1. Introduction

The Kerr effect is at the basis of many important device concepts like all-optical switching [1], optical limiting [2] and soliton mode-locking of lasers and microresonators [3,4]. The capability to accurately measure and model the nonlinear refractive index changes associated with the Kerr effect is crucial for improved device operation, where a specifically tailored nonlinear refractive index is required, e.g. for the intensity-dependent Kerr lensing. Several measurement schemes have been developed in the past for the characterization of the nonlinear refractive index [5–7], with the Z-scan technique [8] being undoubtedly the most prominent one due to its simplicity and high sensitivity. This method continues to be of high experimental value with the rise of novel material classes like graphene and other two-dimensional semiconductors, which often exhibit a very strong refractive nonlinearity [9–11].

Kerr-lens mode-locked Ti:Sapphire lasers have dominated the field of ultra-short high-power mode-locked lasers since their initial discovery nearly three decades ago [12,13]. In these lasers, the intensity-dependent refractive index of the gain crystal leads to self-focusing of the laser beam for high intensities. When part of the continuous-wave (cw) beam profile is suppressed by inserting a slit into the cavity or reducing the pump spot on the crystal, an artificial ultrafast saturable absorber can be formed and can lead to very short pulse emission down to a few fs in duration [14,15].

In recent years, a similar behavior has been observed in saturable-absorber-free mode-locked vertical-external-cavity surface-emitting lasers (VECSELs) [16–20] with very good mode-locking properties being reported for quantum-well-based (QW) [21] as well as for quantum-dot-based VECSELs [22]. These “self-mode-locked” VECSELs could lead to cheap and compact, high-power pulsed sources with sub-GHz to multi-GHz repetition rates for frequency metrology, spectroscopy and nonlinear imaging, rendering expenses for and limitations of saturable-absorber mirrors obsolete. Thus, one can expect self-mode-locking to open up new application scenarios for these techniques. However, there has been a controversy in this field whether self-mode-locking in VECSELs has been properly demonstrated, particularly addressed by [23], and whether it represents a technique capable of delivering stable mode-locking, as evidenced in the history of this field [24]. The lack of understanding concerning its possible driving mechanisms remains the key motivation behind ongoing attempts to identify and quantify intrinsic VECSEL-chip related nonlinearities that could promote self-mode-locking, in particular the nonlinear refractive index properties, which are essential for Kerr-lens mode-locking.

Here, we present wavelength-dependent measurements of the nonlinear refractive index around the resonance of a multi-quantum-well VECSEL gain structure designed for lasing at around 960 nm. We perform Z-scan measurements in the range of 930 nm to 975 nm in order to reveal the interplay of nonlinear refraction and absorption in this spectral range. Changing the angle of incidence allows us to discuss the role of the longitudinal confinement factor, i.e. the cavity resonance, in the observed enhancement of the nonlinearity. We report a negative nonlinear refraction up to 5x10⁻¹² cm²/W in magnitude close to the laser design wavelength. This leads to a defocusing lens which is sufficient to perturb the cavity beam and possibly give rise to Kerr-lens mode-locking. Figure 1(a) displays a possible cavity configuration to exploit this measured Kerr-lens for mode-locking where the defocusing lens in the gain chip leads to beam narrowing at the end mirror, thus, favoring mode-locking when a slit is inserted there. A similar cavity has been used in [21] to obtain self-mode-locking and has been investigated numerically in [25] to support in principle Kerr-lens mode-locking for a defocusing nonlinear lens in the gain chip.

 

Fig. 1 VECSEL chip characteristics: (a) Illustration of a Kerr-lens mode-locked VECSEL with the dashed lines indicating the cw beam profile while the shaded area traces the beam profile when altered by the nonlinear defocusing lens in the VECSEL chip. Such an intensity-dependent lens can lead to beam narrowing at the end mirror of the cavity and possibly favor mode-locking when a slit inserted there adds losses to the cw beam by truncating it laterally. (b) Chip structure with standing-wave electric field at the design wavelength of 960 nm. (c) Band-gap configuration of the different materials used in the gain chip compared to the stopband of the chip and the investigated wavelength range (930 – 975 nm). (d) Measured reflectivity spectrum, surface and edge photoluminescence (PL), and calculated longitudinal confinement factor (LCF) of the investigated VECSEL chip.

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2. Investigations on the self-mode-locking effects in VECSELs

The claim of the observed “self-mode-locking” behavior being governed by Kerr-lens mode-locking [17] has been supported by investigations of the nonlinear refractive index in VECSEL gain structures [26–28], reporting refractive index changes sufficiently strong to be able to perturb the cavity beam profile strongly. Although not being a definite proof that self-mode-locking is indeed caused by Kerr-lensing these results are suggestive and call for a more detailed understanding of the nonlinear lensing properties in VECSELs. In particular, prior measurements were exclusively performed at arbitrarily selected (experimentally available) single wavelengths. Such experiments were not taking into account the microcavity resonance as well as the strong dispersion of the nonlinear refractive index around the band edge, which is characteristic for contributions to the nonlinear refractive index, both, from the bound-electronic Kerr effect (BEKE) existing below the bandgap [29] and from free-carrier-related nonlinearities (FCN) existing above the band edge [30].

While the nonlinear optical properties of semiconductors below the band gap have been extensively studied and modeled using the Z-scan method [29,31,32], nonlinear optical processes above and around the band gap have mostly been investigated by pump-probe techniques [30,33] or linear absorption measurements [34]. These methods cannot access the nonlinear refractive index directly but use nonlinear Kramers-Kronig relations to calculate it from nonlinear absorption spectra. Very few reports on direct measurements of the nonlinear refractive index properties of QW structures exist so far, which solely can account fully for nonlinear pulse propagation of a single pulse through the medium [35,36].

Although giving a rough estimate of the strength of the resonant nonlinearity in multiple-quantum-well structures, these measurement results [33–36] cannot arbitrarily be transferred to other sample designs as the nonlinear refractive index constitutes an effective one, composed of different material nonlinearities and their respective interaction containing both BEKE and FCN. Thus, the direct determination of the nonlinear refractive index by Z-scan measurements becomes inevitable whenever its precise value for a particular semiconductor heterostructure is of interest. We define the effective nonlinear refractive index n₂ as

Complementary to the definition of the intensity dependent refractive index n₂ the nonlinear absorption coefficient β can be defined as

The goal of this work is the determination of the effective n₂ of a VECSEL chip as well as of the nonlinear absorption coefficient β acting as possible loss mechanism.

3. VECSEL chip design and Z-scan measurements

The investigated VECSEL chip sketched in Fig. 1(b) consists of 20 (InGa)As/Ga(AsP) quantum wells embedded within GaAs spacer layers to provide partial overlap of the antinodes of the standing wave electric field in the microcavity with the quantum wells. Subsequent 30 GaAs/AlAs distributed Bragg reflector (DBR) pairs provide the high reflectivity needed for efficient laser operation together with external mirrors. After the DBR a thick GaAs spacer layer is grown with the intention of achieving a double-resonance for both the laser wavelength at normal incidence and an 888-nm pump under an incident angle of 30°. Since at this wavelength only the QWs are pumped, the absorption length is drastically reduced compared to a barrier pumping scheme. To compensate for this, the DBR stopband is designed to cover also the pump wavelength, and the field enhancement due to the resonance leads to an increased pump absorption. The chip is designed for lasing operation at around 960 nm. Figure 1(c) shows the band gap alignment of the various materials composing the gain chip. It can be seen that the probe wavelengths, which range from 930 nm to 975 nm, are scanning across the quantum-well band gap, thus, our measurements are particularly sensitive to the nonlinear refractive index changes which are characteristic for near-resonance probing. The wavelength-dependent longitudinal confinement factor (LCF) characterizes the strength of a microcavity’s standing-wave electric field at the location of the quantum wells (see Appendix B) and is depicted together with the reflectivity spectrum, the surface as well as the edge photoluminescence (PL) in Fig. 1(d).

For Z-scan measurements, the chip structure was mounted on a piece of copper without any active cooling. A wavelength tunable, mode-locked Ti:Sapphire laser emitting pulses of approximately 150 fs pulse length and a spectral width of about 8 nm was used to probe the chip structure at different center wavelengths around the design wavelength of the gain chip.

For each center wavelength, the pulse length was determined with a frequency-resolved optical gating (FROG) device and the laser spectrum was monitored with an optical spectrometer.

A lens with 50 mm focal length was used to focus the probe beam down to a spot size radius of 12 µm which was confirmed by a beam profile measurement around the focus of the probe beam. After reflection from the sample and a folding mirror, which were both mounted on a translation stage, a beam splitter and two detectors were used to record the position-dependent transmittance of the probe laser when the sample was moved through the focus of the lens. One detector was covered by a partially closed aperture, which was aligned for maximal transmission. The second detector collected the whole probe beam using a focusing lens. The trace recorded with the first detector will be referred to as closed scan, which is sensitive to, both, nonlinear lensing and nonlinear absorption. The trace recorded with the first detector will be referred to as open scan, which is only sensitive to nonlinear absorption. Following [8], we obtain the final Z-scan curve by dividing the closed scan by the open scan, performing a background subtraction with a low-intensity scan and normalizing to a transmittance of 1 for transmission at positions far away from the focus. Figure 2(b) shows exemplary Z-scan measurements at a center wavelength of 955 nm for different peak probe intensities at 20° angle of incidence together with the respective fits. They show good agreement between model and experiment including an approximately symmetric decrease of peak and valley transmission for decreasing peak probe-power density. The origin of the slight asymmetry of the Z-scan measurements with respect to peak and valley magnitude is not completely understood but might stem from a not-ideal Gaussian probe beam profile or aberrations induced by the focusing lens. The decrease in transmission close to the focus in the open scan (Fig. 2(c)) is attributed to two-photon absorption.

 

Fig. 2 Experimental Z-scan setup and data: Different neutral-density (ND) filters are used to systematically vary the incident probe power on the VECSEL chip. The sample is translated through the focus of the 150-fs pulsed probe beam, which is chopped and analyzed by two detectors using a lock-in scheme. (b) Exemplary Z-scan measurements at 955 nm for different peak probe intensities irradiated at an angle of 20° are shown, with the corresponding open scan recorded by detector 1 (c). The solid lines are the respective fits to the measurement data according to the models from [8].

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The angle of incidence of the probe beam on the VECSEL chip was varied in the course of the experiment from 10 to 20 and 30°, in order to reveal the influence of the microcavity on the measured nonlinearity. At each angle of incidence, Z-scan measurements were performed for different probe center wavelengths around the chip’s design wavelength and varying probe intensities.

4. Experimental results and discussion

In order to determine very accurately the value of nonlinear absorption and refraction in the VECSEL chip, we performed Z-scan measurements for different probe-peak intensities at center wavelengths from 930 to 975 nm. If a refractive (or an absorbing) third-order nonlinearity is present, the nonlinear on-axis phase shift ΔΦ₀ (or the normalized nonlinear on-axis absorption q₀) will vary linearly with peak probe power density. In such case, the strength of the nonlinearity is proportional to the slope of the nonlinear phase shift (or the normalized nonlinear absorption) with respect to the peak probe power density.

Figure 3 shows, both, the normalized nonlinear on-axis absorption q₀ (Fig. 3(a)) and the nonlinear on-axis phase shift ΔΦ₀ (Fig. 3(b)) as a function of the incident peak power density extracted from the open and closed Z-scan curves. The very good agreement with the linear fit shows that measured nonlinear absorption and nonlinear phase shift are indeed mainly caused by a third-order nonlinearity. It is important to note that nonlinear-refractive-index changes caused by two-photon-absorption-generated free carriers would appear as an effective fifth-order nonlinearity [8] causing a deviation from the linear fit. Due to the good agreement to the linear fit, this mechanism can be neglected for our measurements.

 

Fig. 3 Coefficients for the extraction of nonlinear absorption and refraction: (a) Normalized nonlinear absorption q₀ and (b) nonlinear on-axis phase shift ΔΦ₀ as a function of the incident peak probe intensity for different probe center wavelengths irradiated at 20°. The errors for the respective extracted magnitudes represent the fit tolerance and were determined by varying the fit parameter (q₀ or ΔΦ₀) until the fit model did not match the measurements anymore (similar to [37]).

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Using the relations β = dq₀/dI₀ 1/L_eff and n₂ = dΔΦ₀/dI₀ 1/(kL_eff), values for the nonlinear absorption β and refraction n₂ can be extracted. These are plotted in Fig. 4 for the incidence angles of 10, 20 and 30°.

 

Fig. 4 Wavelength and angle-dependent nonlinear absorption and nonlinear refraction: (a) Nonlinear absorption β and (b) nonlinear refraction n₂ as a function of the wavelength (left axes), measured for the incidence angles of 10, 20 and 30°. The lines between the measurement points may serve as guides to the eyes. Arrows atop the diagram indicate the wavelength of the QW PL (cf. Fig. 1(d)) as well as the different angle-dependent LCF peaks. The corresponding surface PL at given angle of incidence, which is subdue to the microcavity resonance, is plotted in both graphs for comparison (shaded plots) as well as the corresponding reflectivity spectra (line plots), both normalized to 1 and with respect to the right axes. An inset represents the probe geometry with respect to the VECSEL chip. The errors were determined by performing a linear least square fit to the normalized nonlinear absorption q₀ and nonlinear phase shift ΔΦ₀ as a function of probe intensity (as displayed in Fig. 3) and taking the 95% confidence interval of the fit as error.

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For both nonlinear absorption and nonlinear refraction, a strong enhancement of the nonlinearity can be observed at wavelengths between 950 to 960 nm. Strikingly, this corresponds very well to the field enhancement at the QWs due to the microcavity resonance, which is represented by the surface photoluminescence peak at that on-chip angle and is also plotted into Figs. 8(a) and 8(b) for comparison. To verify that the microcavity resonance, i.e. the Fabry-Pérot cavity’s filter function, is responsible for this effect, we have performed the measurements for different angles of incidence and find that the peak of the nonlinearity spectrum follows the corresponding surface PL spectrum peak for different angles. This unambiguously demonstrates that the microcavity resonance is strongly contributing to the enhancement of the respective nonlinearity. This enhancement takes place slightly below the linear absorption band edge, which can be estimated from the reflectivity spectra plotted in Fig. 4 for comparison. We consider the contribution of the GaAs spacer layer negligible with respect to the contributions of the (InGa)As/Ga(AsP) gain region, as the theoretical nonlinear refractive index of GaAs [29] is in the order of −6x10⁻¹³ cm²/W in the investigated wavelength range and thus significantly smaller than the effective n₂ values measured here. The absolute magnitude of the effective nonlinear refractive index rises up to 5x10⁻¹² cm²/W, which is an order of magnitude larger than reported in [28]. This can be attributed to the fact that the authors did not use a probe wavelength close to the maximum of the microcavity resonance. Moreover, the chip exhibited nearly half the number of QWs compared to our chip design.

The strong tunability of the nonlinearity with the microcavity resonance indicates that the (InGa)As/Ga(AsP) material system, which showed best self-mode-locking behavior so far [21] can be possibly operated in the self-mode-locking regime over a large wavelength range exceeding 10 nm thus supporting both broad-band operation and tunability. This is consistent with the results of [35] for a multiple-quantum-well structure which showed a comparable bandwidth of strong nonlinear refraction (up to 8.5x10⁻¹⁰ cm²/W in magnitude) where no microcavity effect was present.

In order to evaluate our measurement results with respect to possible self-mode-locking in VECSELs due to the Kerr effect, we plot the ratio of real and imaginary part of the third-order nonlinearity χ⁽³⁾,

 

Fig. 5 Ratio of real and imaginary part of the measured third order nonlinearity χ⁽³⁾ and typical focal length of a Kerr-lens: The ratio of nonlinear refraction with respect to nonlinear absorption is plotted as a function of the wavelength in the left column of the diagram for the three different angles of incidence. Additionally, the focal length, which would occur in a typically self-mode-locked VECSEL configuration, is plotted here in the right column. The error bars stem from the errors in nonlinear absorption and refraction as depicted in Fig. 4.

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Here, a nearly constant ratio of real and imaginary part of χ⁽³⁾ shows that both n₂ and β are equally affected by the cavity-resonance enhancement. In contrast to this, in pure quantum-well structures without any cavity, this ratio varies considerably as a function of the wavelength around the bandgap [35]. However, optical loss by two-photon absorption is quite significant in our measurements when rising up to more than 15% at the highest probe intensity in Fig. 2(c). Nevertheless, this effect might be reduced in a pumped system, in which population inversion is maintained by the typical above-band high-power continuous-wave pumping scheme. Also, longer pulses – typical pulse lengths of self-mode-locked VECSELs are around 1 ps [21,22]– might reduce the occurrence of two-photon absorption [38].

The typical focal length of a Kerr-lens, which would occur in a self-mode-locked VECSEL cavity as reported in [21], with the here measured value of nonlinear refraction is plotted in the same diagram (right axis of Fig. 5). For sub-ps pulses with 1-kW peak power, the focal length of the induced Kerr nonlinearity should be in the order of magnitude of the laser cavity length, which ranges usually between 10 to 30 cm. Our calculated values are between a few cm up to 30 cm and more. This information supports the attribution of the observed self-mode-locking being Kerr-lens mode-locking and is key to designing a cavity where the angle of incidence of the laser mode on the gain chip is, apart from geometric constraints, a free design parameter in a V- or Z-cavity.

5. On the origin of nonlinear lensing and the implications for mode-locking

The respective contributions of FCN and BEKE to the total, measured nonlinearity would define the pulse-shaping mechanism in a Kerr-lens mode-locking scenario due to the different response times. FCN has a slower response time, which is mostly due to the ps relaxation time of excited carriers, compared to BEKE, which can be regarded as instantaneous with respect to fs pulses. Both FCN and BEKE exhibit characteristic dispersions when regarded separately as depicted in Figs. 6(a) and 6(c), with both effects showing strongest changes close to the band gap.

 

Fig. 6 Illustration of possible saturable absorber mechanisms shaping the pulse: A dominating bound-electronic contribution (BEKE) in the nonlinear response, with its dispersion given in (a) as adapted from [29], would lead to a fast saturable absorber mechanism (red solid line in the schematic time trace of action shown in (b)) similar to traditional Kerr-lens mode-locking. In contrast, a dominating free-carrier-related nonlinear response (FCN), with its dispersion around the band gap given in (c) for different carrier densities as adapted from [30], would lead to slow saturable absorber action (green dashed line in (b)) where pulse formation relies on its combination with fast gain saturation (black solid line in (b)).

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In the theory described by [29], nonlinear refraction is inherently related to nonlinear absorption by the nonlinear Kramers-Kronig relations. Although in most resonant excitation-and-probe scenarios, carrier-related nonlinearities are believed to dominate the third-order nonlinear response [39], bound-electronic contributions even continue to play a role closely above the band gap when no significant excitation or deexcitation of carriers occur [40]. Strong deexcitation of free carriers could be described with an additional use of the Bloch equations. This has been done in [41] and showed that a probe pulse will indeed experience a significant phase shift both from FCN and BEKE with the same order of magnitude when the semiconductor material is excited with a high-intensity pump pulse. These findings have also been supported by comparison to experiments [42].

This indicates that also in mode-locked VECSELs a non-negligible amount of BEKE might be present. The occurrence of two-photon absorption in our measurements, also above the linear absorption edge, further supports this argument as it indicates that probe-power densities are sufficiently high to significantly perturb pulse propagation by ultrafast χ⁽³⁾ effects, both from its imaginary part resulting in two-photon absorption as well as its real part resulting in nonlinear refraction, which as mentioned already above, are linked by nonlinear Kramers-Kronig relations. It is worth noting that also the authors of [17] attribute their self-mode-locking results with a VECSEL to the ultrafast Kerr effect of bound electrons, without having performed any experimental characterization of the nonlinearity.

However, our measurements did not investigate the effect of cw-pumping, which takes place in actual laser operation. The resulting generation of free carriers, which are probed by the laser pulse and lead to strong stimulated deexcitation of carriers might change the magnitude of the nonlinearity as well as the relative contributions from FCN and BEKE. Preliminary investigations concerning optical pumping and probe-induced nonlinear-refractive-index changes suggest little influence in the absolute value of the nonlinear refraction [28]. The effect of the microcavity resonance on the nonlinearity enhancement revealed by this work will persist in a pumped system.

To ultimately distinguish the possible two contributions from each other, that are the ultrafast bound-electronic Kerr effect (i.e. BEKE) and the non-instantaneous free-carrier nonlinearities (i.e. FCN), time-resolved Z-scan measurements would be required [43]. In addition, the variation of the probe-pulse duration while maintaining equivalent peak powers could provide insight into the nature of the carrier dynamics including the effect of non-equilibrium many-body effects, which become important at sub-ps time-scales [44]. This will be the subject of our future work.

If BEKE is the dominating contribution to the nonlinear lensing, self-mode-locking of VECSELs, if caused by nonlinear lensing, would have to be modeled by substituting the slow, highly nonlinear saturable absorber term with a fast saturable absorber term in the common mode-locking model [45] (c.f. Fig. 6(b) on pulse shaping). Additionally, self-phase modulation would have to be incorporated into the model, for which the value of the effective n₂ would be obtained from Z-scan measurements. As a consequence of this, the required group-delay dispersion (GDD) for shortest pulses and stable Kerr-lens mode-locking might differ from the slightly positive GDD values required for optimal SESAM-VECSEL mode-locking [45], which could become a key advantage of self-mode-locked devices towards ultra-short pulse generation after more detailed investigations triggered by the findings reported here.

In contrast, FCN might act as a slow saturable absorber with timescales similar to SESAMs and correspondingly similar mode-locking characteristics. It is important to point out that this would not be due to real saturable absorption occurring in the gain chip. Instead, the refractive-index change induced by changes in the excited-carrier densities would lead to nonlinear lensing and only lead to some kind of artificial saturable-absorber action (c.f. Fig. 6(b)) when employed in an appropriate cavity, where nonlinear lensing reduces cavity losses, as indicated in Fig. 1(a).

If both effects were present simultaneously with comparable strength, a combined action of slow and fast saturable absorber would have to be incorporated into a mode-locking model. Also, a situation as in soliton mode-locking might occur in this case [3].

A precise knowledge of the different timescales involved in the effective third-order refractive nonlinearity n₂ would ultimately enable one to explain and model the mechanisms behind Kerr-lens induced self-mode-locking, with important implications for future chip designs.

Finally, we note that the above discussion assumed that self-mode-locking is caused by a Kerr-lens in the VECSEL chip supported by the strong nonlinear refractive index measured here. However, the pure existence of a strong nonlinear lens in the gain chip is not a definite proof yet that self-mode-locking is caused by Kerr-lens mode-locking, which would require observation of self-(de)focusing during self-mode-locked operation.

6. Conclusion

We have presented unique, systematic and direct, wavelength-resolved measurements of the effective nonlinear refractive index and nonlinear absorption in a vertical-external-cavity surface-emitting-laser chip using the Z-scan technique. In support of recent self-mode-locking achievements, strong enhancement of the Kerr nonlinearity is observed in the spectral region of the quantum-well band gap in clear correlation to a microcavity effect, which can shape Kerr lensing in the active region significantly. Negative nonlinear refraction up to 5x10⁻¹² cm²/W close to the laser design wavelength is obtained. Our results show that the resonance feature dominates any material-related dispersion of n₂ which might render it exploitable as design element in future Kerr-lens mode-locked VECSEL devices. This would affect the operation wavelength and spectrum of such VECSELs in addition to other known factors, which govern these properties, such as the material-PL/LCF overlap. While two-photon absorption poses a loss channel for lasing operation, the refractive index changes are sufficiently strong to perturb the cavity beam profile in favor of self-mode-locking. Moreover, by comparing our results to existing theories about the nonlinear-refractive-index changes in semiconductors, we believe that the ultrafast electronic Kerr effect contributes significantly to the nonlinearity while the share of free-carrier nonlinearities compared to the amount of the ultrafast Kerr effect yet remains to be investigated for a pumped system. We expect our results to lead to a better understanding of self-mode-locking in VECSELs and to pave the way to reliable, ultra-short, higher-power pulsed semiconductor-laser sources based on this method.

Appendix A

The semiconductor structure used was fabricated by metal-organic vapor-phase epitaxy (MOVPE) in an AIXTRON FT 3x2” closed coupled shower-head reactor using the standard sources trimethyl-gallium, trimethyl-indium, trimethyl-aluminum, arsine and phosphine. Deposition occurred at pressures of 100 mbar on 6° misoriented GaAs substrates. Following a 30 pair AlAs/GaAs DBR the GaAs spacer layer as well as the active region are grown and the whole structure is completed using a GaAs capping layer.

The active region comprises 20 (InGa)As quantum wells grouped into five packages which are separated using (AlGa)As spacer layers. The compressive strain induced by the QWs is mostly compensated by the surrounding tensile-strained Ga(AsP) barriers and the QW packages are placed according to a resonant periodic gain design at the antinodes of the standing wave electric field.

Appendix B

To calculate the electric-field distribution in the structure for a particular wavelength we use the matrix formalism of [46]. The overlap of the electric field with the position of the quantum wells is then described by the longitudinal confinement factor,

(3)$${\Gamma }_z=\frac{1}{N_q}\frac{{{\displaystyle {\sum }_q\left|A_q^{+}\exp (i{k}_q{z}_q)+A_q^{-}\exp (-i{k}_q{z}_q)\right|}}^2}{{\left|A_0^{+}\right|}^2+{\left|A_0^{-}\right|}^2},$$

where A_q⁺ and A_q^- are the forward and backward traveling wave amplitudes of the vector potential of the electric field at the q^th quantum well (QW), respectively. k_q is the respective wave vector, N_q the total number of QWs and z_q the position of the q^th QW. A₀⁺ and A₀^- are the vector potential amplitudes before the first material layer. The matrix formalism is solved for TE-polarization in accordance with the polarization of the probe laser in the experiment.

Appendix C

In order to extract the nonlinear absorption coefficient β, we fit the open scan data with the model from [8],

(4)$$T(z)={\displaystyle {\sum }_{m=0}^{\infty }\frac{{\left[-q_0(z)\right]}^m}{{(m+1/2)}^{3/2}}},$$

with q₀(z) = βI(z)L_eff = (βI₀L_eff)/(1 + z²/z₀²) being the normalized nonlinear on-axis absorption and I(z) = I₀/(1 + z²/z₀²) the z-dependent peak-probe power density as a function of z and I₀, L_eff and z₀ being the on-axis peak probe power density, the effective length of the sample and the Rayleigh length of the focused probe beam, respectively. The effective length L_eff = (1-exp(-α₀L))/L takes into account the different strength of linear absorption α₀ at different wavelengths. The total length L is here the accumulated thickness of the different material layers including the penetration depth into the DBR and taking into account the double-pass through the chip with the respective angle of incidence. This leads to lengths L from 4.4 to 5.1 µm for angles of incidence from 10 to 30°. Saturable absorption, which becomes important above the linear absorption edge at wavelengths below 950 nm is phenomenologically included to the model applied on the open-scan data by adding the simple model for saturable absorption in transmission,

(5)$$T_{sat}(z)=1-\frac{{\alpha }_0}{1+\frac{I(z)}{I_s}},$$

to Eq. (4).

Here, α₀ is again the linear absorption at the respective center wavelength of the probe laser spectrum, and I_s the saturation intensity which is determined by the fit. Equation (5) is normalized to 1 at the edges of the scan, added to Eq. (4) and subtracted by 1 to align the total open-scan fit function to a transmission of 1 for z positions far away from the focus. An exemplary fit of an open scan at 940 nm for 20° angle of incidence is displayed in Fig. 7(a). The resulting normalized nonlinear absorption (the positive one is attributed to two-photon absorption) exhibits a good linear trend as shown in Fig. 7(b), thus validating our method.

 

Fig. 7 Example of extraction of normalized nonlinear absorption in the presence of saturable absorption: (a) Open scan at 940 nm and 30° angle of incidence for different probe powers. The fit (solid line) is obtained as described in Appendix C using $I_s$ = 0.2 GW/cm² and ${\alpha }_0$ = 1.2x10⁻⁵ m⁻¹ for all probe intensities. (b) Resulting nonlinear on-axis absorption as a function of the peak probe density with linear fit (solid line).

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We extract the nonlinear phase shift ΔΦ₀ from the Z-scans by fitting the normalized transmission to the simple model from [8],

(6)$$T(z)=\text{}1-\frac{4\Delta {\Phi }_0\frac{z}{z_0}}{({(\frac{z}{z_0})}^2+9)({(\frac{z}{z_0})}^2+1)},$$

where ΔΦ₀ = kn₂I₀L_eff is the nonlinear on-axis phase shift induced by the effective nonlinear refractive index n₂. The sign of the nonlinear phase shift can be easily determined by the position of peak and valley in the Z-scan trace around the focus, which in our measurements always leads to a negative, thus defocusing, nonlinearity.

Appendix D

The angle-dependent characteristics of the Bragg reflector and the resonant-periodic-gain structure are displayed in Fig. 8. Figure 8(a) shows the full stopband of the DBR for different angles of incidence together with the corresponding calculated longitudinal confinement factors. For an increasing angle of incidence, both the stopband and the cavity resonance experience a significant blue shift.

 

Fig. 8 Angle-dependent chip characterization: (a) Measured reflectivity spectra for different angles of incidence (solid lines, left axis) as well as calculated longitudinal confinement factor (LCF) for the same angles (dashed lines, right axis). (b) Zoom on the longitudinal confinement factor (dashed lines) around the relevant wavelength range in comparison to the measured surface photoluminescence (PL) at the same angles (shaded areas).

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The surface PL spectra in the Fig. 8(b) (the same as in Figs. 4(a) and 4(b)) are recorded by placing the collection optics (collimating and focusing lens aligned in a lens tube) at the respective angle of incidence with respect to the gain chip and by fiber coupling the signal to an optical spectrum analyzer. These spectra correspond well with the respective calculated longitudinal confinement factors, thus, validating the use of the surface PL for determining the spectral position of the microcavity resonance.

Appendix E

To estimate the uncertainty in the determination of the fit parameter q₀ or ΔΦ₀, we vary them in Eqs. (4) and (6) until there is a significant deviation from the measurement data. Upper and lower bounds are taken as equally spaced from the fit. An example of the limiting cases is displayed in Figs. 9(a) and 9(b) for open scan and Z-scan of Fig. 2 (shown in (a) and (b), respectively). The bounds of q₀ and ΔΦ₀ are then taken into account in a weighted linear least square fit where the weights are the inverse of the respective variances meaning they are proportional to the inverse of the square of the error bounds. The resulting fits are plotted in Fig. 9. The subsequent error of the value of nonlinear refraction n₂ or β as displayed in Fig. 4 is then the 95% confidence interval of this fit.

 

Fig. 9 Measurement data with fit model as well as manually-determined symmetrical upper and lower bounds of the fit: This is a zoom of Fig. 2(b) and 2(c). (a) Open scan and (b) Z-scan with fit (solid line) and upper and lower limit of the respective fits (dashed lines) for the estimation of the data-evaluation uncertainties.

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Funding

Deutsche Forschungsgemeinschaft (DFG), Grant No. RA2841/1-1.

Acknowledgments

The authors thank S.W. Koch and M. Gaafar for helpful discussions and O. Mohiuddin as well as M. Alvi for technical assistance.

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