We demonstrate the coherent transfer of optical orbital angular momentum (OAM) to the center of mass momentum of excitons in semiconductor GaN using a four-wave mixing (FWM) process. When we apply the optical vortex (OV) as an excitation pulse, the diffracted FWM signal exhibits phase singularities that satisfy the OAM conservation law, which remain clear within the exciton dephasing time (~1ps). We also demonstrate the arbitrary control of the topological charge in the output signal by changing the OAM of the input pulse. The results provide a way of controlling the optical OAM through carriers in solids. Moreover, the time evolution of the FWM with OAM leads to the study of the closed-loop carrier coherence in materials.
© 2009 Optical Society of America
Optical vortices (OVs) have attracted great interest and have been utilized for various applications, such as manipulating the orbit of trapped particles  including Bose-Einstein condensates (BECs) [2, 3], and spatially-resolved imaging and fabrication beyond the diffraction limit . One unique characteristic of the vortex beam is its ability to carry the orbital angular momentum (OAM), which arises from the phase circulation around the singularity (the helical wave front along the direction of the propagating beam) . This phase singularity also characterizes the beam intensity profile as a dark hole. Unlike the spin angular momentum, which consists of just two dimensions (right and left circularly polarized states), optical OAM can reach higher dimensions according to the number of times ℓ of the 2π phase variation along the azimuthal direction φ (so-called topological charge of the OV). Since this infinite-dimensional degree of freedom of OAM can be addressed as multiple channels of information, OVs also have potential for optical communications and information processing both in the classical and quantum regimes [6–10]. In these applications, time variable OV pulses are necessary for carrying the information. In general, the active control of OAM in OVs has been realized by using a spatial light modulator (SLM). However the update rate of the OV pulse is highly restricted due to the low refresh rate of the liquid crystals. One promising way to realize fast optical modulation/switching is to use electrons in solids. But most previous research on the coherent excitation of electrons using OVs has been performed in atomic vapor systems [2,11]. For practical use, the OAM transfer to the electrons in solids is important  but has not been well investigated. Moreover, recent progress in materials science has allowed us to create ring-shaped materials (structures), in which the electrons are expected to have a closed-loop coherence, which we will be able to measure by using coherent excitations with OV pulses [13, 14].
In this paper, we demonstrate the coherent OAM transfer from OV pulses to the center of mass momentum of excitons in semiconductor GaN. We show that the output signal from the four-wave mixing (FWM) process with OV pulse excitation is spatially coherent within the exciton dephasing time (~1ps) and exhibits phase singularities that satisfy the OAM conservation law. We also demonstrate the time evolution of OAM transfer in the emitting signal during the single input pulse, and optical gate control for the OAM by changing the input OAM. Our demonstrations represent a new type of coherent spectroscopy using OV pulses and provide a way of controlling sequential OV pulses through carriers in solids.
In our experiments, we realize the coherent excitation of excitons with OAM via self-diffracted FWM using a pair of OV pulses (k 1, ℓ 1 and k 2, ℓ 2) with a delay time τ. Here, we consider the electric fields of the incident δ -function pulses with co-linear polarizations given by , where ω 0 is the center frequency of the pulse. An OV with a paraxial approximation is characterized by the Laguerre-Gaussian mode (LG0 ℓ, and hereafter denoted simply as LGℓ) having a radial index of 0 with the form exp(iℓnφ). When ℓ=0, the beam is the fundamental Hermite-Gaussian (HG00, and hereafter denoted as HG) mode. The two incident pulses interfere via some nonlinearities of the sample and produce nonlinear polarization waves that contain P (3) 2k2-k1(P (3) 2k1-k2) in the direction 2k 2-k 1 (2k 1-k 2) . For semiconductor materials, the manifestation of optical nonlinearity arises from the phase-space filling and Coulomb interactions, both of which are responsible for the anharmonicity of the exciton transitions. Within the framework of optical Bloch equations, one of the P (3) is described by
Here µx, h̄ωx and γx are the exciton dipole matrix element, energy and relaxation rate, respectively, and Θ is the Heaviside step function. The equation clearly indicates that the output photons are characterized by the wavevector (k FWM=2k 2-k 1) and OAM (ℓ FWM=2ℓ 2-ℓ 1). Within the paraxial approximation for OVs, these two parameters are independently controllable. As a result, we can generate OVs with an arbitrary OAM in the same direction by changing the OAM in the one of the incident pulses. In the resonance condition (ω 0=ωx), is nonzero when t≥τ≥0 and it decays with τ. It is important to note that µx may exhibit spatial variations in terms of spatial coherence, which reflects the spatial phase relationship among excitons and with the electric fields creating them. If the spatial dephasing such as diffusion or spatially-dependent scattering is efficient, the magnitude of the total µx, i.e.,P (3) also decays with τ.
Figure 1 (a) shows a schematic of the FWM setup with OV pulses. The measurement was performed using a frequency-doubled mode-locked Ti:sapphire laser with a spectral width of 10 meV (~1.0 nm). Without spatial modulations to generate the OVs, the pulse duration is estimated to be ~130 fs. A center energy of ~3.495 eV (~354.8 nm) was selected for the GaN exciton transitions. The ultraviolet (UV) laser pulse is split into two collinearly polarized pulses with the same intensity. The delay time τ between the two pulses was determined by the variable spatial delay line.
The OV pulses were created from the HG laser beams by rotating the phase of the wavefront. Here, the mode conversion was realized by a computer generated holographic (CGH) technique using a two-dimensional (2D) spatial light modulator (SLM), in which all the optical elements are optimized for the UV wavelength. The phase patterns without fringes were used to gain a high conversion efficiency (upper part of Fig.1(b)). A conversion efficiency up to 70% was achieved in our measurements. Such phase patterns also enable us to avoid the spatial chirp induced by the diffraction . Figure 1(b) shows the intensity profiles of the incident OV pulse with various values of ℓ together with the corresponding phase patterns. The temporal chirp still remains but does not significantly affect the spatial and temporal profiles of the OV pulse owing to the relatively small spectrum bandwidth. The effect of the unmodified portion of the HG beams also remains and will be discussed in detail later.
Two pulses whose time-averaged powers just before focusing were ~300 µW/pulse were overlapped on the sample using a lens (f=200 mm), and the diffracted FWM signal was collected in the reflection geometry (Fig.1(c)). The sample was mounted on a cold finger of a liquid helium cryostat and all the data were obtained at 10 K. A simple evaluation of the phase singularity of the beams is realized by measuring the dark holes in the spatial intensity profile. The FWM signal was recorded using a charge-coupled device (CCD) camera equipped with an image acquisition system. The total intensity was evaluated by the spatial integration of the beam profile. A more reliable way of checking the topological charge of the OV in FWM is to observe the interference pattern, which was realized by overlapping the output signal with a reference pulse with the HG mode generated by another frequency-doubling of the laser. For the spectrally-resolved FWM, the diffracted signal was introduced into a monochromator by a flipper mirror. The spectral resolution was better than 0.1 nm.
The sample studied here consists of undoped GaN film (thickness: 3 µm) grown on the (0001) c-plane of a sapphire substrate. The sample properties are detailed in Ref. . We comment that the imperfection of the phase-matching for two-pulse degenerate FWM is negligible since the signal is created in the vicinity of the sample surface in our reflection geometry . Figure 2(a) shows a typical FWM spectrum obtained at τ=0. In the spectrum, the lower- and higher-energy peaks are identified as two of three allowed free-exciton transitions (A- (XA: 3.4909 eV) and B-exciton (XB:3.4992 eV) transitions, respectively). The simultaneous excitation of the two transitions results in a quantum beat (QB) in the time-domain FWM as described in detail in the next section.
3. Results and discussions
Figure 2 (b) shows the time evolutions of FWM signals as a function of τ in the signal directions 2k 2-k 1 (blue solid circles) and 2k 1-k 2 (red open circles). Here a combination of the HG pulse (ℓ 1=0) in the k 1 direction and LG1 (ℓ 2=1) pulse in the k 2 direction was used for coherent excitation. The crossing point between two data indicates the zero position of the delay τ. Both data show a QB oscillation arising from the simultaneous excitation of XA and XB transitions, where the beating period (TQB) of 0.51 ps is consistent with the energy separation of 8.3 meV (Δν~THz). Figure 3(a) shows the two sets of spatial FWM images obtained at each QB oscillation peak in the signal directions 2k 1-k 2 (left) and 2k 2-k 1 (right). Each data set shows that the output FWM beam exhibits a characteristic intensity profile and its intensity decays with increasing |τ|.
We first focus on the difference between the beam profiles of the data sets; the FWM in the 2k 1-k 2 direction exhibits a single dark spot at the center while the FWM in 2k 2-k 1 exhibits two dark holes. From Eq.(1), the OAM of the FWM polarizations with 2k 1-k 2 and 2k 2-k 1 are 2ℓ 1(=0)-ℓ 2(=1)=-1 and 2ℓ 2(=1)-ℓ 1(=0)=2, respectively. The exact number of phase dislocations will be verified later using an interference measurement (see Fig. 4). Note that an LG beam with a charge ℓ, even with a higher |ℓ|>1, exhibits a single dark spot. The split of the dark spot with ℓ into |ℓ| spots occurs when there is a weak perturbation of the HG component (HGbg), and this has widely been observed in LG beams generated by nonlinear processes [18–20]. To clarify the formation of the splitting, we display the results of theoretical calculations for the phase and intensity profiles of the LG2 with and without HGbg in Fig. 3(b) and (c), respectively. In Fig. 3(b), we assume that the fraction of HGbg to LG2 is 0.2. In the phase profile of LG2+HGbg (upper data in Fig. 3(c)), the phase dislocation splits into two dislocations with ℓ=1, and each dislocation shifts outwards in a radial direction. As a result, the single dark spot splits into two separate dark spots in the intensity profile (lower data in Fig. 3(c)), which agrees qualitatively with the results of ℓ FWM=2 (right part of Fig. 3(a)). The HG background mainly arises from the unmodified portion of the LG1 in k 2 pulse during the mode conversion process with SLM. Because of the difference of the mode profile between LG and HG beams, the FWM conversion efficiency between the remnant HG in k 2 and HG in k 1 is larger than that between the LG2 in k 2 and HG in k 1, giving rise to increase the fraction of HGbg to LG2 even if the unmodified portion of the incident LG (i.e. HG) is small. Another plausible origin is the imperfection of the phase-matching for FWM. It is important to remember that perfect phase-matching for two-pulse degenerate FWM can be realized in a signal produced in the vicinity of the sample surface . The FWM in thick samples thus meet the imperfect phase matching condition. However, this effect should be small in our reflection geometry under resonant excitation condition. It is also important to note that the HGbg can be removed by introducing an additional HG pulse whose phase and amplitude are adjusted to cancel out the output HGbg .
We now consider the time evolutions of the OAM in FWM. The QB in Fig.2(b) shows a clear oscillation, whose visibility is almost unity, suggesting that the time resolution is much better than TQB/2~0.25 ps. From Eq.(1), the time integrated FWM signal is given by I FWM ∝∫dt|P (3)|2, as a function of τ. If we assume that we have homogeneously broadened excitons, the FWM signal with a QB oscillation can be expressed by
where we neglect the Coulomb interactions between excitons . From the fitting to the data, we obtain a dephasing time of T 2=1.40 ps (T 2=2.80 ps for the inhomogeneous case), which is in good agreement with that obtained from the FWM without the mode conversion scheme (i.e., conventional excitation with HG pulses) [17, 21]. This consistency is quite reasonable because the contributions of spatial dephasing such as diffusion or spatially-dependent scattering are very small in such a short time region. Let us remember that µx in Eq.(1) is a position dependent function, and its magnitude reflects the spatial phase relationship between excitons and with the incident pulse. If the spatial dephasing is efficient, the magnitude of the total µx decays with τ. The dark spot(s) in Fig.3(a) are clearly visible within the dephasing time and the contrast between the dark and bright portions of the OV remains almost constant (compare the top and bottom signals in Fig.3(a)). Therefore, we conclude that the dephasing of OAM in the present study is limited by the T 2 of spatially-coherent excitons. We stress the importance of the fact that the short dephasing of OAM is useful for the fast switching applications. We also note that the application of the OAM memory can be developed by using exciton systems with a longer dephasing time, such as excitons in semiconductor quantum dots . Furthermore, the OAM dephasing observed here is responsible for the spatial coherence characterized by the azimuthal direction. Coherent spectroscopy using OVs allow the spatially-dependent coherent properties of carriers in solids such as closed-loop electron coherence in quantum ring structures [13, 14].
To evaluate the capability for switching applications, we next carried out the arbitrary control of OAM in FWM. In this experiment we treated the k 1 pulse as a gate pulse for controlling the output OAM and detected the FWM in the 2k 2-k 1 direction. The OAM in the k 2 pulse was fixed at ℓ 2=1 as a target signal. Fig.4 (a)–(c) show the results of the intensity profiles of the output FWM (τ≈0 ps) with ℓ 1=0,1, 2, respectively. On the basis of the OAM conservation law, the output FWM exhibits ℓ FWM=2,1, 0, whose variations are seen in the different numbers of dark spots. For further corroboration, we checked the interference patterns (Fig.4 (d)–(f)) by overlapping the signal with a tilted HG00 reference beam. The parallel fringe pattern produced phase dislocations (fork-like structures) at singular points if the output FWM beam included finite topological charges [5, 19]. Theoretically calculated interference patterns are also displayed in Fig.4 (g)–(i) for comparison. In Fig.4 (d), we can see two phase dislocations with the same topological charge of 1, which are well reproduced by the simulation (Fig.4 (g)), where the contribution of HGbg is taken into account. As a result, the overall topological charge is 2 (ℓ FWM=2). For comparison, Fig. 4 (e) shows one phase dislocation with ℓ FWM=1. In contrast, no phase dislocation can be seen in Fig.4 (f) showing ℓ FWM=0. These results agree well with those predicted under the conservation law and are responsible for the fact that the arbitrary OAM control is possible in FWM using a gate OV pulse. It should be noted that conversion to the HG mode is important for separating the target signal from the composed vortex field, which can easily be realized by using a pinhole/single-mode fiber in front of the detection devices [6, 7].
In summary, we successfully demonstrated the OAM transfer from photons to the free excitons of GaN in coherent manner (in a time region of several picoseconds). The coherent transfer was realized with a FWM scheme, showing that the phase singularities in the output OVs follow the OAM conservation law and were clearly visible within the exciton dephasing time. Using the FWM process, we also demonstrated the optical gate control of OAM in the output OVs. If OVs are to be practically employed for optical communications and information processing, the time variable OVs must be sent subsequently. Our demonstrations thus mark progress in the generation and control of sequential OV pulses, and pave the way for the future optical communications using multiple OVs. Moreover, in terms of coherent spectroscopy, our time evolution measurement ofFWMsignals with OAMprovides a new way for studying the closed-loop coherence of carriers in solids, which should play an important role in the topological materials such as quantum rings.
YT acknowledges the Sumitomo Foundation, the Foundation for Opto-science and Technology. RM acknowledges Grant-in-Aid for Scientific Research (B), 2008-2010, No.20360025 from the Japan Society for the Promotion of Science (JSPS).
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