ and $\eta _{sp}$ are the internal quantum efficiency of stimulated and spontaneous emission, respectively; $f_{sp}$ is the ratio of spontaneous emission photons left the active section, to the all produced spontaneous photons; $d_A$ is the thickness of the active layer; $\rho _{e,i}$ is the electric resistivity. The values of parameters are: $V_{j-a}=0.83$ V, $V_{j-p}=0.95$ V, $\eta _{sp}=0.4$, $f_{sp}=0.59$ and $\eta _i=0.8$, when $I_{th}=25$ mA. The other parameters are listed in Tables 1 and 2. In our model, the average is taken over the laser waveguide and chip cross-section ($x-z$) with a plane perpendicular to the laser propagation axis $y$. By employing the two-dimensional divergence theorem, Eq. (1) in terms of the average temperature $\bar {T}_l(y,t)$ over the $x-z$ cross-section can be written as:In Eq. (6) the term $m^2( \bar {T}-T_a)$ expresses the heat losses through the side surface via heat convection. $m^2=\sum _{i=1}^4 h_il_i /(kS_{xz})$ where $l_i$ is referred to each side of the $x-z$ cross-section of the laser and $h_i$ is the convection coefficient of each lateral boundary of the laser [31–33]. Suppose $m^2$ is constant with time and coordinate.
By introducing $\theta _l=\bar {T}- T_{a}$, the temperature relative to the ambient temperature, the heat conduction equation, Eq. (5) is rewritten as follows:
We call Eq. (7) the reduced heat conduction equation, similar to the heat conduction in the extended surfaces problems [33].
2.2 Steady-state solution of reduced heat conduction equation
Suppose that the laser consists of $n$ different sections (e. g., front and rear mirrors, active and phase sections) and some functional elements (e. g., SOP and EAM). Therefore, the steady-state reduced heat equation, $\partial \theta _{l}/{\partial t}=0$ is written as:
For simplicity, let the injection electric current ($I_i$, $i=\{f,a,p,r\}$) in each section be constant, where $f$, $a$, $p$ and $r$ referred to front, active, phase and rear, respectively; i.e., $\bar {Q}_{xz}(y)$ is piecewise constant along the laser structure. For the $i$-th region, Eq. (8) can be solved as follows:
As shown in Fig. (2), $y_i$ and $y_{i+1}$ are the first and end coordinates of the $i$-th section. where $\bar {Q}_{ixz}$ is the average heat generation rate over $x-z$ plane, because of injected currents in $i$-th section. The boundary condition equations at the interface $y_i$ are stated as:
Due to the heat conduction between neighborhood sections, $A_i$ and $B_i$ depend on all injected currents into laser sections. The steady-state temperature distribution in the $i$-th section can be obtained by Eq. (9).
The position of the local maximum or minimum temperature ($y_{mi}$) in the $i$-th region is the root of the derivative of Eq. (9) versus $y_i$:
For positive $A_i/B_i$, the root $y_{mi}$ is real. Here, $y_{mi}$ belongs to the interval $[y_i,y_{i-1}]$ when $A_i/B_i \in [e^{my_{i-1}},e^{my_i}]$; otherwise, the corresponding maximum or minimum can not be found in the temperature distribution. As an example, if $Q_i$ is less than the energy injection to its neighbor sections ($Q_{i-1}$ and $Q_{i+1}$), and $(y_{i+1}-y_i)$ is large enough such that the heat transfer from the i-th section to the ambient is greater than the heat transfer from the adjacent sections to the i-th section, thus, the i-th section can be cooled. Accordingly, the minimum temperature will be observed in the $[y_i,y_{i+1}]$ interval; otherwise, the temperature remains increasing on the $[y_i,y_{i+1}]$ interval. In the absence of TEC for industrial laser parameters, no minimum can be found in the temperature profile along the laser structure. Our numerical calculations verify analytical results.
The cooling process of the laser by TEC is investigated via coupling between laser and TEC heat equations. The heat transfer from the element $dy$ of the laser system to the TEC by using a thermal resistance circuit is $dq_l=dG(\bar {T}_l(y)-T_c)$, where $\bar {T}_l(y)$ as mentioned before is the average temperature of the laser structure over the $x-z$ cross-section plane and $T_c$ is the cold temperature of the TEC. Thermal conductance is $dG= l_xdy/ \sum _{i}( d_i/ k_i)$ where $k_{i}$, is the thermal conductivity, $d_{i}$ ($i=c,S_1,S_2$) is the thickness of the i-th layer, and $l_x$ is the width of the laser structure. The $c$, $s_1$ and $s_2$ indices are employed for the laser chip, first and second submounts, respectively. When laser is mounted on the submounts, heat exchanges between interfaces via conduction instead of convection, hence, the convection from the bottom side of the laser in Eq. (5) must be replaced by the conduction. Therefore, employing $q_l=\int dq_l=G_{eq}(\hat {T}-T_c)$ $(W)$ as the replacement of convection from bottom side on the RHS of Eq. (5), yields modification of the damping factor ($m$) and ambient temperature ($T_a$) by $m'$ and $T_A$ as follows:
The laser equation coupled to the TEC is written as follows:
As can be seen from Eq. (20), it is possible to solve this equation using Eq. (9) without solving the TEC heat equation, i.e., the solution of Eq. (20) is independent of the TEC heat solution. A closer inspection of the solution shows that although $\theta _l$ is independent of the TEC heat equation solution, but laser temperature $\bar {T}_l(y)=T_A+\theta _l$, depends on the cold TEC temperature through effective ambient temperature ($T_A$). Equation (9) is employed to obtain the volume-averaged laser temperature:
2.3 TEC thermal coupling to the laser system
Section 2 introduced a one-dimensional analytical model for the average temperature variation along the SG-DBR laser. This section considers the thermal behavior of the laser system mounted on a TEC. Laser and TEC are packaged into a 14-pin butterfly-package tunable laser module.
TEC module contains two ceramic substrates and some thermocouples. When the DC electric current passes through the n-type leg and flows into the p-type leg, the heat ($Q_c$) is pumped from the top ceramic, and the heat $Q_H$ is delivered to the bottom ceramic as shown in Fig. (3). Heat flows in the same direction as the electric current in the p-type leg and the opposite direction in the n-type leg. After a while, the temperature difference is produced between the top and bottom ceramics. The bottom ceramic settles on the heat sink, and its temperature is assumed constant and denoted by $T_h$. In addition, it can be mounted on the other TEC module for better performance.
The coefficient of performance of a TEC (COP) depends on the Peltier coefficient ($\Pi$). This coefficient can be expressed by the Kelvin relation in terms of the Seebeck coefficient as : $\Pi =\alpha _sT$ [9]. Consequently, the heat equation of a TEC in the presence of Peltier effect is:
Taking the average of temperature over the cross-section of the legs and considering the convection and radiation heat loss from the legs, the steady-state heat transfer equation in every leg can be expressed as follows [16]:
The differential Eq. (23) shows four different mechanisms that happen when the electric current is applied to the thermoelectric elements. The first term on the LHS of Eq. (23) is due to the heat conduction; the second is Peltier Effect; and the third is the radiation and convection from legs. The RHS of Eq. (23) shows the Joul heat generation.
2.4 TEC boundary conditions
The cold and hot side ceramics are located at $z=0$ and $z=l$, respectively as shown in Eq. (3). In a steady-state situation, suppose that the heat generated by the laser system is absorbed by the top ceramic of the TEC at $z=0$. For this to happen, the unused surface of the TEC must be isolated. In other words:
whereThe volume-averaged temperature of the laser ($\hat {T_l}$) can be written versus the laser temperature $\hat {T}_l=(1/ l_y)\int _{0}^{l_y}\bar {T}_l(y)dy$. It follows that $\hat {T}_l$ is linear versus $T_c$:
$\hat {A}$ and $\hat {B}$ are obtained from Eq. (21). The laser heat flow into the TEC is also linear versus $T_c$:
Both $T_c$ and $\hat {T}_l$ are unknown and must be obtained from the solution of the TEC equation with the following boundary condition:
To obtain $q_{TEC}(z=0)$ versus $T_c$, Eq. (23) is solved with $\theta _i(z=0)=\theta _c$ and $\theta _i(z=l)=\theta _h$ boundary conditions, for both $i=n,p$. Solutions are written as:
Here $A_i=N_iA_T$. where $i=n,p$. In Eq. (31) the coefficient of $T_c$ is denoted by $\hat {C}$ and all other terms are called $\hat {D}$. In other words,
Employing Eq. (27) and Eq. (31) the cold TEC temperature is obtained as:
2.5 Simple method for $T_c$ and $\hat {T_l}$ estimation
In the absence of forced convection, our calculations in Section 3 shows that the effect of $m_i$ term in Eq. (23) which corresponds to the heat convection is negligible relative to other heat sources [10]. In this approximation, the laser-TEC system can be considered a closed-system, and energy conservation is written as follows [10]:
The laser dissipation power $q_l=\sum _{i}q_i$ is a function of laser parameters and electric currents, where $q_i$ can be directly obtained by Eq. (2)–(4). The hot side temperature of the TEC ($T_h$) is the heat sink temperature, which is approximately equals to the environment temperature. $T_c$ can be determined by Eq. (33). By substituting $T_c$ and $q_l$ in $q_l= (\hat {T_l}-T_c)/R_{eq}$, the average temperature of laser $\hat {T_l}$ is determined.
Besides, for a given $T_c$, Eq. (34) can be solved for TEC electric current ($I_T$). Equation (34) is a quadratic equation versus $I_T$. The condition for real solution is :
where $R=R_{e,n,p}/R_{t,n,p}$. This inequality is only satisfied for:Equation (37) shows $T_c$ has a theoretical lower bound. The error due to the neglecting of convective heat loss was obtained by comparing the results with those determined by Eq. (33).
3. Numerical results and discussions
Our model is based on the averaged temperature of the laser on the $x-z$ cross-section. In this section, for convenience, we assume that laser has been made of four sections. Injection currents of the front mirror, active medium, rear mirror, and phase section are $I_f=10$, $I_a=100$, $I_r=10$ and $I_p=5$ mA, respectively, unless otherwise stated. Laser and TEC parameters for numerical calculations are summarized in Tables (1) and (2).
The average of the three-dimensional steady-state temperature distribution of an SG-DBR laser over the $x-z$ plane and along the laser was calculated. To verifying our one-dimensional approach in the absence of TEC, we compare our results with those are obtained by FEM in Fig. (4) [36]. As expected, the results are in good agreement with our analytical results.
Fig. 4. Laser cross-sectional average temperature $\bar {T}_l$; analytical and numerical (FEM) for laser structure without submount and TEC. FEM: finite element method; TEC: thermoelectric element cooler.
According to the proposed solution, we conclude that the system of coupled ordinary differential equations (ODE) of laser and TEC has been reduced to a system of two linear algebraic Eq. (27) and Eq. (32). For given currents and parameters of laser and TEC system, the cold temperature of TEC and averaged laser temperature can be obtained.
Variation of $T_c$ and $\hat {T}_l$ versus the TEC electric current are plotted in Fig. (5). Due to the competition between the Joul heating, conduction, and Peltier effect, a minimum of $T_c$ and $\hat {T}_l$ can be found versus the TEC current. The maximum cooling power corresponds to the TEC current for minimum $T_c$ temperature. The current corresponding to the minimum of $T_c$ can be approximately determined by the roots of the derivative of the Eq. (34). The cold temperatures ($T_c$) obtained by Eq. (33) is compared with that is obtained by Eq. (34). The results showed that the error is negligible, i.e., the convection heat loss is negligible relative to other thermal heat sources in the laser and TEC system.
Fig. 5. Effective ambient temperature ($T_A$), the cold temperature of TEC ($T_c$) and laser average temperature $(\hat {T}_l)$ versus injection electric current into TEC ($I_T$). Also, cooling power versus electric current is plotted. The maximum cooling power happens at $I_T=2.2$ A. TEC: thermoelectric element cooler.
The minimum $T_{c,min}=213$ k and $T_{l,min}=261$ k, which are corresponding to the optimum TEC current $I_T=2.2$ A, respectively, are used in Eq. (21) and Eq. (29). Next, the temperature distributions along the laser and TEC are obtained. The results are plotted in Fig. (6) and Fig. (7), respectively.
Fig. 7. Temperature distribution of the SG-DBR laser mounted on the TEC, when $I_f=10$ mA, $I_a=100$ mA, $I_p=5$ mA, $I_r=10$ mA, and $I_T=2.2$ A. SG-DBR : sampled grating distributed Bragg reflector; TEC: thermoelectric element cooler.
In this study, an SG-DBR laser was designed by choosing the operating laser temperature $\hat {T}_l$. $\hat {T}_l$ is employed in Eq. (26) to obtain $T_c$ temperature. For $T_c$ satisfying the inequality Eq. (37), the TEC current is obtained by Eq. (34); otherwise, the desired temperature cannot be obtained by the single stage TEC cooler. For a multistage optimal TEC, the maximum TEC hot temperature of the first stage is determined by the inequality Eq. (37). The result is used in Eq. (34) to determine the first stage TEC current. The hot temperature of the first stage TEC can be employed as the cold temperature of the second stage and continue the process until the hot temperature becomes equal or greater than the environment temperature.
A noteworthy point about this method is that the laser governing equation in the presence of the TEC is the same as that of in the absence of TEC, except that the damping factor ($m$) and environment temperature ($T_a$) are replaced by $m'$ and $T_A$ defined by Eq. (18).
The effective ambient temperature is the weighting average of $T_c$ and $T_a$ with weighting factors $\beta$ and $1-\beta$. The $\beta$ factor is defined in Eq. (19). For $\beta =0$ and $\beta =1$, the effective environment temperature is $T_a$ and $T_c$ respectively. Owing to presence of TEC, both effective ambient temperature and damping factor are reduced. Decreasing $T_A$ causes the increasing the laser cooling power due to the heat conduction, whereas decreasing the $m'$ causes the decreasing of laser cooling power due to the convection. The effective environment temperature ($T_A$) versus TEC current by our model are calculated and presented in Fig. (5). Theoretical results are in good agreement with those reported by theoretical and experimental publications, e.g., [35].
4. Conclusion
This paper set out to provide an analytical model to explain the thermal behavior of an SG-DBR laser when cooled by a thermoelectric cooler (TEC). The heat conduction for average temperature on the $x-z$ plane cross-section along the SG-DBR laser is investigated. The analysis shows that the steady-state solution of heat conduction is a piecewise exponential distribution. The decay rates of this distribution depend on the heat convection, and effective ambient temperature is a weighted average of ambient and TEC cold temperature. The system of coupled differential equations is reduced to a system of two linear algebraic equations versus laser volume averaged temperature and cold TEC temperature. Our analytical results are in good agreement with those obtained by numerical FEM. The presented method is employed to design multistage cooler for SG-DBR laser.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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