High-power broad-area（BA）diode lasers have found a wide range of industrial，medical，scientific，space and military applications ^［1］. In the 976-nm region，much attention has been attracted as pump sources for erbium-fiber amplifiers，optical data storage and displays ^［2-4］. A supper-large-optical-cavity（SLOC）waveguide structure ^［5］ has the benefit of lower power density on the facet ^［6］，lower vertical far-field angle，and lower internal loss，thus high-power，high-efficiency and high reliability diode lasers. Furthermore，the lateral beam quality can be greatly improved due to the low differential modal gain of the SLOC structure ^［7］.

However，the number of higher-order modes increases with the vertical waveguide thickness. Higher-order transverse modes in a SLOC can deteriorate the far-field characteristics ^［8］ and reduce the power efficiency，since the lasing of higher-order modes will introduce a higher internal optical loss. Many reports focused on the design of high-power diode lasers based on the large-optical-cavity（LOC）structure ^［9-12］，and some reports discussed the suppression of higher-order modes ^［12］. However，it is difficult to suppress the lasing of higher-order modes in a SLOC by changing the position of the quantum well（QW）or by the modal loss discrimination^［13］.

Furthermore，when a thick waveguide is adapted，electrons may accumulate around the p-waveguide，decreasing the injection efficiency of the carriers，which is the main cause of a lower electro-optical efficiency in a SLOC structure device ^［10-11］. B.S.Ryvkin^［10］ found that the broadening of the waveguide leads to a drastic reduction in the internal quantum efficiency due to the increased carrier density in the p-waveguide. Efforts were taken to overcome the decreasing of internal efficiency in the SLOC structure. Pietrzak et al. ^［14］ reported an 8.6 μm thick SLOC with the vertical far-field of 15° and internal loss of 0.68 cm^-1 with an injection efficiency of 0.70. Hasler et al. ^［15］ obtained an injection efficiency of 0.91 and internal loss of 0.56 cm^-1 using a 2.8 μm SLOC，but the vertical divergence is 50.8°. Andre. M ^［16-17］ obtained an internal efficiency of 0.99 and internal losses of 0.8 cm^-1 with a vertical far-field angle of 14.4° at full width at half maximum（FWHM）.

The output power is also limited by mechanisms such as longitudinal spatial hole burning（LSHB）^［18］，which has been experimentally proved through the non-pinning of the carriers and spontaneous emission above lasing threshold in both GaAs-based^［19-20］ and InP-based^［21］ diode lasers. Narrow-stripe InP lasers showed around 25% improvement of rollover power using flared designs attributed to reduction of the LSHB effect ^［21］. There have been theoretical studies ^［22］ showing the effect of LSHB on the output power in GaAs-based laser diodes. Reducing the cavity length can suppress the LSHB effect but the increase of thermal resistance degrades the output power at high injected currents. R. B. Swertfeger^［23］ recently found that higher current densities（and hence higher local recombination rates and higher losses）arise near the front facet due to spatial hole burning and that the non-uniformity is strongly affected by laser geometry，which is more severe for longer resonators and less severe for higher front facet reflectivity. Abdullah D. ^［24］ reported an unfolded cavity，where the reflectivity of the rear and front facets were set equivalent with R=10%，to decrease the LSHB effect. It was reported that the unfolded cavity decreases the LSHB penalty by 10 times. However，the coupling or shaping of the laser beam might be complicated since the unfolded cavity has two equivalent output facets.

The goal of this paper is to design a SLOC structure diode laser with low vertical divergence angle and high power-efficiency. The refractive index，band energy，and thickness of the epitaxial layers were carefully optimized to effectively suppress the lasing of higher-order modes and carrier leakage. Taking into account the LSHB effect，the carrier density，gain profile and power along the cavity were calculated and analyzed based on the diode laser rate equations. The 4-mm-long and 100 μm wide broad-area single emitter with the SLOC waveguide structure was designed to obtain a high power-efficiency of about 71% at an output power of 21 W.

For the vertical waveguide simulation，the band，carrier concentration，gain，index profile，mode intensity，confinement factor，mode power，voltage，external efficiency，and far-field patterns were calculated. The 2D laser diode model was built with infinite lateral boundary condition. The cross sections due to the absorption by free electrons and holes in the materials were set to α_n=4×10¹⁷ cm² and α_p=12×10¹⁷ cm²，respectively ^［5］. The calculation of the LSHB effect was based on the carrier and photon rate equations with the basic material and device parameters obtained from the results. The cavity length and emitter width were set to 4 mm and 100 μm in all the calculations. In the vertical waveguide model，the calculations were done under fixed lattice temperatures，while in the longitudinal calculations thermal heating effect was considered with a thermal resistance of 2.0 K/W，which is a typical value with p-down package on submount^［5］.

The vertical structure starts from a reference structure as shown in Table 1 The total wavegueide thickness is 3 500 nm with asymmetric n-type and p-type cladding layers. The single QW is set at the vertical position of around 4 000 nm with a p-waveguide thickness of 1 500 nm，where the confinement factor of mode0 is higher than all the higher-order modes，to ensure single transverse mode lasing ^［25］. The confinement factors of mode0，mode1 and mode2 are 0.43%，0.10% and 0.08%，respectively. However，the relative thick p-waveguide leads to strong electron leakage，with a maximum concentration of about 4×10¹⁷ cm^-3 as Fig. 1（b）shows. It is obvious that the leakage will decrease the carrier injection efficiency and increase the optical loss due to free carrier absorption. The leakage can be suppressed by increasing the p-type doping level，as shown in Fig. 1（b）. A p-type doping concentration of 4×10¹⁷ cm^-3 reduces the maximum electron concentration to about 1×10¹⁶ cm^-3. However，high-level p-type doping in the waveguide introduces large optical loss which degrades the power efficiency as the data decipited in Fig. 2.

A decreased thickness of the p-waveguide can suppress the carrier leakage but high-order mode will lase. Mode control layers were designed underneath and up the QW layer to maintain a fundamental mode with thinner p-waveguide ^［26］. As shown in Fig. 3，50 nm Al_0.08Ga_0.92As layers were designed as the mode control layers，with which the thickness of the p-waveguide can be reduced to about 900 nm. The confinement factors of mode 0，mode1 and mode2 are 0.69%，0.22% and 0.08%，respectively. However，high-order modes can lase if the p-waveguide thickness was further reduced according to the calculations.

The carrier leakage can be further suppressed by a grading p-waveguide which increases the potential energy of the electrons，as the structure shown in Table 2. As plotted in Fig. 4（a），the Al content grades from 0.15 to 0.6 in the p-waveguide within a thickness of 600 nm. Single transverse mode lasing is achieved，where the confinement factors of mode0，mode1 and mode2 are 0.52%，0.39% and 0.28%，respectively. The electron concentration in the grading and without grading p-waveguide structure were compared and plotted in Fig. 4（b）. The electron concentration was reduced to 5×10¹⁵ cm^-3 at 20 A，about two orders lower compared to 4×10¹⁷ cm^-3 in the reference structure.

The suppression of electron leakage in the p-waveguide improves the slope efficiency，especially in the current range of above 4 A，as shown in Fig. 5（a）. The slope efficiency decreases from 1.095 W/A around 1 A to 0.886 W/A around 20 A in the reference structure，while it is 1.100 W/A to 1.039 W/A in the grading structure. From the slope efficiency，an internal quantum efficiency of 0.954 at 1 A and 0.917 at 20 A can be deduced. Since the mode overlap with the p-doped layers is reduced in the grading structure，the doping level of the grading p-waveguide layer can be designed much higher. As shown in Table 2，a graded doping profile from 2×10¹⁷ cm^-3 to 2×10¹⁸ cm^-3 is used in the grading p-waveguide layer，which reduces the voltage by 0.142 V at 20 A compared to the reference structure as decipited in Fig. 5（b）. The power efficiency is thus improved from 61% to 71.6% at 20 A as plotted in Fig. 6（a），due to the improvement of output power and the decrease of voltage. The vertical far-field pattern is plotted in Fig. 6（b）. The full width at half maximum（FWHM）vertical far-field angle is about 17.4° for both structures.

Typically，the junction temperature of the laser diode will increase with the injected current. The temperature dependence of threshold，I_th，and slope efficiency S，are calculated as a function of lattice temperature T，using the simple empirical expressions with

and

where T0=300 K，I_ref and Sref is the threshold current and slope efficiency at 300 K，respectively. In the temperature range of 300 K to 360 K，the extracted T0 values are 174 K and 194 K for the reference and grading structure with relatively small difference. However，the extracted T1 values are 228 K and 483K for the reference and grading structure. The improved T1 value in the grading structure results from the suppression of carrier leakage.

Table 3 summarized the main parameters of the reference and grading structures. The key point is that the increase of modal loss and the decrease of internal quantum efficiency were suppressed in the grading structure.

Rate equations for diode lasers describe the relation of the density of photons and carriers（electrons）with injection current and optical gain. General rate equations assume uniform longitudinal photon density，carrier density and optical gain. However，under the effect of LSHB，rate equations must be modified ^［27］，with local carrier density N（z），local photon density N_p（z），and local gain g（z）varying with longitudinal position z. The detailed calculation processes of the power，gain and carrier profile was referenced from Ting Hao ^［28］，with the material and device parameters in Table 4.

The output powers of the chips with and without the LSHB effect were compared in Fig. 8（a）. The output power degraded with the LSHB effect with the increase of injected current. Figure 8（b）shows the power and gain profile in the chip at 20A. It is obvious that carriers accumulated on the high-reflection（HR）side because of lower optical power density. The accumulation of carriers leads to strong leakage and spontaneous emission resulting in a lower output laser power.

To suppress the LSHB effect，a linear current profile was proposed，as shown in Fig. 9（a）. The current density was linearly increased from the HR facet to the anti-reflection（AR）facet，while keeping the total injected current as the same with a uniform current density. A linear current profile can be realized by a tapered electrode contact along the cavity or a patterned ohmic contact ^［29］. The corresponding power and gain distribution in the chips were plotted in Fig. 9（b）. The gain was much lower at the HR side since the injection current density was low. The maximum modal gain reduced from 39 cm^-1 to 16 cm^-1 compared to the uniform current injection，since the maximum carrier density was reduced from 6.4×10¹⁸ cm^-3 to 2×10¹⁸ cm^-3，as shown in Fig. 10（a）.

The suppression of the LSHB effect improves the output power，as plotted in Fig. 10（b）. The maximum output power increased by 1.0 W at 20 A when linear current profile was used. Concerning the thermal effect with a thermal resistance of 2.0 K/W，the 4-mm-long and 100 μm wide broad-area single emitter with the SLOC waveguide structure was designed to have a high power-efficiency of about 71% at an output power of 21 W under continuous current injection at 25°C，as shown in Fig. 11.

Numerical study of the epitaxial and longitudinal design of a SLOC structure diode laser is presented in this paper. Mode control layers were used to maintain a single transverse mode with reduced p-waveguide thickness. A graded p-waveguide layer can effectively suppress the leakage of electrons and reduce the voltage. The optimized SLOC structure has an internal loss of 0.66 cm^-1，a confinement factor of 0.517%，an internal quantum efficiency of 0.954，a character temperature of T₀=194 K and T₁=483 K in temperature range of 300∼360 K，respectively，and a full width at half maximum vertical far-field angle of 17.4°. A liner current profile along the cavity can suppress the LSHB effect，which increases the output power by 1.0 W at 20 A concerning the thermal effect with a thermal resistance of 2.0 K/W. As a result，4-mm-long and 100 μm wide broad-area single emitter with the SLOC waveguide structure was designed to have a high power-efficiency of about 71% at an output power of 21 W under continuous current injection at 25 °C.