1. Introduction

Advanced photonic structures on a (sub-) wavelength scale provide the novel technology basis necessary for truly high-density optical integration. Perhaps the most familiar technology is that of Photonic Crystals (PCs) [1,2], which offer the possibility of creating the next generation of multi-functional integrated components with footprints significantly smaller than those conventionally achieved. The objective of obtaining full photonic band gap behavior from three-dimensional periodic photonic crystals is, of course very attractive: in the present context, however, it is natural to work with technological approaches which are normally described as planar. A celebrated example is provided by two-dimensional PCs (2D PCs), which have recently been subject of widespread theoretical and experimental investigations [3–6], since they can be fabricated on semiconductor layers using mature microelectronic and opto-electronic technologies. In this case, the vertical confinement is insured by the index contrast between the core and cladding layers: an efficient vertical optical index confinement is achieved with “suspended” semiconductor membranes, either in air or bonded to low index material.

We have recently proposed [7] a major extension of planar technology through exploitation of the third (‘vertical’) dimension by using a multi-layered approach, where the lateral high index contrast patterning of layers could be combined with the vertical 1D high index contrast patterning: it is here more appropriate to think in terms of ‘2.5 dimensional’ photonic structures in which an interplay between guided confined photons and photons propagating through the planar layer structure occurs. A simple illustration of this approach is the use of a plain Photonic Crystal Membrane (PCM) as a wavelength selective transmitter / reflector: when light is shined on a PCM in an out-of-plane (normal or oblique) direction, sharp resonances in the reflectivity spectrum can be observed [8]. These resonances arise from the coupling of external radiation to the guided modes in the structures, whenever there is a good matching between the in-plane component of the wave vector of the incident wave and the wave vector of the guided modes. More recently, Pottage et al. [9] have shown that photonic crystal membranes can exhibit vertical resonances in its reflectivity spectrum. Tibuleac and co-workers [10] have proposed an accurate description of the optical behavior of dielectric multi-layered structures with infinite lateral size, under normal incidence, with some of the dielectric layers being patterned laterally, thus forming a one dimensional (1D) diffraction grating. Very recently, Sooh and co-workers [11] have proposed a theoretical study of two suspended and tunable 2D PCM with infinite lateral size, to be used for various types of micro-mechanically tunable optical devices. In our above mentioned contribution [7], we have analyzed the effect of the limited lateral size of the device and proposed a simple modeling approach derived from the coupled mode theory [12,13] and based on the transfer matrix method (TMM), allowing a very fast simulation of the reflectivity / transmission spectral characteristics of any stack of photonic crystal slabs: it was demonstrated that the optical characteristics are, to a large extent, controlled by the coupling rate (time constant τ_c) of guided and radiated modes through the periodic corrugation and by the in-plane lateral escape rate (time constant τ_g) of the guided photons out of the corrugated membrane illuminated area; in particular, the lateral escape of photons should be considered as a loss mechanism in devices specifically designed for “vertical” operation. The reflectivity bandwith (Δω_p) of the PCM alone is approximately given by Δω_p=1/τ_c+1/τ_g. The ability of high index contrast PC to slow down photons, allows a very good control over the lateral escape losses and results in very compact devices: this is achieved especially at high symmetry points (or extreme points) of the dispersion characteristics, where the group velocity approaches zero and guided modes are stationary; we showed that, in these conditions, the relevant parameter is the second derivative at the extreme point, which should be made the smallest possible, for the most efficient lateral confinement of guided modes interacting with radiated modes.

In this contribution, we propose a new concept of laser combining a PC membrane (PCM) and a Bragg reflector: the operation of the structure relies on the coupling between guided and vertical resonant modes, the later being induced by the reflecting properties of both PCM and Bragg stack. The concepts and tools developed in our previous work are fully used for the design of the structure. The analysis is restricted to TE waves (the electric field is located along the y direction). Light is always emitted vertically in free space, but when the structure is perfectly tuned (with the PCM’s reflectivity approaching 100%), a major portion of the light is emitted in-plane (in the PCM).

2. Design of the vertical patterning of the multi-layered structure

The basic structure is illustrated in Fig. 1. For the PCM, we choose an optical thickness which is an integer N of half of its resonant wavelength (λ_p) corresponding to the coupling between radiated modes and the PCM under normal incidence (λ_p is defined when the PCM is immersed in air, without the rest of the laser structure). In these conditions, the PCM behaves like a reflector whose spectral response exhibits a Gaussian shape, with a frequency bandwidth on the order of 1/τ_c, when lateral losses are negligible. For the rest of the paper N=2 : this choice will be justified later when further details of the PCM design are provided. The Bragg stack is constituted of four pairs of quarter wavelength air/semiconductor layers. The intermediate section in between the PCM and Bragg stack includes an air gap and a semiconductor active layer, which is meant to host the gain material (e.g., quantum heterostructures such as quantum wells, dots, etc.).

 

Fig. 1. Basic laser structure

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TMM is now used to complete the design of the vertical patterning of the device, which does not include yet any gain material in the active layer : for that purpose the transmission spectrum of the structure is determined as shown in [7], and, in addition, the field distribution of the “vertical” radiated modes can be obtained for each wavelength. The material in the semiconductor layers is assumed to be Indium Phosphide (InP). In all cases, light is incident from the semiconductor substrate, which is assumed to be infinite for calculation purposes. The design aims at obtaining a strong “vertical” resonance, i.e., a strong build-up of vertical radiated photons. This is achieved at, or close to, the resonance wavelength λ_p of the PCM and corresponds to phase conditions for constructive vertical interference. Also, the structure should be defined in such a way that the electric field is maximized in the active region, for an efficient operation of the laser. Based on these criteria, a maximum photon build-up, for the simple case where lateral losses are negligible, can be obtained when the operation wavelength λ coincides with λ_p and for h ₂=3λ_p/4 (or also λ_p/4 : we have chosen a three quarters wave thickness for h ₂ in order to prevent any significant evanescent coupling of the gain material into the resonant guided modes in the PCM) and n_InPh₃=λ_p/2, where n_InP is the refractive index of InP. τ_C is assumed to be, for example, 0.3 ps (2πτ_cc/λ_p=Q_c≅365, c is the speed of light in free space): this choice corresponds to a reflectivity bandwidth for the PCM (Δω_p) significantly larger than the bandwidth of the resonance of the laser structure (Δω_r). In the spectral response of the phase of the reflected field, it is observed a fast variation of phase near resonance. Based on the spectral width of this phase change, it is possible to estimate the bandwidth of resonance, Δω_r. The resonance lifetime is then defined as τ_r=1/Δω_r. An extra index ‘o’ is added to τ_r, to indicate the maximum value of τ_r, situation that occurs when the PCM presents no losses (in this case, τ_r=τ_ro). The vertical field distribution in the laser structure is shown in Fig. 2(a) at resonance (in this case, the resonance wavelength of the laser structure (λ_r) coincides with λ_p). The field in the active membrane is strongly enhanced (factor 300) within respect to the incident field in the substrate: this enhancement corresponds to a quality factor Q_ro of the resonance in the order of 1.2×10⁶, which gives τ_ro=1 ns (2πτ_roc/λ_p=Q_ro). No transmission of photons is possible across the structure at and around the resonance : this results from the fact that the PCM behaves like a perfect reflector for λ=λ_p (in absence of lateral losses); in these specific conditions, the lifetime of the resonance τ_ro is essentially controlled by the escape rate of photons across the Bragg stack. If the air gap is slightly de-tuned (2% decrease in the air gap thickness), a resonance is still observed, but λ_r is shifted toward a neighboring wavelength which meets vertical phase matching conditions; the PCM does not behave anymore as a perfect reflector, and transmission of photons is now allowed across the structure at and around the resonance, as shown in Fig. 2(b). The lifetime of the resonance is reduced since photons can also escape across the PCM: this results, accordingly, in a reduction of the field in the active layer at the operation wavelength corresponding to the shifted resonant wavelength.

Coming back to the case of a tuned structure (that is for h ₂=3λ_p/4), we examine now the impact of the lateral losses. It is illustrated in Fig. 3(a) where the field in the active layer is plotted versus the lateral escape time in the PCM, τ_g. It is observed that if τ_g<τ_ro, the field decreases sharply; this is expected since the lifetime of the resonance τ_r coincides now withτ_g, and the lateral losses in the PCM prevent the build-up of photons fed into the active layer across the Bragg stack. A plot of τ_r versus τ_g is presented in Fig. 3(b). For low values of τ_g (less than 80 ps), it is noticed that τ_r≈τ_g. Further increasing τ_g (implying reduction of lateral losses), makesτ_r slowly converge to τ_ro, case in which the linewidth of the structure is limited by losses in the Bragg layers.

 

Fig. 2. Electric field distribution (based upon TMM) at resonance for no lateral losses. In case (a), h ₂=3λ_p/4 and in (b), h ₂=2.94λ_p/4.

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One has to remark that TMM should be considered as a preliminary design tool for the final laser structure, which cannot be fully described at this stage of the design procedure. In an actual laser device, light is generated internally and, in case of quantum heterostructures, the radiation pattern is quite different from a plane wave : in any case, for example, the electric field distribution provided by TMM cannot be considered as the actual field distribution in the laser device. However, TMM has given a fast access to vertical patterning of the structure and provided preliminary basic information about how the electric field will build up in the active layer (information which can be used to determine the optimum position of the quantum well, wire or dot gain material) and, importantly, on key factors which control the lifetime of the resonance in the multi-layered structure: these factors can be summarized in terms of the escape rates of photons (vertical radiated modes) across the Bragg stack and PCM reflector and the lateral losses of guided modes in the PCM escaping out from the beam area before having a chance to be coupled back to the radiated vertical modes.

3. Design of the PCM

In order to minimize the lateral losses, it is necessary to design the PCM in such a way that the largest possible lateral escape time τ_g is achieved. Also the design should result in a large coupling rate (small enough τ_c) between stationary guided modes in the PCM and vertical radiated modes (Γ point in the dispersion characteristics), in order to achieve comfortably large bandwidth reflective characteristics: in that respect, the choice of N=2 in the previous section, i.e. with an optical thickness of the PCM equal to the resonance wavelength λ_p is motivated by an improved coupling rate between radiated modes and guided modes as compared with the case N=1 (half wavelength optical thickness), owing to a more favorable overlap integral of radiated and guided modes at resonance [7].

 

Fig. 3. Electric field in the active layer (based upon TMM) at resonance versus lateral escape time(a). Resonance lifetime as a function of lateral escape time (b).

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For a given lattice constant of the PCM, the other key parameter besides the optical thickness of the PCM which one has to optimize, is the filling factor (FF) of the Photonic Crystal. It is adjusted in such a way that the related stationary guided mode should couple efficiently to radiated modes and show a strong lateral confinement, i.e., be the result of a strong in-plane (2^nd order) diffraction coupling of guided modes propagating forward and backward. This coupling opens up a band gap, whose edges coincide with the extreme points at Γ point. As shown in [7] and recalled in the introductory section, the controlling factor for the most efficient lateral confinement of stationary guided modes interacting with radiated modes is the second derivative α at the extreme points of the dispersion characteristics where coupling occurs. If D is the lateral spot size diameter of the PCM illuminated area, then it can be shown [7] that τ_g≅D ²/4α : as a result, α should be made the smallest possible, to limit the impact of lateral losses.

Without loss of generality, our analysis is restricted to one dimensional Photonic Crystals: this is motivated by the need of relaxing simulation and modeling constraints. Moreover, the principal conclusions can be derived from this simpler case. In this case, it is assumed that the structure is infinite in the y direction, leading to no significant variation of the electromagnetic fields in this direction [13]. Therefore, two dimensional finite-difference time-domain method (2D FDTD) can be used to analyze this structure, which is far less time consuming than 3D FDTD. The initial design of the PCM is based upon FDTD analysis.

In the PCM design, three parameters need to be defined: the lattice constant (Λ), the filling factor (FF) and the PCM thickness (h ₁). In order to begin with the PCM design, Λ is “arbitrarily” set to 1 µm; this initial value will be modified later when the whole designed structure will be re-scaled to operate at λ=1550 nm. The lateral size of the PCM structure is set to about 30 µm, in such a way that it includes several grating periods in the x direction. Absorbing layers are placed at the boundaries of the computation area in the FDTD simulation and the membrane is embedded in air (without the rest of the laser structure). The source is a Gaussian waveform with spot size diameter (D) of about 20 µm. Line detectors are placed behind and in front of the PCM in order to evaluate its reflectivity and transmission. Whenever spectral information is necessary (for example, to calculate the reflectivity spectrum of the PCM), the temporal variation of the source is made Gaussian, otherwise it is assumed to be sinusoidal. Several PCMs are analyzed with different filling factors (FF).

For a given FF, the effect of h ₁ is investigated. Starting from a small initial value, h ₁ is increased systematically in FDTD analysis. The reflectivity spectrum of the PCM changes from a Fano shape to a Gaussian shape (for a half wavelength optical thickness in an effective index approximation). Further increasing the thickness, it changes again into a Fano shape and later on back to a Gaussian shape. At this point, the PCM has an effective full wavelength optical thickness. This procedure to determine h ₁ is repeated for different filling factors. The aim is to choose parameters in such a way that the PCM’s reflectivity peak is maximized. In order to provide a better understanding of the designed structures, band diagrams are generated by MIT’s photonic bands software [14]. Based upon band diagrams, we can observe that the second derivative at Γ point is quite low for FF=50% and even lower for FF=60%. However, FF=50% provides higher reflectivity than FF=60%, as revealed by FDTD analysis. That is due to the fact that although for FF=60%, the guided waves in the PCM look to be more stationary, the coupling of energy to these modes is worse, which leads to lower reflectivity. Therefore, FF=50% is chosen for the final design. Its band diagram is illustrated in Fig. 4(a), where the Γ point corresponds to a wave vector $\overrightarrow{k}$=0, while the X point orresponds to a wave vector $\overrightarrow{k}$=π/Λâ_x, with â_x being the unit vector in the x direction. In this band diagram, mode 1 is chosen because it provides the largest reflectivity, while mode 2 cannot couple efficiently with the incoming vertical waves. After choosing a filling factor of 50%, the whole structure is re-scaled for operation close to λ=1550 nm, in this case, the period (Λ) and the thickness (h ₁) are set to 1200 nm and 670 nm, respectively. For the final designed structure, the peak reflectivity is greater than 97% and the linewidth Δλ is about 80 nm, as illustrated in Fig. 4(b). Based upon the reflectivity spectrum, we obtain τ_c=0.016 ps (Q_c≈19.6) and, based upon the band diagram, we obtain τ_g=2.2 ps (τ_g≈D²/4α and Q_g=2πτ_gc/λ_r≈2700) clearly demonstrating that τ_c≪τ_g for a good operation of this surface emitting device. Using these parameters, we have obtained (using TMM) the field distribution at resonance in the multi-layered structure, as shown in Fig. 4(c): the buildup of photons in PCM is clearly observed.

4. FDTD simulations of the multi-layered laser structures.

Once designed the PCM, the next step is to insert it in the laser structure. The internal source used to simulate the emission of quantum heterostructures is a dipole oriented in the y direction with a radial emission pattern (see Fig. 1). TMM has shown that the best place to insert the point-source in FDTD simulations is in the upper boundary of the active layer, where the “vertical” radiated field is maximum. The point source is also positioned in the middle of the structure within respect to x axis. In these conditions, the dipole will couple essentially to the vertical radiated resonant mode and the direct (evanescent) coupling with the guided resonant mode will be negligible.

Absorbing boundary layers are placed in the boundaries of the computation area of the FDTD simulation. A point detector is positioned close to the upper edge of the air gap (in respect to z direction) and in the middle of the structure within respect to x axis. In order to obtain the spectrum of the radiated light, a Gaussian temporal pulse is emitted by this source and by using fast-Fourier transform algorithms it is possible to analyze the spectral behavior of the structure.

 

Fig. 4. Band diagram (a) and reflectivity spectrum (b) for the designed PCM. Electric field distribution, at resonance, for the laser structure (c).

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We first consider a perfectly tuned structure as defined in the TMM section. The spectrum of the electric field is shown in Fig. 5(a). The major resonant peak of the laser structure occurs close to 1550 nm, with a quality factor Q_r≅2900 : this value corresponds to τ_r=2.4 ps which closely relates to τ_g defined in the previous section devoted to the design of the PCM. This is consistent with the results of the TMM section, and corresponds clearly to the situation where the lifetime of the resonance coincides with the lateral escape time in the PCM (τ_r=τ_g in the TMM section). The electric field (oriented the y direction, E_y) profile is shown in Fig. 5(b), at resonance, for a continuous wave (cw) excitation. A large buildup of the electric field is observed in the active region and the air gap.

 

Fig. 5. Electric field spectrum at a point in the air gap (a). Electric field distribution at resonance for h ₂=3λ_p/4(b).

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For the “perfectly” tuned device, both Bragg reflector and PCM are strongly reflective and, in an ideal situation, would not allow photons to be radiated to free space. However, due to finite losses of the PCM, some photons are allowed to be radiated to free space while others escape laterally. In this case, as the lateral escape time is small compared with the vertical one, photons can reach the edges of the PCM and be partially reflected back. The main peak in the electric field spectrum represents the fundamental lateral mode in the PCM. This perfectly tuned situation is the most favorable for in-plane emission, since it is difficult to emit light vertically and the easiest way for photons to escape laterally.

In order to make the device to further emit light vertically, two options are possible: first, the Bragg reflectivity can be made smaller, to the extent where the escape rate across it becomes larger than the lateral escape of photons in the PCM; second, keeping the Bragg reflector unchanged, the device can be slightly de-tuned as explained in the TMM section, thus reducing the reflectivity of the PCM (while shifting the resonance wavelength for phase matching requirements) and promoting further escaping of photons in the vertical direction across the PCM reflector at the expense of lateral insertion. This can be achieved by moving, via electrostatic actuation, judiciously chosen semiconductor layers of the multi-layered structures (either within the Bragg stack to reduce its reflectivity for the first option, or in the region where the vertical field is strong, for the second option). Figure 6(a) shows the spectrum for the case in which the air gap is reduced by 2%. The resonant peak (λ_r) is shifted to 1527 nm and the quality factor (Q_r) is reduced to 1500. Fig. 6(b) shows the electric field distribution in the laser structure, at resonance, for cw excitation. It can be noticed that, although the field enhancement in the laser device has decreased, there is more light being radiated out of the device compared with the electric field in the active region. Note also that the envelope of the field distribution shows that the field presents a smaller lateral extension, which indicates that most of the photons are emitted vertically before they can reach the edges of the PCM: therefore they do not have a chance to be reflected at the boundaries. A power analysis of the FDTD simulation indicates that the ratio of the radiated power to the power lost laterally is about twenty (20). Further increasing the air gap thickness leads to lower Q values and to lower buildup of electric field in the active region, what is not desirable.

 

Fig. 6. Electric field spectrum at a point in the air gap (a). Electric field distribution at resonance for h ₂=2.94λ_p/4.

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In any case it is clear that minimizing lateral losses is essential for an improved operation of the device in that more freedom will be left for operation in the different above described regimes. In that respect, making use of a two dimensional (2D) PCM should be more favorable, since one can expect to achieve a more efficient lateral confinement of photons than with a 1D PCM.

Typical micro-cavities embedded in two Bragg mirrors have quality factors in the range of 2000 and 10000. Moreover, these mirrors have reflectivities in the range of 97% and 100%. These values are comparable with those obtained in our hybrid PCSEL design.

5. Summary

In summary, we have presented a hybrid version of a laser which combines a photonic crystal membrane and a Bragg stack. The proposed approach results in compact devices with relaxed technological constraints and versatile functionality. For example, via micro-opto-electro-mechanical actuation, not only the wavelength can be changed, but also the direction (vertical or in-plane) where most of the photons will be emitted. Also, multiple emission of different wavelengths can be achieved by using photonic crystals with different lattice constants in the same wafer. Devices based on similar concepts may find applications in switches, tunable filters, etc.