1. Introduction

Photonic crystals (PCs) utilize Bragg reflection of electromagnetic waves in periodic dielectric media for controlling light propagation and light-matter interaction [1–3]. The photonic band structure formed in such crystals offers means for designing novel optical materials [4–6] for micro- and nano-optics systems [7–9]. Many concepts related to PCs have originated from analogy to semiconductor crystals [10]. Solid-state band structure models have been successfully employed to identify PC configurations that exhibit photonic bandgaps, i.e. frequency ranges in which light propagation and optical emission are inhibited [2,11]. Moreover, the notion of localized states formed at PC impurities or defects [3] has been utilized to trap or guide photon modes with potential applications in novel lasers [8] and integrated optics elements [7,12]. However, another important concept of modern solid-state physics, namely, the semiconductor heterostructure [13], has not been elaborated yet with PCs operating in the optical frequency range [14]. Here, we develop the notion of photonic crystal heterojunctions and heterostructures (PCHs) using phase-coupled arrays of vertical cavity surface emitting lasers (VCSELs) [15] as a model system. We show that PCHs offer a powerful approach for tailoring the envelope functions of photon modes propagating in quasi-periodic dielectric structures and illustrate their use in controlling the lasing modes of coherent VCSEL arrays.

In semiconductor homostructures, consisting ideally of perfectly periodic crystals, the band structure is set by the crystal potential, and the states of charge carriers consist of periodic Bloch functions modulated by plane wave envelopes. In semiconductor heterostructures, several such homostructures are joined together, stitched by heterojunctions constituting the interfaces between the periodic domains. In each domain, the material properties (band gap, carriers effective masses, …) are fixed by the “local” crystalline structure. At the heterojunctions, the envelope functions of the wavestates are altered by reflections due to the band offsets. Semiconductor heterostructures thus provide means for controlling the envelope functions. Moreover, in semiconductor heterostructures incorporating lower-bandgap domains of dimensions comparable to the Fermi wavelength of the carriers, quantum confinement effects set in, yielding discrete energy spectra and localized envelope functions. Such semiconductor quantum structures made possible numerous fundamental studies of low-dimensional systems and form the basis for important device technologies, e.g., quantum well lasers and high mobility electron transistors [13].

By analogy, PCHs consisting of PC domains joined at photonic heterojunctions of well-defined interface configurations offer exciting possibilities for tailoring photonic envelope functions. Similar to low-dimensional semiconductors, low-dimensional PCHs, e.g., photonic quantum wells, have been proposed [16–18]. The quantization of the photonic band was predicted and indeed observed in the transmission spectra of photonic quantum wells in the millimeter wavelength range [14]. Here, we implement the concept of the PC heterojunction and show how it can be designed to achieve photonic envelope function confinement in photonic crystal heterostructures. Using this notion, we present experimentally and theoretically the control of such confinement in two-dimensional (2D) PCHs based on arrays of phase-coupled VCSELs operating in the near-infrared region of the optical spectrum.

2. Heterojunctions and heterostructures implemented with VCSEL-crystal

A 2D PCH based on phase-coupled arrays of VCSELs is depicted in Fig.1(a). These structures represent a special class of 2D PCs incorporating laterally coupled optical resonators [19–21] or waveguides (e.g., photonic crystal fibers [9]). In this case, the electromagnetic modes propagate nearly perpendicular to the periodic index modulation such that only a small in–plane component k _⊥ of the propagation vector k undergoes Bragg reflections [Fig.1(b)]. Thus, effective in-plane Bragg reflection occurs in a 2D PC of periodicity much larger than the wavelength. For structures utilizing coupled resonators, the longitudinal component k _// is fixed by the cavity resonance condition. In VCSEL-based structures [Fig. 1(b)], k _// also undergoes Bragg reflection along z axis in each of the distributed Bragg reflectors (DBRs), however its value is fixed across the uniform VCSEL wafer by the length of the vertical cavity and does not influence the in-plane effects we discuss here. The elements of the VCSEL PC (the pixels) are defined by patterning the reflectivity of the top DBR [15] of a VCSEL wafer. Square-shaped, higher-reflectivity Au pixels surrounded by a lower-reflectivity Cr grid define the single-mode VCSEL resonators [Fig. 1(b)]. Since the round trip optical gain is low, a reflectivity modulation of the order of 1% is sufficient to define the position of the lasing microcavities.

A PC heterojunction obtained by interfacing two different, square-lattice VCSEL PCs, designated “type-A” and “type-B”, is illustrated in Fig. 1(b). In this particular case, the two PCs have the same lattice parameter Λ and differ only by the fill factor FF, defined as the ratio of the Au pixel area to that of the unit cell. The impact of the PC heterojunction is related to the discontinuity produced in the photonic band edge at the interface [Fig.2(a)]. To evaluate the photonic band offset, the band structures of the PCs constituting the heterojunction were calculated using our non-Hermitian model Hamiltonian [22]. The real parts of the Hamiltonian’s eigenvalues correspond to the photon energy and the imaginary part represents the modal loss of the state. In our VCSEL crystal, the metallic pattern on the top DBR results in reflectivity modulation with low phase contrast. Therefore, the photonic energy band structure is close to that of the empty lattice, and no gaps are opened in the energy bands [22]. On the other hand, the photonic state with the lowest optical cavity loss [22] (the lasing mode) defines the photonic band edge in the loss domain. In this domain, there are no photonic states with lower losses, and hence a photonic bandgap exists below this band edge.

 

Fig. 1. Photonic crystal heterostructure (PCH) based on 2D photonic crystal (PC) VCSELs. (a) Schematic of the wafer structure showing the metal-patterned top DBR with a core region PC of type A and a cladding region PC of type B. The highlighted region of the heterojunction is detailed in (b): the two lattice-matched PCs differ by the size of gold pixels (top panel); the longitudinal component of the propagation vectors is fixed by the vertical cavity and the transversal components are affected by the heterojunction (bottom panel). (c) 2D Brillouin zone.

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Fig. 2. Analogy between photonic crystal and semiconductor heterojunctions: simplified band diagrams (left panels) and potential/loss profiles (right panels). (a) Photonic crystal heterojunction between two VCSEL PCs with the same lattice constant (Λ=6 μm) and fill factors of 0.69 and 0.44. (b) GaAs/AlAs heterojunction. The band offset ΔΓ at the PC heterojunction (grey area) acts as a barrier for photons and is equivalent to the potential barrier for charge carriers formed by the band misalignment at the semiconductor heterojunction.

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For a lattice with square symmetry, the reciprocal lattice has square symmetry as well, and the irreducible Brillouin Zone (BZ) [1] is a triangular domain Δ-Z-T [Fig. 1(c)]. The lowest-loss state |T ₅⟩ is located at the T-point of the BZ (notation as in Ref.[22]). The calculated loss-dispersion curves corresponding to the T ₅ band of two lattice-matched VCSEL PCs A and B with different fill factors are shown in Fig. 2(a) along the Z-T line (k_y = π/Λ) of the BZ assuming infinite in-plane extent of the PCs (“bulk” 2D crystals). The losses per vertical roundtrip of the cavity are plotted as a function of k_x for a photonic Bloch wave with k _⊥ on the Z-T line of the BZ [the direction Z-T of the crystal is normal to the heterojunction, as indicated in Fig. 1(b)]. To emphasize the order of parameters involved, only the losses related with the disperssive features of the patterned top DBRs are plotted in Fig.2; the material losses (~0.1%) and the output coupling losses on the bottom DBR (~0.16%) are equal in PCs A and B, and are not accounted for in Fig. 2. For the type-B crystal, the analytic continuation of the loss dispersion curve into the complex k _⊥ -plane is calculated along the line Re(k_x ) = Re(k_y ) = π/Λ and Im(k_y ) = 0 (along this line Im(k_x ) ≠ 0).

The reduction in the FF clearly modifies the band structure: the modal loss increases and the bandgap becomes wider. The juxtaposition of two “bulk” PCs with different FF at a heterojunction thus results in a photonic band offset. As in semiconductor heterojunctions, where the misalignment of the valence and conduction bands impedes the transport of carriers [Fig. 2(b)], the band offset at the photonic heterojunction alters the propagation of the photonic Bloch waves. The step-like variation in the modal losses ΔΓ is analogous to a potential barrier (Fig. 2(a), right panel). A given photonic state of the type-A crystal (that is, a state of a given frequency and losses) can be either partially or totally reflected at the boundary between the two PC domains, depending merely on whether its modal loss is higher or lower than the band edge of the barrier PC (the energy dispersion curves of A and B PCs with metal patterned DBRs coincide). For example, the state represented by the yellow point in Fig. 2(a) falls within the gap of the type-B crystal; its Bloch wave cannot propagate on the B-side of the photonic heterojunction. This is manifested by the imaginary part of k _⊥, yielding an evanescent envelope function in the heterojunction barrier.

3. Modeling of photonic crystal well

We evaluated the effects of the photonic heterojunctions and geometry on the envelope function confinement in 2D VCSEL-based “photonic well” such as the one shown in Fig. 1(a). The field patterns and modal losses of the PCH states were modeled utilizing the finite difference element presentation of our system Hamiltonian [22] in the effective mass approximation. This formulation is analytically similar to the scalar coupled mode theory [23] with effective propagation and coupling constants derived from the photonic dispersion curve of each sub-lattice in the vicinity of the T point.

 

Fig. 3. Impact of the depth and geometry of a “photonic well” heterostructure on the confined photon states (numerical simulations). (a) Left panel: modal losses versus the FF contrast (FF=0.69 at the core) for a PCH containing 10×10 core unit cells. The dispersion curves of the states are represented by the coloured solid curves; the black solid curves indicate the dispersions of the photonic band edges. The inset shows the cross sections of the (full) photonic wavefunction for selected cases. Right panel: Near field patterns of the states indicated by the corresponding points on the dispersion curves. (b) Left panel: modal losses for three different states versus the aspect ratio of the core a/b (number of lattice periods a varies from 1 to 15 and b is fixed to 10) for a constant photonic well depth (FF contrast of 0.25). Right panel: near field patterns of the states labelled A, B and C in the left panel.

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We first examined the effect of the photonic band offset on the envelope function confinement [Fig. 3(a), the resonant wavelength of the VCSEL cavity is assumed to be 940 nm]. The band edge of the type-A PC, constituting the core of the photonic well, was kept constant (FF=0.69) whereas in the barrier (type-B PC) it was varied by changing the FF between 0.39 and 0.59 (respectively, two solid black curves in Fig. 3(a), left panel). The dispersion curve of the lowest-loss state of the photonic well |$T_5^{\left(00\right)}$⟩ always lies below the band edge of the barrier PC, and its envelope function is hence always confined. On the other hand, the dispersion curve of the next lowest-loss states |$T_5^{\left(10\right)}$⟩ and |$T_5^{\left(01\right)}$⟩, doubly degenerate due to the square shape of the well, lies below the band edge of the barrier material only for sufficiently large FF contrast (in this case, ΔFF > 0.17). The envelope function of these higher order states may thus be propagating or evanescent at the PCH barrier, depending on the FF contrast. These features are illustrated in the right panel of Fig.3(a), which shows the near field patterns of the |$T_5^{\left(10\right)}$⟩ state for two different FF values.

As in low-dimensional semiconductor heterostructures, another way of controlling envelope functions is by changing the shape of the well keeping the barrier height fixed. Figure 3(b) shows the influence of the aspect ratio a/b of a rectangular PCH core on the dispersion curves and the near field patterns of the |$T_5^{\left(00\right)}$⟩, |$T_5^{\left(10\right)}$⟩ and |$T_5^{\left(01\right)}$⟩ states. The lower symmetry of the rectangular configuration removes the degeneracy of the |$T_5^{\left(10\right)}$⟩ and |$T_5^{\left(01\right)}$⟩ states. Moreover, their relative modal loss can be adjusted by proper selection the aspect ratio of the core.

4. Experimental results and discussion

The impact of the band offset at a PC heterojunction was investigated experimentally in VCSEL - based PCHs. A series of square - lattice PCHs ( lattice constant Λ = 6 μm ) incorporating coupled VCSELs emitting at 940 nm wavelength (the design wavelength of the vertical structure of the VCSEL wafer) were fabricated and tested. Details of the VCSEL wafer structure are given in Ref. [24]. Each PCH contained 16×16 unit cells in which a 10×N (N=4 or 5) core with FF=0.69 was defined. The barrier Au pixels had widths ranging from 3 to 4 μm, corresponding to FF contrasts of 0.25 to 0.44.

 

Fig. 4. Comparison of measured and calculated features of the |$T_5^{\left(00\right)}$⟩ state in a PCH based on coupled VCSEL arrays. (a) Near field intensity patterns and cross sections (along the dashed white line) in lasing 16×16 VCSELs PCH with 5×10 unit cells in the core measured for three different values of the FF contrast: ΔFF=0.25, 0.35 and 0.44, from top to bottom. (b) Numerical simulation of the near field intensity patterns for the three measured structures. (c) Measured (triangles) and calculated (circles) in-plane propagation constants in the core (top) and attenuation constants in the cladding (bottom) versus the effective size parameter N(ΔFF)^1/2 of cores containing N×10 unit cells, with N=4 or N=5 lattice periods. Red dashed lines show the influence of ΔFF for constant N and black lines show the impact of N for a constant ΔFF.

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The measured near field patterns of the lowest-loss photonic state, lasing under pulsed electrical pumping, are shown in Fig. 4(a) for different values of the FF contrast. The observed near field patterns are characteristic of the confined |$T_5^{\left(00\right)}$⟩ Bloch states, comprising an envelope function modulated by a periodic “out-of-phase” supermode with maxima of intensity at the gold pixels locations [15]. The envelope function is cosine-like in the core and evanescent in the PCH barrier. The evanescent tails extend more into the barrier for small FF contrast, which is expected due to the reduction in photonic band offset with decreasing FF contrast. This evolution of the envelope function is in agreement with the calculated intensity patterns presented in Fig. 4(b).

A quantitative comparison between the calculated features of the PC heterojunction and the observed characteristics of the photonic envelope function is shown in Fig. 4(c). In this figure we plot the normalized transverse wave vector component along the lattice rows k_x in the PCH core and barrier as a function of the effective size of the core N(FF_A - FF_B )^1/2 The measured (triangles) and calculated (circles) values of k_x are in good agreement. The discrepancies can be partly explained by boundary effects [25] caused by the finite size (16×16) of the fabricated structures. The solid lines in Fig. 4(c) indicate the general trends of the evolution of the envelope function: stronger confinement is obtained with increased size and/or increased effective “depth” of the PCH core.

5. Summary

We have shown that the concepts of heterojunctions and heterostructures developed for semiconductors can be implemented with photonic crystal heterostructures. By joining several periodic photonic crystal domains at photonic heterojunctions of well-defined photonic band offsets, it is possible to control the characteristics of the photonic envelope functions and thus achieve new means for controlling photonic states in quasi periodic media. Particular PCHs implemented with 2D arrays of coupled VCSELs were employed to illustrate the use of this concept in achieving confinement of lasing modes in VCSEL-based PCHs. These concepts should stimulate further development of novel PCHs for control of photon propagation and confinement.