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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15998–16006
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The output power and beam divergence behaviors of tapered terahertz quantum cascade lasers

YanFang Li, Jian Wang, Ning Yang, Junqi Liu, Tao Wang, Fengqi Liu, Zhanguo Wang, Weidong Chu, and Suqing Duan  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15998-16006 (2013)
http://dx.doi.org/10.1364/OE.21.015998


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Abstract

We report on Terahertz quantum cascade lasers with tapered waveguide structure operating at ∼ 103 μm. The tapered waveguide effect on the output power and the laser beam divergence are experimentally studied with the tapered angle ranging from 0° to 8°. It is found that the peak output power of the devices with same length reaches the maximum at about 5° ∼6° tapered angle. Meanwhile, the horizontal divergence angle of the laser beam can be greatly reduced. The existence of such optimal tapered angle is explained by the finite-element simulation with the consideration of the self-focusing effect for the devices with larger tapered angle.

© 2013 OSA

1. Introduction

In this paper, we present a study on the characteristics of tapered THz QCL devices. By comparing the output powers and the far-field divergence angles of the devices with different tapered angles, we obtain the optimized tapered angle of 5° ∼6° for our THz QCLs with which the peak output power is maximized and the beam divergence is minimized. The finite-element simulation can well explain the measured data with the consideration of the self-focusing effect.

2. Materials and fabrication

The active region of the tapered THz QCL devices presented in this paper is based on bound-to-continuum (BTC) transition design with 120 active periods of GaAs/Al0.15Ga0.85As heterostructures which was grown by EPI Gen II solid source molecular beam epitaxy on a SI GaAs substrate. The layer sequence of the active region are as follows: 125/15/114/25/114/33/126/44/93/6/169/9/166/10/143/12Å (The numbers in bold are the thickness of the Al0.15Ga0.85As barriers, in normal are GaAs wells and underscore doped with Si to 3 × 1016 cm−3). The designed laser wavelength is 103 μm (∼ 2.9 THz) under a bias of 2.55kV/cm. The growth started with 0.8 μm highly-doped (Si, 2.5 × 1018 cm−3) GaAs cladding, followed by 120 periods of the active region unit. Finally, a 0.2 μm highly-doped (Si, 5 × 1018 cm−3) GaAs was grown as the top contact layer.

The cross-sectional schematic of the device is illustrated in Fig. 1(a). Samples were processed by optical lithography and wet chemical etching to form the ridged and tapered regions with varied tapered angles. An Ge/Au/Ni/Au (26/54/15/150 nm) layer was evaporated on top of the ridge (two 25 μm-wide narrow metal bars on each side of the ridged and the tapered region) as well as the bottom contact layer by thermal annealing for 60 s at 365°C under nitrogen atmosphere to provide Ohmic contacts. The Ge/Au/Ni/Au layers on the top of the ridge are away from corresponding edge of the device by 20 μm [see the inset in Fig. 1(a)]. Then, a Ti/Au layer (3/100 nm) was deposited to cover both the contact layer and the entire ridge except the two 20 μm-wide stripe regions on the edges of the ridge. This 20 μm-wide uncovered stripe region on each side of the ridge along with the Ge/Au/Ni/Au Ohmic contact layers form the absorption regions which strongly suppresses the high-order transverse modes in the cavity. After thinning down the substrate to about 200 μm, the Ti/Au layer was deposited to the back of the device to facilitate die mounting. For measurement, the laser bars were cleaved to the desired length, followed by deposition of Al2O3/Ti/Au as high-reflection coating on the rear facet. The SEM image and the tapered angles for the tapered THz QCLs are shown in Fig. 1(b). The devices are composed of same ridged region with 0.5 mm length (Lridged) and 103 μm width and varying tapered regions. At last, the devices were soldered onto the Cu heat-sinks and wire-bonded. In all set of experiments, the devices were driven by 2μs pulse at 5 kHz repetition rate.

Fig. 1 (a) Cross-sectional schematic of the tapered THz QCL, inset: zoom in of the 25 μm-wide Ge/Au/Ni/Au ohmic contact layer and 20 μm-wide uncovered area. (b)the SEM image of the tapered THz QCL with tapered angles equal to 0°, 3°, 5° and 8°.

3. Measurements and discussions

In Fig. 2(a), the current-voltage (I–V) and the pulsed light-current (L–I) characteristics are shown for the 5°-tapered device with various heat-sink temperatures. The collected peak optical power of 58.2 mW is reached at ∼ 2000 mA (the collection efficiency of 22%). The device lases up to a heat-sink temperature of 95K with a peak power of 7.1 mW. The maximum slope efficiency of this device is 145 mW/A at 10 K. The emission spectrum of the device with 5°-tapered angle at 10K is around 103 μm [see inset of Fig. 2(b)].

Fig. 2 (a) Voltage and output power as a function of current at different heat sink temperatures for the 5°-tapered device. (b) Output powers versus current density at 10K of the tapered THz QCL with a total length of 2.5 mm and tapered angles 0°, 3°, 5° and 8° (measured width of output facets corresponding to 103 μm, 210 μm, 275 μm, and 375 μm, respectively). The inset shows the emission spectrum of the 5°-tapered device.

Figure 2(b) comparatively shows the light-current density (L–J) characteristics of four tapered devices with same total length 2.5 mm and varying tapered angles of 0°, 3°, 5° and 8°, whose output facet width are 103 μm, 210 μm, 275 μm and 375 μm, respectively. From Fig. 2(b), one can see that the threshold current density of the lasers decreases from 269 A/cm2 to 156 A/cm2 as the tapered angle increases from 0° to 8°. The devices with 0°-tapered angle reaches its emitted peak power at 538 A/cm2, the corresponding value for other devices gradually decreases with the increase of the tapered angle. For the 8°-tapered device it is 426 A/cm2. These fact is mainly due to the increasing pumped volume of the active region.

Unlike the monotonic decreasing of threshold current density, the peak output powers of the devices do not monotonic increase. The peak output power increases with the tapered angle when it is smaller than 5°, and reaches the maximum 58.2 mW for the device with 5° tapered angle. Then the peak output power of the device with 8°-tapered angle drops to 53 mW. If we compare a “power density” which is defined as the output power divided by the area of the device, we will see this decrease more apparently. The “power density” at the peak power is 100 mW/mm2 for the 8°-tapered device which is much smaller than the value 135 mW/mm2 for the 5°-tapered device, and even smaller than the value 112 mW/mm2 for the 0°-device. We can also compare this “power density” at smaller current density. For example, when J=300 A/cm2, the “power density” for 8°-tapered device is 45.3 mW/mm2 which is not only larger than the value 8.9 mW/mm2 for 0°-device but also larger than the vale 34.9 mW/mm2 for the 5°-tapered device. These observations suggest that with the increasing current density, some effect limiting the optical power may occurs in the device with larger area.

To further investigate the output power characteristics of the tapered THz QCL, in Fig. 3 we show the peak output power of the devices with various angles and lengths of tapered region given the same ridged region length Lridged =0.5 mm. From Fig. 3, it is obvious that the peak output power of tapered devices increases with increasing Ltapered for any tapered angles. However, all devices with Ltapered ranges from 1.0 mm to 2.42 mm approach their maximum power at same tapered angle 5°, which is similar to that of Fig. 2(b). Further increasing the tapered angle to 8°, the peak output power of the devices decreases instead. This result indicates that there is an optimized tapered angle for the peak output power of tapered THz QCLs, which is about 5° ∼6° indicated by the curve fitting results in Fig. 3.

Fig. 3 Peak output power versus tapered angle of the devices with Ltapered equals to 1.0 mm, 1.3 mm, 2.0 mm and 2.42 mm. The scatters are the experimental data and the lines are curve fitting data.

Another important characteristic of tapered THz QCLs is their horizontal beam divergence. Figure 4 shows the angular distributions of the normalized intensity of the far-field laser beam of the tapered THz QCL with 0°, 5° and 8° tapered angles. The full width at half maximum (FWHM) of the horizontal laser beam of the devices with 0°, 3°, 5° and 8° tapered angles are also presented in Table 1. The measured FWHM angles decreases from 31.72° to 19.05° as the tapered angle increases from 0° to 5°. Similarly to the non-monotonic change in peak output power, an abnormal increase of the FWHM angle is observed with the further increase of the tapered angle, noting that the FWHM of 8° tapered device is 21.51° [see Fig. 4(c) and Table 1]. This result further proves the existence of an optimized tapered angle. And, the abnormal change of the output powers and the far field behavior in the devices with larger tapered angles could be probably attributed to the same physical effect in the tapered waveguide section. The increase width of the far-field distribution implies that the optical mode is tightened up in the waveguide. Thus, we deduced that the self-focusing effect occurs with the widening of the tapered region.

Fig. 4 The measured normalized intensity (°) of laser beam in the x direction of the tapered THz QCL with tapered angle equals to 0°, 5° and 8°, respectively. The ridged and tapered region lengths are 0.5 mm and 2.0 mm, respectively. The theoretically simulated results with the finite-element method are also presented with the solid lines. Inset: the division of the tapered region of the device within the simulation.

Table 1. The measured and theoretically simulated full width at half maximum (FWHM) angle of the horizontal far field laser beam of the tapered THz QCL with tapered angle equals to 0°, 3°, 5° and 8°. The ridged and tapered region lengths are 0.5 mm and 2.0 mm, respectively.

table-icon
View This Table

Self-focusing effect of laser beam can be understood by considering the diffraction of a laser beam in material exhibiting an intensity-dependent or temperature-dependent refraction index [14

14. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965) [CrossRef] .

, 15

15. C. C. Wang, “Length-dependent threshold for stimulated Raman effect and self-focusing of laser beams in liquids,” Phys. Rev. Lett. 16, 344–346 (1966) [CrossRef] .

]. With the increase of the current density, i.e. the output power, the thermal accumulation can greatly change the refraction index of the laser material and finally influence the device performance. In order to analyze the behaviors of FWHM of the far-field beam and output power of the tapered THz QCL, we simulate the optical mode propagation within the realistic waveguide with the mode coupling theory and the finite-element method [16

16. J. Wang, W. D. Wu, X. L. Zhang, and S. Q. Duan, “Analysis of terahertz quantum cascade laser beam,” Information and Electronic Engineering 9, 365–368 (2011).

]. In the calculation, the tapered region of the waveguide is divided into a series of rectangle subregions with gradually increased width as sketched in the inset of Fig. 4. Within each rectangle region, the field distributions and corresponding propagation constants of all the optical modes are calculated from the Helmholtz equation ×(εr1×H)k02μrH=0, where εr and μr are conductivity and permeability of solved regions, which include the active region, substrate, plasma layer, Au, Ge/Au/Ni/Au layer as well as an air layer. εr = n2, where n is the refractive index. k0 is the wave number in free space. According to the continuity condition of the tangential electric field at the interface, the coupling coefficient of the optical modes in two neighbor rectangle regions can be obtained. Then the optical mode distribution at the output facet of the tapered region can be deduced from the optical mode imported from the ridged region. And the effective refractive index of each mode can be also obtained.

The far-field distribution of the laser beam can be obtained from the optical mode distribution at the output facet according to the Rayleigh-Sommerfeld diffraction integral theory [17

17. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002) [CrossRef] .

] as
Ex(x,y,z)=12πEx(x,y,0)z(eikRR)dxdyEy(x,y,z)=12πEy(x,y,0)z(eikRR)dxdyEz(x,y,z)=12π[Ex(x,y,0)x(eikRR)+Ey(x,y,0)y(eikRR)]dxdy
(1)
where Ex,y(x′,y′,0) are the optical field at the output facet, Ex,y,z(x, y, z) are the far-field distribution, and R=(xx)2+(yy)2+z2 Finally the FWHM can be obtained from the far-field laser beam.

Before the simulation of the optical mode propagation within the tapered waveguide, we first calculate the refractive index of each layer of the devices. For example, the refractive index of the active region is 3.5639–0.03i. And the indexes of other layers can be found in [16

16. J. Wang, W. D. Wu, X. L. Zhang, and S. Q. Duan, “Analysis of terahertz quantum cascade laser beam,” Information and Electronic Engineering 9, 365–368 (2011).

]. As described in section 2 and shown in the inset of the Fig. 1(a) the uncovered 20 μm-wide stripe region and the Ge/Au/Ni/Au stripe layer on the top of the device effectively suppress the high-order transverse modes, and the tapered angles are smaller than the diffractive angle of the fundamental transverse mode (about 11°), so only the propagation of the fundamental transverse mode within the tapered devices need to be considered. Besides, the reflection optical field from the output facet is also important for the defining of the cavity mode, however, since the backward field at the output facet is always R times of forward field (R is the reflectivity of the output facet), it does not change the relative field distribution at the facet which determines the FWHM of the far-field laser beam. Then only the propagation of the optical field along +z-direction need to be considered when we calculate the FWHM of the far-field beam.

In Fig. 4, we present the simulated angular distributions of the normalized far-field laser beam intensity of three devices. The simulated FWHM angles of the devices with 0°, 3°, 5° and 8°-tapered angles are also listed in Table 1. For smaller tapered angles, i.e. 0°, 3°, 5°, the simulated results agree with the experimental data accurately. However, for the larger tapered angle, i.e. 8°, the simulated divergence angle is 16.64° which deviates from the experimental value 21.51° greatly. In fact, for the devices with larger tapered angle, the larger active region area induces the thermal energy accumulation within the center of the devices, especially when the high power operation. The rise of the temperature in the device will changes the refractive index of the center and then gives rise to the self-focusing effect. The self-focusing effect makes the real part of the refractive index of the active region increase while the imaginary part decrease with the increase of the optical power. And the self-focusing effect mainly occurs in the region with high optical power density, i.e. the center region near the output facet. So for the convenience, we consider a effective change of the refractive index of active region in the Nth subregion with the form
n=3.5639*[1+Δn1e(|x|/x0)]0.03i*[1+Δn2e(|x|/x0)],
(2)
where x0 = 117μm is the x-position where the optical power within the Nth subregion decreases to the 1/e of its maximum. Δn1 and Δn2 indicate the degree of the influence of the self-focusing effect to the refractive index. Fitting with the experimental far-field distribution, we find that the calculated FWHM of the device with 8°-tapered angle can be 21.70° with Δn1 = 0.04 and Δn2 = −0.05.

Not only the increase of the beam divergence, but also the decrease of output power of the 8°-tapered device can be attributed to the self-focusing effect. According to our simulation, the effective index of refraction for the fundamental transverse mode in the 0°, 3° and 5°-tapered devices are 3.6256 – 0.0050i, 3.6331–0.0070i and 3.6347–0.0074i, respectively. On the contrary, for the device with 8°-tapered angle, with same formula of the modification of the the refractive index of the active region in Eq. (2), we get the effective index of refraction for the fundamental transverse mode of the device is 3.6701+0.0092*i. The positive imaginary part of the effective index of refraction means the decay of the optical power within the tapered region. It should be pointed out that the self-focusing gradually takes effect with the increasing of the device’s power. When the optical power is small, self-focusing effect can be ignored. so in Fig. 2(b), the threshold current density of 8°-tapered device is the smallest.

Finally, it should be noted that the realistic description of the self-focusing effects is more complicated. The change of the refractive index depends on the temperature distribution and the carrier distribution, which calls for self-consistent solutions of the cavity mode, the temperature profile and the carrier distribution with the consideration of the reflection optical field due to the facets [18

18. S. Mariojouls, S. Morgott, A. SChflitt, M. Mikulla, J. Braunstein, G. Weimann, F. Lozes, and S. Bonnefont, “Modeling of the performance of high-brightness tapered lasers,” Proc. SPIE 3944, 395–406 (2000) [CrossRef] .

]. Such simulation which will give further information on the performance of tapered THz QCL is highly desired in the future studies of the tapered THz QCLs and tapered THz amplifiers.

4. Conclusion

In summary, we have investigated the effect of the tapered waveguide region on terahertz quantum cascade lasers. Although the tapered sections have no influence on the emitted spectral characteristics, the tapered angle has a prominent effect on the output powers and far-field laser beam of devices. The peak output power is higher and the horizontal far-field beam divergence of the tapered THz QCLs are narrower than those of ridged waveguide lasers without tapered region. Remarkably, there is an optimal tapered angle for our devices at about 5° ∼6°, which corresponds to the maximum peak output power and the minimum beam divergence. The theoretical simulation suggests that the self-focusing effect may play an important role in the devices with larger tapered angle. Thus, a appropriate tapered waveguide angle for THz QCL is required to avoid the occurrence of self-focusing effect which may worsen the device’s performance. The studies on the tapered lasers in this paper may inspire future investigations of THz power-amplifier and extend the applications of THz QCL.

Acknowledgments

This work is supported by Special-Funded Program on National Key Scientific Instruments and Equipment Development ( 2011YQ13001802), National Basic Research Program of China ( 2013CB632801/5) and CAEP under Grant No. 2012B0401061. The authors acknowledge Jianyan Chen, Ping Liang and Ying Hu for their help in device processing and measurement.

References and links

1.

J. Faist, F. Capasso, Sivco, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994) [CrossRef] [PubMed] .

2.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417, 156–159 (2002) [CrossRef] [PubMed] .

3.

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1, 517–525 (2007) [CrossRef] .

4.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. A. Nolde, D. C. Larrabee, I. Vurgaftman, and J. R. Meyer, “Broad-stripe, single-mode, mid-IR interband cascade laser with photonic-crystal distributedfeedback grating,” Appl. Phys. Lett. 92(7), 071110 (2008) [CrossRef] .

5.

J. N. Walpole, “Semiconductor amplifiers and lasers with tapered gain regions,” Optical and Quantum Electronics 28623–645 (1996) [CrossRef] .

6.

C. Pfahler, M. Eichhorn, M. T. Keleman, M. Mikulla, J. Schmitz, and J. Wagner, “Gain saturation and high-power pulsed operation of GaSb-based tapered diode lasers with separately contacted ridge and tapered section,” Appl. Phys. Lett. 89, 021107 (2006) [CrossRef] .

7.

H. Wenzel, K. Paschke, O. Brox, F. Bugge, J. Fricke, A. Ginolas, A. Knauer, P. Ressel, and G. Erbert, “10 W continuous-wave monolithically integrated master-oscillator power-amplifier,” Electron. Lett. 43(3), 160–161 (2007) [CrossRef] .

8.

D. Vijayakumar, O. B. Jensen, R. Ostendorf, T. Westphalen, and B. Thestrup, “Spectral beam combining of a 980 nm tapered diode laser bar,” Opt. Express 18(2), 893–898 (2010) [CrossRef] [PubMed] .

9.

L. Nähle, J. Semmel, W. Kaiser, S. Höfling, and A. Forchel, “Tapered quantum cascade lasers,” Appl. Phys. Lett. 91, 181122 (2007) [CrossRef] .

10.

S. Menzel, L. Diehl, C. Pflügl, A. Goyal, C. Wang, A. Sanchez, G. Turner, and F. Capasso, “Quantum cascade laser master-oscillator poweramplifier with 1.5 W output power at 300 K,” Opt. Express 19, 16229–16235 (2011) [CrossRef] [PubMed] .

11.

A. Lyakh, R. Maulini, A. Tsekoun, R. Go, C. Kumar, and N. Patel, “Tapered 4.7 μm quantum cascade lasers with highly strained active region composition delivering over 4.5 watts of continuous wave optical power,” Opt. Express 20(4), 4382–4388 (2012) [CrossRef] [PubMed] .

12.

P. Rauter, S. Menzel, A.K. Goyal, B. Gökden, C.A. Wang, A. Sanchez, G.W. Turner, and F. Capasso, “Master-oscillator power-amplifier quantum cascade laser array,” Appl. Phys. Lett 101, 161117 (2012) [CrossRef] .

13.

J.D. Kirch, J.C. Shin, C.-C. Chang, L.J. Mawst, D. Botez, and T. Earles, “Tapered active-region quantum cascade lasers (λ = 4.8 μm) for virtual suppression of carrier-leakage currents,” Electron. Lett. 48(4), 234–235 (2012) [CrossRef] .

14.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965) [CrossRef] .

15.

C. C. Wang, “Length-dependent threshold for stimulated Raman effect and self-focusing of laser beams in liquids,” Phys. Rev. Lett. 16, 344–346 (1966) [CrossRef] .

16.

J. Wang, W. D. Wu, X. L. Zhang, and S. Q. Duan, “Analysis of terahertz quantum cascade laser beam,” Information and Electronic Engineering 9, 365–368 (2011).

17.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002) [CrossRef] .

18.

S. Mariojouls, S. Morgott, A. SChflitt, M. Mikulla, J. Braunstein, G. Weimann, F. Lozes, and S. Bonnefont, “Modeling of the performance of high-brightness tapered lasers,” Proc. SPIE 3944, 395–406 (2000) [CrossRef] .

OCIS Codes
(140.3070) Lasers and laser optics : Infrared and far-infrared lasers
(230.0230) Optical devices : Optical devices
(260.5950) Physical optics : Self-focusing
(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 22, 2013
Revised Manuscript: June 6, 2013
Manuscript Accepted: June 13, 2013
Published: June 27, 2013

Citation
YanFang Li, Jian Wang, Ning Yang, Junqi Liu, Tao Wang, Fengqi Liu, Zhanguo Wang, Weidong Chu, and Suqing Duan, "The output power and beam divergence behaviors of tapered terahertz quantum cascade lasers," Opt. Express 21, 15998-16006 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15998


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References

  1. J. Faist, F. Capasso, Sivco, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264, 553–556 (1994). [CrossRef] [PubMed]
  2. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417, 156–159 (2002). [CrossRef] [PubMed]
  3. B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics1, 517–525 (2007). [CrossRef]
  4. C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. A. Nolde, D. C. Larrabee, I. Vurgaftman, and J. R. Meyer, “Broad-stripe, single-mode, mid-IR interband cascade laser with photonic-crystal distributedfeedback grating,” Appl. Phys. Lett.92(7), 071110 (2008). [CrossRef]
  5. J. N. Walpole, “Semiconductor amplifiers and lasers with tapered gain regions,” Optical and Quantum Electronics28623–645 (1996). [CrossRef]
  6. C. Pfahler, M. Eichhorn, M. T. Keleman, M. Mikulla, J. Schmitz, and J. Wagner, “Gain saturation and high-power pulsed operation of GaSb-based tapered diode lasers with separately contacted ridge and tapered section,” Appl. Phys. Lett.89, 021107 (2006). [CrossRef]
  7. H. Wenzel, K. Paschke, O. Brox, F. Bugge, J. Fricke, A. Ginolas, A. Knauer, P. Ressel, and G. Erbert, “10 W continuous-wave monolithically integrated master-oscillator power-amplifier,” Electron. Lett.43(3), 160–161 (2007). [CrossRef]
  8. D. Vijayakumar, O. B. Jensen, R. Ostendorf, T. Westphalen, and B. Thestrup, “Spectral beam combining of a 980 nm tapered diode laser bar,” Opt. Express18(2), 893–898 (2010). [CrossRef] [PubMed]
  9. L. Nähle, J. Semmel, W. Kaiser, S. Höfling, and A. Forchel, “Tapered quantum cascade lasers,” Appl. Phys. Lett.91, 181122 (2007). [CrossRef]
  10. S. Menzel, L. Diehl, C. Pflügl, A. Goyal, C. Wang, A. Sanchez, G. Turner, and F. Capasso, “Quantum cascade laser master-oscillator poweramplifier with 1.5 W output power at 300 K,” Opt. Express19, 16229–16235 (2011). [CrossRef] [PubMed]
  11. A. Lyakh, R. Maulini, A. Tsekoun, R. Go, C. Kumar, and N. Patel, “Tapered 4.7 μm quantum cascade lasers with highly strained active region composition delivering over 4.5 watts of continuous wave optical power,” Opt. Express20(4), 4382–4388 (2012). [CrossRef] [PubMed]
  12. P. Rauter, S. Menzel, A.K. Goyal, B. Gökden, C.A. Wang, A. Sanchez, G.W. Turner, and F. Capasso, “Master-oscillator power-amplifier quantum cascade laser array,” Appl. Phys. Lett101, 161117 (2012). [CrossRef]
  13. J.D. Kirch, J.C. Shin, C.-C. Chang, L.J. Mawst, D. Botez, and T. Earles, “Tapered active-region quantum cascade lasers (λ = 4.8 μm) for virtual suppression of carrier-leakage currents,” Electron. Lett.48(4), 234–235 (2012). [CrossRef]
  14. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev.137, A801–A818 (1965). [CrossRef]
  15. C. C. Wang, “Length-dependent threshold for stimulated Raman effect and self-focusing of laser beams in liquids,” Phys. Rev. Lett.16, 344–346 (1966). [CrossRef]
  16. J. Wang, W. D. Wu, X. L. Zhang, and S. Q. Duan, “Analysis of terahertz quantum cascade laser beam,” Information and Electronic Engineering9, 365–368 (2011).
  17. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun.202, 17–20 (2002). [CrossRef]
  18. S. Mariojouls, S. Morgott, A. SChflitt, M. Mikulla, J. Braunstein, G. Weimann, F. Lozes, and S. Bonnefont, “Modeling of the performance of high-brightness tapered lasers,” Proc. SPIE3944, 395–406 (2000). [CrossRef]

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