1. Introduction

Improvement of active optical elements such as lasers and amplifiers is expected from better control of photon modes: photonic crystal structures with a strong index contrast are especially considered in this respect. One feature that has been addressed since a decade is the band-edge enhancement of gain in periodic systems [1,2], due to the reduced group velocity, a phenomenon also tackled for two-dimensional structures [3]. While some of the physical ingredients exist in conventional DFB lasers (gain, periodicity of both index and gain), a regime of strong enhancement of gain over wider spectral regions arises in systems with a wider index contrast. In the photonic crystal literature, lasing action has often been demonstrated but spectral gain has never been measured in 2D photonic crystal semiconductor systems, to our knowledge.

Intuitively, electron-hole recombination have to be channeled in only a few competing optical modes for optimal use of the photons for amplification and lasing. One-dimensional photon systems, confined in two dimensions[4,5], are promising candidates in this respect, since radiation is inhibited over a wide fraction of k-space. We study here the appearance of gain in such a system. Specific to one-dimensional system is the divergent singularity of the photon density-of-states (DOS) at zero-group velocity points (be it from confinement or periodicity). This implies a stronger concentration of the light-matter interaction than in, e.g., 2D systems, where the photon DOS has only non-diverging van Hove singularities. The full physics of gain in photonic crystal structures (bulk or defects) has however not been clarified yet, in particular the implication of a divergent DOS, which may affect also electron-hole radiative lifetimes (the rate of light-matter interaction, and the Purcell enhancement effect [6]) and not only the optical wave group velocity. A good model system for gain studies is a photonic crystal channel waveguide (PCCW), opened at both edges, which embodies an optical amplifier. The periodicity of the guide boundaries causes the fundamental mode to slow down due to mode coupling.

In view of these elements, we have devised a method to actually measure optical gain in such photonic crystal waveguides. The method is derived from the “internal light source” (ILS) technique, depicted in great details for InP-based samples with active quantum wells [7]. Two pump beams of unequal intensity are used, one to create a rectangle-shaped gain area, the other to create probe light to be amplified along the gain area. We apply this technique to the measurement of so-called “W3” PCCW on a InP(200nm)/InGaAsP(400nm)/InP(substrate) heterostructure, containing two layers of two different quantum wells, emitting at 1.48µm and 1.55µm. “W3” denote guides of three missing rows along ΓK in a triangular lattice as shown in Fig. 1(a) and Fig.1(b). We find that the spectral shape of the TE-polarized gain, besides the expected features from the active quantum wells, exhibits a clear double peak around the frequency of the so-called “mini-stopband”, the mode coupling phenomenon [8] that couples the fundamental guided mode to a higher-order mode, causing a zero-group velocity point in the dispersion diagram. This phenomenon is investigated here around transparency.

2. Theory

Figure 1(c) shows the dispersion diagram of a three missing rows waveguide for TE polarization. It is calculated in two-dimensions, using the effective index of the heterostructure, n_eff=3.21 to account for planar confinement the third dimension. The air filling factor is f=35%. The fundamental mode of this waveguide, in red, anti-crosses modes of the same parity, hence here only mode 5 within the 2D photonic band gap. This occurs around the normalized frequency u=a/λ=0.26. The relative width and position of the anticrossing depend on f and neff. Here these quantities are Δu/u=0.01 and u=a/λ=0.266.

In theory, group velocity goes to zero at these points, and in a simple view, optical gain goes to infinity: assuming for simplicity that the number of recombination per unit time, unit volume, and unit photon energy is unchanged in this small spectral region (smaller than the typical gain accidents from electron DOS, e.g., the plateaus of the joint DOS due to the quantum well energy levels), the wave is slowed down and interacts for much longer time at a given location, gathering more gain per unit length.

Several complexities blur this simple picture: gain or losses means that k or ω do not have a one-to-one relation, as broadening is present (finite lifetime and/or finite wave extension). Also, in the real experiment, radiation loss is unavoidable and leads to spoil some of the photons gained, in a way analogous to the so-called “intrinsic losses” α_i familiar to laser diode practitioners. Next, and more fundamentally, the overall joint DOS at these photon energies may be fundamentally altered by the divergent absolute photon DOS, to an amount depending also on losses/gain (as the cavity Q in the microcavity Purcell effect). Hence, it is delicate, with a simplified theory, to predict what is the actual gain enhancement.

 

Fig.1. (a) Two-dimensional photonic crystal etched through a vertically monomode semiconductor InP-based heterostructure ; (b) Dispersion relations of the W3 waveguide folded into the first Brillouin zone; (c) Micrograph of a three-missing-rows PCCW (top view) with period a=400 nm in a InP heterostructure containing quantum wells.

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3. Gain measurement method

We use two laser beams, one to photopump a gain region, “g”, the other to induce a probe light beam, “p”. Figure 2(a) depicts the setup and Figure 2(b) a front view of the sample and beams with either an unpatterned area or a photonic crystal channel waveguide. Compared to a standard ILS method, the two pump beams are combined by a polarization beam splitter. This allows a low-loss coupling of the two laser beams with orthogonal polarizations, whatever their wavelengths. We used 655nm laser diode for the probe beam and either a second 664nm laser diode (up to 30 mW output) or a fibered single-mode 980 nm laser diode (up to 75 mW output) to induce large carrier densities, close to transparency.

In order to shape and locate the gain region independently from the probe beam, the second source is mounted on a XYZ motion, and a cylindrical lens CL (f=100 to 300 mm) is used before the collimating lens L₂ (f=15mm) to form the image of the laser output junction on the sample as a focal line of well defined length and relatively homogenous intensity: measurement of front-emitted spontaneous emission at various points along this focal line was used to assess the flatness of the excitation profile. Typical variations were below 30 % on 60µm out of the center of the focal line. The inhomogeneity of gain translates into an uncertainty of the about 30% on the gain determination, i.e., modeling the actual experiment by a rectangle-shaped excitation function.

 

Fig. 2. (a) Scheme of the setup with two pump beams (Cam=camera; Sp=Spectrometer; Fib=optical fiber; BS=beamsplitter; P=polarizer; MO=mirror objective; S=sample; LO=lens objective; F1 to F4=filters, L=lens; CL=cylindrical lens; P=probe laser diode; G=gain laser diode); (b) scheme of the sample and excited areas for gain measurement in an unpatterned region (left) and in a PCCW (right); (c) principle of the gain soustraction method.

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The geometry of gain measurement is sketched in Fig. 2(c). We measure the guided light spectrum at the cleaved edge as on the ILS technique, but now for three different configurations : probe alone (p), gain-inducing laser alone (g), and both together (pg). By making the ratio A(λ)=(I_t-I_g)/(I_p-0)=I_pg/I_p, we measure the quantity exp(-(α-α_o)L), where L is the pumped area length, α is the absorption (negative above transparency) in the pumped area, and α_o the unpumped absorption coefficient. In the limit case of full inversion, the ratio is exp(2α_oL), since the full inverted gain g is, basically, equal to the unpumped absorption α_o, or in other words, the lowest achievable value of α is -α_o. The last step is to measure A(λ) for several lengths L, in order to extract (α-α_o) (by linear fit or just averaging). In unpatterned areas, the data for various pairs of lengths are checked for consistency. Advantages of this class of method, where the light collection remains unchanged, have appeared in the recent controversy over gain measurement in silicon-based samples [9]

We measured gain of unpatterned regions and that of W3 PCCW. For an unpatterned quantum well gain measurement, the beams have their natural in-plane divergence up to the cleaved edge. For gain in a PCCW, we are obliged to induce gain both in a fraction of the relatively short PCCW sample (here 60a periods long) a confined region, and in the remaining area from the PCCW exit to the cleaved edge. There is hence less room to vary L and the amplification outside the PCCW has to be taken into account when deducing the gain within the photonic crystal waveguide, but the method still works. The PL pictures from an unpatterned area also reveal that the width of the PL locus, when carriers are allowed to diffuse, is about 4µm. Similar pictures for a PCCW shows that PL is essentially restricted to the guide, only very weak PL comes from the surrounding photonic crystal, likely due to poor carrier diffusion beyond the first row of holes. Our excitation optics, however, is sufficiently resolved to excite predominantly inside the PCCW. The difference in local carrier density due to different diffusion effects will be discussed below.

As for the particular heterostructure used here, it was grown by metal-organic chemical vapor epitaxy. It is analogous to the best ones in [7,10], with a waveguide made of InGaAsP quaternary clad by InP. Inserted in the core are two thin strain-compensated quantum wells+barrier sandwiches, of different widths. Their fundamental transitions (n=1 electro-heavy holes) are at λ=1480 nm and 1550 nm, and we focus our analysis to below the n=2 transition. Hence their overall joint DOS g_j(λ) has two steps at these wavelengths, but actually due to the two quantum wells, not to the second level of a well. Thermal equilibrium between the two wells is further assumed (they have the same quasi Fermi level).

4. Results and discussion

Figure 3(a) shows I_pg(λ) and the bare spectra I_p(λ) and I_g(λ) for a measurement in an unpatterned area, here for L=30µm. Figure 3(b) shows the spectrum of (α-α_o) (gain difference) extracted from these data. The grey line is the naïve expectation from basic gain theory[11], whereby the difference (α-α_o) is expected to scale like the product of the unbroadened joint DOS g_j(λ) by the difference of the Fermi-Dirac electron occupancy factors (f_c-f_v), for the proper quasi Fermi levels that ensure carrier neutrality (we neglect light holes and focus on the dominating TE polarization). Here, we are close but below inversion, and the plateaus are relatively flat. This is an advantage in order to separate purely photonic features from electron-hole joint DOS variations. The uncertainty of 30% mentioned above holds for the absolute value of the measured gain, but the uncertainty on the relative intensity of the peaks on both sides of the mini-stop-band, where the signal is at its best (unlike in the low gain region), is much less and, from comparison of different measurements and samples, is around 10%.

Figure 3(c) shows the same quantity (α-α_o) now for a 60a-long PCCW. The photonic crystal patterns used here were obtained by ICP-reactive-Ion-Etching [12,13]. The PCCW period is here a=400 nm. We have separately measured on this and similar samples of the same run that the air-filling factor is about f=35%, and that the losses, as characterized by the effective imaginary part ε” of the dielectric in air holes in a 2D approach, are low (ε” is on the order of 0.03 to 0.04 depending on the band edge)[10].

Comparison of Figs. 3(b) and (c) reveals that (α-α_o)L is generally higher in the case of the PCCW than that measured for the unpatterned region: due to the inhibition of the lateral carriers diffusion outside the waveguide, carriers experience a better confinement within the PCCW, which results as a higher gain. Moreover, as can be seen, there is a weak drop of the gain difference in the ministopband, at the wavelength λ=1500nm (normalized frequency u=a/λ~0.266), but also two clear peaks with 40–60% higher gain on both sides of the mini-stopband. There are no special features elsewhere, only those of the quantum well joint DOS. Similarly, there are no features for a PCCW at other periods, in agreement with the fact that the fundamental mode has no special feature. Let us insist that, as widely checked by our and other teams, the multimode aspect of the waveguide does not raise any specific problem here : what we measure is relative to the PCCW fundamental mode. In the middle of the ministopband, this mode disappears, but at the edges, it is first slowed down.

We thus observe the enhancement in light-matter interaction (here, around transparency) due at least partly to the slowed down group velocity. The possible contribution of enhanced radiative rate due to the DOS divergence remains to be assessed, a delicate issue in our view.

Finally, one might be surprised that we do not study a classical counter-propagating feedback regime. It turns out that this regime is not in the photonic gap for our PCCW orientation and practical photonic crystal parameters. A third-order gap indeed occurs for counter-propagative geometry (that is at k x period=mπ, where m is an integer) if one uses the other crystallographic guide direction, ΓM, with a periodicity a√3 along the boundary. Lasing action in this geometry at the band edge has also been demonstrated recently [14]. The fact that we observe the reinforcement of the interaction for a slightly more complex wave interaction confirms, however, that the same basic phenomena are operating in these conditions.

 

Fig. 3. (a) Bare spectra for the unpatterned measurement region, as indicated in Fig. 2(c), so that n I_pg (top spectrum), I_g was already subtracted ; (b) Gain difference spectrum from the above measurements, with a comparison to the expectation from basic gain theory with quantum well square-shaped joint DOS ; (c) Gain difference spectra for the three-missing-row 60a-long photonic crystal waveguide. The dip is the mini-stopband region. Note the two peaks on each side; (d) Plot of the group index n_g=c(∂k/∂ω) of the fundamental mode as a function of wavelength, calculated by using the plane wave expansion method, and negative for consistency with Fig. 1(b). Note the divergence of n_g at the edges of the mini-stop-band.

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5. Conclusion

We have proposed a method to measure gain spectra in heterostructure-based photonic crystal structures and waveguides carved in them. We have found, for an InP-based structure and a three-missing row waveguide, peaks with 30–40% higher gain at singularities of the photon dispersion relations, that we attribute, at least partly, to the slowed down group velocity.