## Abstract

We propose that self-pumped degenerate four-wave mixing may be used to produce novel diode laser systems where lasing is based on non-linear guiding of the laser beams inside the active semiconductor. The fundamental process responsible for the interaction is spatial hole burning in semiconductor amplifiers. The gain and index gratings created by the modulation of the carrier density in the active gain medium lead to selective amplification of one spatial mode and suppression of all other modes. This mechanism allows the laser system to be operated far above its threshold with an almost diffraction limited output beam. The third order nonlinear susceptibility of the non-linear material, which determines the strength of the induced gratings, depends on the angle between the interacting beams in the four-wave mixing configuration. It is shown theoretically that a narrow range of angles exist where the induced gratings are strong and where mode suppression of higher order spatial modes are obtained simultaneously. Experimental evidence sustaining these findings is given.

©2005 Optical Society of America

## 1. Introduction

Active semiconductors are attractive materials for obtaining strong four-wave mixing interaction. Using collinear, nearly degenerate four-wave mixing, conjugated reflectivities as high as 10^{3}–10^{4} have been obtained [1]. Broad-area lasers with large stripe widths permit optical phase conjugation with a spatially non-degenerate mixing geometry. In this geometry the probe beam may be injected at a small angle with respect to the pump beam thereby causing emission of amplified four-wave mixing signals in different spatial directions [2]. This approach is attractive for generation of phase conjugate signals and leads to phase conjugate reflectivities as high as 165 % [3]. Permanent gratings inside the active semiconductor have previously been used to obtain a single spatial mode output beam from a semiconductor laser [4,5]. This so-called angled-grating distributed feedback (α-DFB) laser comprises a broad-area gain stripe with a permanently embedded grating so that the stripe and the grating are disoriented from the cleaved facets by a substantial angle. The α-DFB laser supplies both feedback and selective spatial filtering, thus enforcing single spatial mode oscillation.

In this paper we propose that four-wave mixing in a broad-stripe amplifying semiconductor may be used to establish a laser cavity where the output beam is a result of diffraction in dynamic gain and refractive index gratings. The term dynamic refers to the fact that the gratings are not permanently written into the gain medium, i.e., they are created through the interaction of the four beams inside the gain medium and, therefore, adapt to changes resulting from changes, for example, in temperature or intensity. Such cavity enables use of a large volume of the diode laser with selective amplification of one spatial mode. Conventional high-power, broad-area diode lasers suffer from poor beam quality since the laser is oscillating in many spatial modes and with detrimental effects such as lateral lasing and filamentation. In the following we show that the presence of the four-wave mixing process in the laser medium eliminates these detrimental effects and, furthermore, makes it possible to use high-power diodes with extraordinarily large stripe widths. We demonstrate theoretically and experimentally that when the diode is operated in the four-wave mixing configuration, the laser system oscillates in a single spatial mode with an almost diffraction-limited output.

The concept of the four-wave mixing laser presented in the present paper is somewhat related to the principles in the α-DFB laser [4,5]. However, in contrast to the α-DFB laser we use dynamic four-wave mixing gratings instead of permanent gratings. The advantages of using dynamic gratings are that the output beam is optimized automatically and, furthermore, that these gratings dynamically compensate for misalignment of the cavity and for thermal effects. As a result, the four-wave mixing laser leads to increased long-term stable operation with high spatial coherence. Finally, the four-wave mixing diode laser provides a simple configuration that effectively couples light out from the semiconductor gain medium.

## 2. Laser action based on four-wave mixing in semiconductor amplifiers

The configuration of the four-wave mixing semiconductor laser cavity is shown in Fig. 1. Four waves interact inside the laser cavity. Interference between the four beams leads to the formation of gain gratings and refractive index gratings and, as a result, the laser beams are diffracted inside the semiconductor.

The four-wave mixing is used in a self-pumped configuration and, therefore, no external pump beams are needed. The beam *A*
_{3} is generated from a reflection at the external mirror M and this beam diffracts in the four-wave mixing gratings into *A*
_{2}. From Ref. [3] it is well known that four-wave mixing in semiconductor amplifiers leads to phase conjugate reflectivities much higher than unity. The phase conjugate four-wave mixing reflectivity is the origin of the laser action in our configuration. The laser configuration in Fig. 1 consists of one external, ordinary mirror (M) and one four-wave mixing mirror with reflection coefficients *R*
_{DFWM}=|*A*
_{4}(*L*) |^{2}/|*A*
_{3}(*L*) |^{2}. Using a high-reflectivity coating at one facet of the diode amplifier, a strong beam *A*
_{1} may be produced from a single internal reflection on the back facet of the laser cavity. The output beam of the laser system *A*
_{1}(*L*) is a result of the amplification and four-wave mixing diffraction processes. Due to the angle- and wavelength selectivity of the four-wave mixing gratings, an output beam with high spatial and temporal coherence is produced. Spatial and temporal filters may be added in front of the external, ordinary mirror in order to increase the spatial mode selectivity further.

## 3. Dynamic gratings in semiconductor amplifiers

The dynamic gratings in the semiconductor amplifier are created by modulation of the carrier density in the active gain medium. This carrier density is governed by the following rate equation [6]:

where *I* is the injected current, *q* is the electron charge, *V* is the active volume, *N* is the carrier density, *τ*
_{s} is the spontaneous recombination lifetime, *D* is the ambipolar diffusion constant, *E*
_{0} is the total optical field, and, finally, *g*(*N*) is the gain that in our analysis is assumed to vary linearly with carrier density, i.e. *g(N)=a(N-N _{0})* where

*a*and

*N*

_{0}are constants. Temperature variations across the stripe width (coordinate ‘y’ in Fig. 1) that may occur at large injection currents have not been taken into consideration. This effect will only have minor influence on the four-wave mixing process in the case of broad area amplifiers. However, in the case of multiple stripe arrays this effect has to be included [7].

In the configuration in Fig. 1 the origin of the gain and index gratings is the modulation of the carrier density due to interference between *A*
_{1} and *A*
_{4} and between *A*
_{2} and *A*
_{3}. Due to diffusion of carriers these transmission gratings are much stronger than the reflection gratings in the four-wave mixing geometry [8]. Thus, the carrier density that leads to the formation of the gratings may be written as:

where Δ*k=k _{1y}-k_{4y}=k_{3y}-k_{2y}* and 〈

*N*〉 is the average carrier density. In the following perturbation analysis it is assumed that Δ

*n*<<〈

*N*〉. Inserting Eq. (2) in Eq. (1) we find after some simple calculations that the induced carrier modulation Δ

*n*is given by:

where |*E _{s}*|

^{2}=(

*ħω*

_{0})/(

*aτ*) is the saturation intensity, and |

_{s}*E*|

_{0}^{2}=|

*E*

_{1}|

^{2}+|

*E*

_{2}|

^{2}+|

*E*

_{3}|

^{2}+|

*E*

_{4}|

^{2}the total intensity, Λ the fringe spacing, and ω

_{0}the optical frequency. In deriving Eqs. (3–4) it is assumed that Δ

*n*<<

*N*

_{0}. The material response is given by the susceptibility [9]:

$$=-\frac{nc}{\omega}\left(\beta +i\right)a\left[\left(\u3008N\u3009-{N}_{0}\right)+\Delta n\mathrm{exp}\left(i\Delta ky\right)\right]$$

where we have inserted *g*(*N*)=*a*(*N-N*
_{0}) together with *N* from Eqs. (2)–(4). The quantity *β* is the anti-guiding parameter, see e.g. [9], and *n* is the index of refraction. The amplitude of the spatially varying part *χ _{4WM}* of the susceptibility responsible for the four-wave mixing process is given by:

Eq. (6) constitutes the result of the susceptibility needed for solving the nonlinear wave equation including the contribution from index and gain gratings, respectively. The strength of the non-linear gratings created in the active laser medium is given by Eq. (6) and for low non-linearities the four-wave mixing diffraction efficiency is proportional to |*χ _{4WM}*|

^{2}. Since Λ=

*λ*/(2sin(

*θ*/2)), Eq. (6) can be recast:

where *χ _{4WM, opt}* is proportional to the optimum value of the non-linear susceptibility when

*θ*=0 and |

*E*

_{0}|

^{2}<<|

*E*|

_{S}^{2}in Eq. (6). Moreover, it should be noted that in deriving Eq. (7) we have assumed that the angle

*θ*between the interacting beams is small. The quantity

*χ*in Eq. (7) determines the strength of the gratings, and in Fig. 2 we have shown

_{4WM}*χ*versus

_{4WM}*θ*calculated from Eq. (7) for different degrees of saturation. The washout of the induced grating due to carrier recombination and diffusion becomes significant as the angle

*θ*increases. The non-linear susceptibility has its largest amplitude for small angles and according to Eq. (7) the amplitude of the spatially varying part of the susceptibility is reduced to half-the-maximum at an angle corresponding to:

Consequently, *θ* should be less than *θ*
_{½} if strong non-linear four-wave mixing interaction in-side the active semiconductor is required.

## 4. Mode suppression factor

The diffraction process in the induced gratings is shown in Fig. 1. The output beam is a result of diffraction in the induced gratings inside the semiconductor amplifier. For small angles the phase difference *δ* between a wave diffracted at the front facet *z*=0 and at the back facet *z=L* is given by, see e.g., [10]:

where *θ* is the diffraction angle in free space. If this phase difference *δ* is much larger than unity only the beam incident at the Bragg angle will lead to a diffracted beam and other laser modes will be effectively suppressed. On the other hand, if δ is much smaller than unity all laser modes are diffracted in the grating with almost the same efficiency. As a result, *δ* plays the role as mode suppression factor. In practice, *δ* must be somewhat larger than 2 π to have effective suppression of different axial modes in the broad-area amplifier.

This, in turn, implies a minimum critical angle at which mode suppression occurs depends on the length of the amplifier and is determined by:

where we have inserted *δ=2π* in Eq. (9).

## 5. Output angle condition

By combining the findings in Sections 3 and 4, respectively, we arrive at the following important conclusion that the output angle *θ* must obey the following condition:

provided mode suppression and strong four-wave mixing interaction are present at the same time.

The implication of the output angle condition can be illustrated by inserting the following data consistent with our experimental setup. Inserting in Eqs. (8) and (11)
*L*=1 mm, *n*=3.4, *D _{a}*=13cm

^{2}/s [6], |

*E*

_{0}|

^{2}=0.5×|

*E*|

_{S}^{2}, and

*τ*

_{s}=1 ns (value for GaAlAs), we obtain

*θ*=4.2° and

_{crit}*θ*

_{½}=8.0° at wavelength λ=810 nm provided the intensity |

*E*

_{0}|

^{2}<< |

*E*|

_{S}^{2}. Accordingly, we conclude that the output angle

*θ*must be larger than 4.2° in order to have good mode suppression and simultaneously it must be less than

*θ*

_{½}=8.0° to have strong gratings in the semiconductor. As the intensity |

*E*

_{0}|

^{2}increases the critical angle

*θ*

_{½}increases and, therefore, it is expected that the optimum angle is moved towards higher values as the output power of the laser increases.

In the previous discussions, it has tacitly been assumed that the total intensity |*E*
_{0}|^{2} is a function of position inside the gain medium. Eq. (11) should, therefore, be considered a first-order approximation. However, by assuming that on average the intensity does not exceed saturation at any position inside the gain medium, we may use Eq. (11) to estimate the angles at which mode suppression and strong four-wave mixing interaction are present at the same time. In Fig. 3, the output angle *θ* is plotted as a function of the degree of saturation (upper boundary in Eq. (11)) using the same parameters as above. The lower boundary limit is also plotted using the cavity length as parameter (this boundary is independent of the saturation). It is important to note that even though the (averaged) saturation is varied from its minimum to its maximum value, the change in upper limit varies within a factor of √2. Hence, using Eq. (11) in the limit of the low intensity approximation leads in the worst case to a maximum deviation of a factor of √2.

## 6. Experiments

Experimental evidence of the theoretical model has been obtained with a configuration similar to Fig. 1. The experiment was carried out with a 1 mm long GaAlAs gain-guided amplifier with a 200 µm wide gain junction. The back facet of the diode amplifier was provided with a coating with a reflectivity of more than 0.99, and the output facet of the diode amplifier was coated with an antireflection coating with reflectivity less than 0.1 %. The laser light was coupled out of the diode amplifier by a lens with a focal length of 4.5 mm and two cylindrical lenses. All lenses have had a broadband anti-reflection coating (R<1%) in order to minimize losses. The mirror M was a dielectrically coated mirror with a reflectivity R>0.999.

The results are shown in Fig. 4 and evidently they show that a narrow laser beam is established on the external mirror M. The angular position of this beam is in agreement with the output angle condition established in Section 5. Fig. 4 shows the angular intensity profile of the output beam, i.e., the intensity in arbitrary units as a function of angle *θ* (unit degrees). The intensity profiles of Fig. 4(a)-Fig. 4(c) have been measured with different pumping levels: In Fig. 4(a), the pump current is *I*=0.95 A, in Fig. 4(b) the pump current is *I*=1.23 A, and in Fig. 4(c) the pump current is *I*=1.40 A. Fig. 4(a)-Fig. 4(c) show a dominant peak and diffraction patterns at both sides of the peak due to diffraction in the induced four-wave mixing grating. In Fig. 4(c) the output power is emitted around an angle *θ*=6.9° and the output power is 620 mW. The full-width-half-maximum of the central peak in Fig. 4(c) is 0.61°, which is close to the diffraction limit. Fig. 4(d) shows the measured intensity profile of the above configuration where the light path between the external mirror M and the laser diode was blocked. The pump current in Fig. 4(d) was *I*=1.40 A. Thus, Fig. 4(d) clearly shows that no signal is observed when the mirror M is blocked, i.e. when the induction of gain and refractive index gratings in the diode amplifier is prevented. The emitted angle *θ*=6.9° in Fig. 4(c) is in good agreement with the theoretical prediction of the optimum output angle condition between 4.2° and 8.0° found from Eq. (11). Furthermore, in Fig. 4(b)-Fig. 4(c) it is observed that the optimum angle is shifted towards higher angles as the output is increased. This observation is also in qualitatively agreement with the theoretical predictions in Section 5.

## 7. Conclusion

We have proposed that self-pumped degenerate four-wave mixing may be used for the development of a new kind of diode laser system. The mechanism is based on dynamic gratings induced by the third order nonlinear material response in the active semiconductor gain material and it may be controlled by external feedback from an ordinary mirror. A theoretical model is established and the optimum output angle is shown to fulfill certain conditions in order to facilitate strong formation of gratings and effective mode suppression of higher order modes simultaneously. Both conditions must be fulfilled in order to have the desired high-power operation in a nearly diffraction-limited beam. Experimental results in a GaAlAs gain-guided amplifier are carried out and they are found to be in agreement with the theory. Finally, these findings lead to further understanding of the fundamental processes inside the lasing medium involved in the operation of for example external feedback lasers.

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