Characteristic analysis of 1.06 μm long-cavity diode lasers based on asymmetric waveguide structures

Ren-Ze ZHAO; Xin GAO; Ding-Yang FU; Yue ZHANG; Peng SU; Bao-Xue BO

doi:10.11972/j.issn.1001-9014.2024.04.016

Introduction

1.06 μm diode lasers are widely used in laser fuzing，laser ranging，laser blinding and target identification. Usually，high power edge-emitting diode lasers are coated with high and low reflectivity films on both cavity facets to enhance slope efficiency. On the other hand，to reduce thermal resistance，a long cavity is adopted for high power diode lasers，which mitigates temperature rise in the active region^［1］. However，the above design in practical devices can exacerbate the non-uniform distribution of carrier density along the cavity length，a phenomenon known as longitudinal spatial hole burning （LSHB）^［2-3］. The LSHB can lead to an increase in threshold current and a decrease in slope efficiency^［4］.

Two-photon absorption （TPA） and free carrier absorption （FCA） are considered the main mechanisms limiting the output power^［5，6］. High power diode lasers tend to have a large waveguide layer^［7］，which is typically a few microns thick. They contain a thin active region consisting of one or more quantum well （QW） layers. The n-side waveguide is thicker than the p-side waveguide （the diffusion coefficient of holes is significantly smaller than that of electrons，so the waveguide loss on the p-side tends to be more severe. A low waveguide loss caused by FCA can be achieved by reducing the p-side waveguide thickness.）

Usually，there are two types of TPA loss：（i） the direct light absorption loss，namely direct TPA loss；（ii） TPA will generate free carriers，which can further attenuate the light field through free carrier absorption，namely indirect TPA loss^［8］. TPA loss with a symmetric waveguide has been studied using experimentally measured parameters^［5，9］. In this paper，waveguide losses including TPA loss in asymmetric waveguide structures are analyzed to optimize the output characteristics of long-cavity diode lasers，while FCA loss due to current injection is also considered.

1　Waveguide structures of lasers

The vertical waveguide structure of a typical high power diode laser is shown in Fig. 1. h is the thickness of the waveguide layer，the position of the QW in the waveguide layer is denoted as $p_{x}$ and Ψ²（x） is the normalized mode field intensity distribution， $\int_{- \infty}^{+ \infty} Ψ^{2} (x) d x = 1$ . Since the QW thickness （10 nm） is much smaller than the waveguide layer in Fig. 1，the optical mode field of the device is mainly determined by the waveguide and confinement layers，which can be expressed approximately by the mode field distribution in a three-layer slab waveguide^［10］：

Ψ (x) = \{\begin{matrix} \frac{1}{C_{0}} c o s (k x - φ), 0 < x < h \\ \frac{1}{C_{0}} c o s φ e x p (γ_{n} x), x < 0 \\ \frac{1}{C_{0}} c o s (k h - φ) e x p (- γ_{p} (x - h)), x > h \end{matrix}

,(1)

where $k = \frac{2 π}{λ} \sqrt[]{n_{w g}^{2} - n_{o}^{2}}$ is the transverse wave vector， $γ_{n} = \frac{2 π}{λ} \sqrt[]{n_{o}^{2} - n_{n}^{2}}$ and $γ_{p} = \frac{2 π}{λ} \sqrt[]{n_{o}^{2} - n_{p}^{2}}$ are the field decrements in the n and p confinement layers，respectively. $φ = a r c t a n (\frac{γ_{n}}{k})$ ，and $C_{0} = \sqrt[]{(h + \frac{1}{γ_{n}} + \frac{1}{γ_{p}}) / 2}$ . The n_n，n_p and n_wg are the refractive indices of the n and p confinement layers and the waveguide layer，respectively. The effective refractive index n_o of the fundamental mode can be obtained from the transcendental equation $k h = a r c t a n (\frac{γ_{n}}{k}) + a r c t a n (\frac{γ_{p}}{k})$ .

$Refractive index and fundamental mode intensity profiles of the vertical waveguide$

Figure 1.Refractive index and fundamental mode intensity profiles of the vertical waveguide

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2　Theoretical model

Direct and indirect TPA losses，FCA loss due to current injection and material loss are considered for performance simulation of long-cavity diode lasers with the effect of LSHB. The direct TPA loss can be expressed as^［11］：

α_{T P A}^{d i r e c t} = \frac{P (z)}{w} \int_{- \infty}^{\infty} β (x) ψ^{4} (x) d x

,(2)

Table 1. Device parameters used

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Table 1. Device parameters used

Symbol	Parameter	Value
λ	wavelength	1.06 μm
n_wg	refractive index of waveguide layer	3.474
n_n	refractive index of n-confinement layer	3.3
n_p	refractive index of p-confinement layer	3.3
β_wg	TPA coefficient of waveguide layer	2×10^-8 cm/W
L	cavity length	5 mm
w	stripe width	100 μm
d	QW thickness	10 nm
R_HR	the rear facet reflectivity	0.99
h	waveguide layer thickness	2 μm
$σ_{e}$	free-electron absorption cross-section	3×10^-18 cm²
$σ_{h}$	free-hole absorption cross-section	1×10^-17cm²
D_e	electron diffusion coefficient	200 cm²s^-1
D_h	hole diffusion coefficient	10 cm² s^-1
$η_{i}$	internal quantum efficiency	0.95
N_cl	doping concentration of confinement layer	1×10¹⁸ cm^-3
$α_{i}$	material loss	0.5 cm^-1
A	SRH recombination coefficient	5.88×10⁸ s^-1
B	spontaneous emission coefficient	1×10^-10 cm³ s^-1
C	auger recombination coefficient	2×10^-30 cm⁶ s^-1
g₀	gain constant	2140 cm^-1
N_tr	transparency carrier density	1.77×10¹⁸ cm^-3
τ	time of carrier capture into the QW	150 fs

here P（z） is the mode power along the cavity，w is the stripe width and β（x） is the TPA coefficient for different materials in the waveguide structure. As shown in Fig. 1，the TPA loss in the confinement layer can be neglected due to the weak mode field in the confinement layer. Previous studies have found that the contribution of QW to TPA is negligible compared to the waveguide regions^［5］. So the total TPA loss is primarily contributed by the waveguide layer （GaAs）. The direct TPA loss can be further simplified as：

α_{T P A}^{d i r e c t} = \frac{β_{w g} P (z)}{w} \int_{0}^{h} ψ^{4} (x) d x

,(3)

here $β_{w g}$ is the TPA coefficient of the waveguide layer. In order to combine the direct TPA loss with one-dimensional （1D） rate equations，the direct TPA loss can be expressed as a function of photon density along the cavity. P（z） is given as a function of photon density N_p（z） by the following equation：

P (z) = \frac{w d}{Γ} v_{g} \frac{h_{1} c}{λ} N_{p} (z)

,(4)

here h₁ is Planck's constant，v_g is the group velocity of light，λ is the wavelength，c is the speed of light in vacuum and d is the QW thickness. The confinement factor $Γ$ is expressed by the following equation：

Γ = \frac{\int_{p_{x} - \frac{d}{2}}^{p_{x} + \frac{d}{2}} ψ^{2} (x) d x}{\int_{- \infty}^{+ \infty} ψ^{2} (x) d x}

.(5)

Substituting Eq. （4） into Eq. （3），we obtain that：

α_{T P A}^{d i r e c t} = \frac{β_{w g} d v_{g} h_{1} c}{Γ λ} \int_{0}^{h} ψ^{4} (x) d x N_{p} (z) = β_{T P A}^{d i r e c t} N_{p} (z)

,(6)

here $β_{T P A}^{d i r e c t} = \frac{β_{w g} d v_{g} h_{1} c}{Γ λ} \int_{0}^{h} ψ^{4} (x) d x$ ，is the direct TPA coefficient. Since the waveguide layer is intentionally undoped，there is quasi-neutrality within the waveguide layer，meaning that the electron and hole have the same density，and then the carrier transport can be described by the ambipolar diffusion equation：

D_{a} \frac{\partial^{2} N}{\partial^{2} x} + G (x, z) - R (N (x, z)) = 0

,(7)

here $D_{a} = \frac{2 D_{e} D_{h}}{D_{e} + D_{h}}$ ，D_eand D_hare the electron and hole diffusion coefficients in the waveguide layer，respectively. The indirect TPA loss is caused by the absorption of light through TPA-generated carriers. The distribution of TPA-generated carrier density can be obtained based on the TPA generation term^［11］：

G (x, z) = \frac{β_{w g} λ}{h_{1} c} {(\frac{P (z)}{w})}^{2} ψ^{4} (x)

.(8)

The indirect TPA loss should be a function of photon density along the cavity length，which means G（x，z） is a function of photon density. Substituting Eq. （4） into Eq. （8） gives：

G (x, z) = \frac{β_{w g} h_{1} c}{λ} {(\frac{d v_{g}}{Γ})}^{2} ψ^{4} (x) N_{p}^{} {(z)}^{2}

.(9)

Neglecting carrier recombination （R=0），the distribution of the TPA-generated carrier density in the waveguide layer is obtained by combining the boundary conditions with Eq. （7）：

∆ N_{T P A} (x, z) = γ N_{p}^{} {(z)}^{2}

,(10)

here $γ$ is a function of x and $p_{x}$ for the n and p side waveguides. The indirect TPA loss is expressed as：

\begin{array}{l} α_{T P A}^{i n d i r e c t} = (σ_{e} + σ_{h}) \int_{0}^{h} ∆ N_{T P A} (x, z) ψ^{2} (x) d x \\ = \int_{0}^{h} (σ_{e} + σ_{h}) γ ψ^{2} (x) d x N_{p} {(z)}^{2} \\ = β_{T P A}^{i n d i r e c t} N_{p} {(z)}^{2} \end{array}

,(11)

here $β_{T P A}^{i n d i r e c t} = \int_{0}^{h} (σ_{e} + σ_{h}) γ ψ^{2} (x) d x$ ，is the indirect TPA coefficient. $σ_{e}$ and $σ_{h}$ are the free-electron and free-hole absorption cross-sections，respectively. Neglecting the carrier recombination，the carrier density distribution in the waveguide layer can be obtained by Eq. （7）^［6］：

∆ N_{j} (x) = \{\begin{matrix} \frac{j}{2 e D_{e}} (p_{x} - \frac{d}{2} - x) + N_{b}, 0 < x < p_{x} - \frac{d}{2} \\ \frac{j}{2 e D_{h}} (x - p_{x} - \frac{d}{2}) + N_{b}, p_{x} + \frac{d}{2} < x < h \end{matrix}

,(12)

here j is the current density，e is the electron charge， $N_{b} \approx \frac{j}{e d} τ$ ， $τ$ is the time of carrier capture into the QW. The FCA loss in the waveguide layer can be expressed as：

α_{j}^{w g} = (σ_{e} + σ_{h}) \int_{0}^{h} ∆ N_{j} (x) ψ^{2} (x) d x

.(13)

The FCA loss in the QW can be expressed as：

α_{j}^{Q W} = Γ (σ_{e} + σ_{h}) N (z) = β_{j}^{Q W} N (z)

，(14)

here $β_{j}^{Q W} = Γ (σ_{e} + σ_{h})$ ，N（z） is the carrier density in the QW. Because of LSHB，the carrier density N（z） is no longer constant but decreases gradually from the rear facet to the front facet. The FCA loss in the confinement layer can be expressed as^［6］：

α_{j}^{c l} = σ_{e} \int_{- \infty}^{0} N_{c l} ψ^{2} (x) d x + σ_{h} \int_{h}^{+ \infty} N_{c l} ψ^{2} (x) d x

，(15)

here N_cl is the doping concentration of the confinement layer. The behavior of diode lasers is determined by the relationship equation between carriers and photons. In the conventional rate equations，a uniform distribution of carrier density along the cavity length is usually assumed. However，this is not the case in high power diode lasers. Considering the effect of the losses mentioned above and the inhomogeneity，the steady state 1D rate equations are modified as：

\frac{η_{i} j}{e d} - R_{s p} [N (z)] - v_{g} g [N (z)] [N_{p}^{+} (z) + N_{p}^{-} (z)] = 0

，(16)

\frac{d N_{p}^{\pm} (z)}{d z} = \pm \{Γ g [N (z)] - α_{i} - α_{j}^{w g} - α_{j}^{c l} - β_{j}^{Q W} N (z) - β_{T P A}^{d i r e c t} N_{p}^{\pm} (z) - β_{T P A}^{i n d i r e c t} N_{p}^{\pm} {(z)}^{2}\} N_{p}^{\pm} (z)

.(17)

Equation （16） describes the change in carrier density due to spontaneous and stimulated recombination under current injection. Equation （17） describes the evolution of the photon density along the laser cavity and takes into account internal losses. α_i_{and η_i are the material loss and internal quantum efficiency，respectively. $N_{p}^{+} (z)$ and $N_{p}^{-} (z)$ are the photon densities for forward and backward propagation，respectively. $N_{p} (z) = N_{p}^{-} (z) + N_{p}^{+} (z)$ . $R_{s p} [N (z)] = A N (z) + B N {(z)}^{2} + C N {(z)}^{3}$ ，is the spontaneous recombination rate. The gain coefficient can be expressed as $g [N (z)] = g_{o} l n [N (z) / N_{t r}]$ . The boundary conditions for Eqs. （16） and （17） at the rear facet （z = 0） and the front facet （z = L） are：}

N_{p}^{+} (0) = R_{H R} N_{p}^{-} (0)

，(18)

R_{A R} N_{p}^{+} (L) = N_{p}^{-} (L)

，(19)

here L is the cavity length，R_AR and R_HR are the front and rear facet reflectivities，respectively. The photon and carrier density distribution along the laser cavity can be calculated by numerically solving Eqs. （16）-（19） by the finite difference method. The threshold lasing condition can be expressed as：

\frac{1}{L} \int_{0}^{L} Γ g [N (z)] d z = α_{t o t a l} + α_{m}

,(20)

here $α_{t o t a l}$ is the total internal loss and $α_{m}$ is the output loss. The threshold current can be expressed as^［2］：

I_{t h}^{} = \frac{e w d}{η_{i}} \int_{0}^{L} R_{s p} [N (z)] d z

.(21)

The front facet output power can be expressed as：

P_{o u t} = (1 - R_{A R}) v_{g} \frac{d w}{Γ} \frac{h_{1} c}{λ} N_{p}^{+} (L)

.(22)

Table 1 lists the main device parameters for the simulation calculation.

3　Results and analysis

Figure 2 shows $β_{T P A}^{d i r e c t}$ and $β_{T P A}^{i n d i r e c t}$ versus the position of QW. It can be seen that both coefficients increase gradually as $p_{x}$ increases. This is because both coefficients are inversely proportional to the confinement factor，which gradually decreases as the QW shifts towards the p-side，as shown in Fig. 4.

Figure 2.The direct and indirect TPA coefficients versus QW positions

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Figure 3.Total internal loss and slope efficiency versus QW positions

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Figure 4.Confinement factor and threshold current versus QW positions

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Since the FCA and TPA losses vary with the carrier or photon density along the cavity，the average loss along the cavity is used to express the magnitude of loss. Figure 3 shows the variation of total internal loss and slope efficiency with QW positions at an injection current I = 20 A and R_AR= 0.01. It can be seen that the total internal loss decreases monotonically as the QW shifts towards the p-side，while the slope efficiency has an opposite trend. The FCA loss decreases as the QW shifts towards the p-side，while the TPA loss increases with the TPA coefficients $β_{T P A}^{i n d i r e c t}$ and $β_{T P A}^{d i r e c t}$ . However，the decrease of FCA loss is greater than the increase of TPA loss.

Figure 4 shows the variation of the confinement factor and threshold current with QW positions at I = 20 A and R_AR= 0.01. It can be seen that the confinement factor decreases and the threshold current increases as the QW shifts towards the p-side. From Eqs. （20） and （21），we can see that the threshold current is approximately an inverse function of the confinement factor. Though the total internal loss decreases as the QW shifts towards the p-side，the confinement factor has a greater effect on the variation trend of the threshold current.

Since the slope efficiency and threshold current exhibit the same trend when the QW shifts towards the p-side，there might be an optimal QW position that maximizes the front facet output power. Figure 5 shows the variation of the front facet output power with QW positions at I = 20 A and R_AR= 0.01. It can be seen that the maximum output power is achieved at $p_{x}$ = 1.39 μm.

Figure 5.Output power versus QW positions

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The front facet reflectivity plays an important role in the performance of high power diode lasers. Figure 6 shows the optimized front facet output power and the corresponding QW position versus different front facet reflectivities at I = 20 A. It can be seen that the output power initially increases and then decreases，while the optimal QW position gradually shifts towards the n-side as the front facet reflectivity decreases. This is because an appropriate reduction in the front facet reflectivity can improve the slope efficiency. However，if the reflectivity is too low，it will cause severe LSHB，resulting in a reduction in output power. The smaller the front facet reflectivity，the higher the threshold current. The optimal QW position needs to shift in the direction of a larger confinement factor to reduce the impact of the threshold current. It can be concluded that the maximum output power can be obtained by adjusting both the QW position and the front facet reflectivity.

Figure 6.The optimized output power and corresponding QW position versus the AR reflectivity

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Figure 7 shows the optimal QW position versus the injection current at R_AR = 0.01. It can be seen that the optimal QW position gradually shifts towards the p-side as the injection current increases. This is because the impact of threshold current on output power decreases as the injection current increases，and the effect of slope efficiency on output power increases relatively. The shift of the optimal QW position tends to be slow with increasing injection current. This is because the larger the value of $p_{x}$ ，the faster the threshold current increases，as shown in Fig. 4.

Figure 7.The optimal QW position versus the injection current

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Figure 8 illustrates the impact of losses on slope efficiency，threshold current and output power at I = 20 A and $p_{x}$ = 1.39 μm. It can be seen that the threshold current and slope efficiency decrease with the increase of the front facet reflectivity in Fig. 8（a）. Figure 8（b） shows that the output power increases monotonically with the decrease of the front facet reflectivity when LSHB is not considered. When considering LSHB，a decrease in output power occurs due to a higher threshold current and a more significant reduction in slope efficiency，as shown in Fig. 8（a）. It is found that LSHB becomes the most significant factor in power reduction when the front facet reflectivity is low.

Figure 8.Threshold current, slope efficiency and output power：(a) threshold current and slope efficiency versus the AR reflectivity; (b) output power versus the AR reflectivity

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4　Summary

A simplified numerical analysis model is proposed to systematically analyze the impact of LSHB，TPA and FCA on the output characteristics of 1.06 μm long-cavity diode lasers. It is found that an appropriate reduction in the front facet reflectivity can increase the output power. But if the reflectivity is too low，it will cause severe LSHB，leading to a significant reduction in the slope efficiency and a notable increase in the threshold current，which in turn reduces the output power. For the same injection current，the QW position in the waveguide can be optimized to maximize the output power. The optimal QW position shifts towards the p-side as the front facet reflectivity or the injection current increases. LSHB becomes the most significant factor affecting the output power when the front facet reflectivity is low.

Category: Research Articles

Received: Aug. 7, 2023

Accepted: --

Published Online: Aug. 27, 2024

The Author Email: Bao-Xue BO (bbx@cust.edu.cn)

DOI:10.11972/j.issn.1001-9014.2024.04.016