Realization of all-optical vortex switching in exciton-polariton condensates

TL;DR

This paper demonstrates a practical method for controlling and switching localized vortices in exciton-polariton condensates using optical techniques, enabling potential applications in data storage and information processing.

Contribution

It introduces a simple, robust approach for optically controlling and switching vortex states in exciton-polariton condensates, advancing their use in information technology.

Findings

Successful optical control and switching of vortex states

Detection of vortex states via emitted light's orbital angular momentum

Potential for practical vortex-based information processing

Abstract

Vortices are topological objects representing the circular motion of a fluid. With their additional degree of freedom, the 'vorticity', they have been widely investigated in many physical systems and different materials for fundamental interest and for applications in data storage and information processing. Vortices have also been observed in non-equilibrium exciton-polariton condensates in planar semiconductor microcavities. There they appear spontaneously or can be created and pinned in space using ring-shaped optical excitation profiles. However, using the vortex state for information processing not only requires creation of a vortex but also efficient control over the vortex after its creation. Here we demonstrate a simple approach to control and switch a localized polariton vortex between opposite states. In our scheme, both the optical control of vorticity and its detection through the orbital angular momentum of the emitted light are implemented in a robust and practical manner.

Equations

$$P(r)=P_0\frac{r^2}{w^2}{e}^{-\left(\frac{r^2}{w^2}\right)}+a{P}_0{e}^{-\frac{(r-r_0{)}^2}{w_p^2}}{e}^{-\frac{t^2}{w_t^2}},$$

$$m=\frac{-i\int {\Psi }^{\ast }(\mathbf{r}\times \nabla )\Psi d\mathbf{r}}{\int |\Psi {|}^{2}d\mathbf{r}}.$$

$$i\hslash \frac{\partial \Psi (\mathbf{r},t)}{\partial t}=$$

$$+\left(g_{\text{r}}+i\hslash \frac{R}{2}\right)n_{\text{A}}(\mathbf{r},t)+g_{\text{r}}{n}_{\text{I}}(\mathbf{r},t)$$

$$+V_{\text{ext}}(\mathbf{r})]\Psi (\mathbf{r},t).$$

$$\frac{\partial n_{\text{A}}(\mathbf{r},t)}{\partial t}=\tau n_{\text{I}}(\mathbf{r},t)-{\gamma }_{\text{A}}{n}_{\text{A}}(\mathbf{r},t)-R|\Psi (\mathbf{r},t){|}^2{n}_{\text{A}}(\mathbf{r},t).$$

$$\frac{\partial n_{\text{I}}(\mathbf{r},t)}{\partial t}=-\tau n_{\text{I}}(\mathbf{r},t)-{\gamma }_{\text{I}}{n}_{\text{I}}(\mathbf{r},t)+P(\mathbf{r},t).$$

Full text

I Introduction

The two fundamental states of a vortex represent different rotation directions and can be used to encode binary information for example in vortex-based random access memories Miyahara et al. (1985). Switching a vortex between its two states is coveted by scientists and the dynamics have been investigated intensively Schneider et al. (2001); Xiao et al. (2006); Hertel et al. (2007); Kim et al. (2008); Clark et al. (2015); Driben et al. (2016); Kong et al. (2017). For example, the manipulation of core magnetization of a vortex in ferromagnetic materials can be achieved by magnetic fields Schneider et al. (2001); Xiao et al. (2006); Hertel et al. (2007) or by an electrical current through the spin-transfer effect Yamada et al. (2007); Caputo et al. (2007). To overcome the so-called electric bottleneck in data transmission, however, all-optical networks have been proposed which requires optical data storage and processing including all-optical switching. Promising platforms to create and utilize vortices in opticsAllen et al. (1992); Swartzlander Jr and Law (1992); Chen et al. (1997); Bozinovic et al. (2012) include plasmonic or dielectric metasurfaces Kim et al. (2010); Genevet et al. (2012); Kildishev et al. (2013); Karimi et al. (2014); Mehmood et al. (2016); Huang et al. (2017) as well as exciton-polaritons in semiconductor microcavities Lagoudakis et al. (2008); Sanvitto et al. (2010); Roumpos et al. (2011); Gao et al. (2018).

A typical planar semiconductor microcavity is sketched in Fig. 1(a). As fundamental exictations of this system, exciton-polaritons are hybrid light-matter objects, composed of both quantum well excitons and cavity photons. In a high quality semiconductor microcavity, the strong coupling between photons and excitons leads to an anti-crossing of the fundamental modes generating spectrally well separated upper-polariton branch (UPB) and lower-polariton branch (LPB) as illustrated in Fig. 1(b). Through their photonic component polaritons can be optically excited and probed and show remarkable coherence properties. The excitonic component leads to strong optical nonlinearity, which is at the heart of many interesting nonlinear phenomena Klembt et al. (2018); Delteil et al. (2019); Zasedatelev et al. (2019); Caputo et al. (2019); Muñoz-Matutano et al. (2019) including the formation of non-equlibrium polariton condensates Kasprzak et al. (2006); Deng et al. (2010) and polariton vortices Dall et al. (2014); Dreismann et al. (2014); Sigurdsson et al. (2014); Ma et al. (2016). This nonlinearity also allows for active optical control of the system dynamics, for example with tailored and spatially structured off-resonant excitation.

In the case of polariton condensation, an off-resonant optical beam creates a reservoir of ”hot” excitons. Through Coulomb and phonon scattering events these then relax towards the so-called bottleneck region close to the top of the LPB. From there, polaritons undergo stimulated scattering towards the ground state as sketched in Fig. 1(b). Due to the finite lifetime of polaritons, even for a macroscopic population and a build-up of long-range coherence in the polariton ground state (i.e., condensation), the coherent polariton system intrinsically is in a non-equilibrium state subject to drive and decay. A persistent external pump with frequency far above the exciton resonance can be used to sustain the population in the condensate. Besides replenishing the condensate, however, the reservoir excitations also play an important role in the dynamics of the condensate due to reservoir-condensate interaction. Moreover, the reservoir inherits the spatial shape from the optical pump as indicated in Fig. 1(a) which can be tailored to specific needs. Here we use a ring-shaped pump that yields a reservoir-induced external potential seen by the condensate as illustrated in Fig. 1(c) (including the perturbation by an additional control beam). This potential is used to trap the condensate. As a result of spontaneous symmetry breaking in the persistently pumped nonlinear dynamical system, this ring-shaped excitation profile also leads to the formation of a vortex inside the ring. Independent from the off-resonant pump beam, this vortex carries a finite orbital angular momentum (OAM). For a system and optical excitation setup with rotational symmetry, two possible vortex states with opposite topological charges are supported by the same pump excitation. Recently, chiral polaritonic lenses Dall et al. (2014) and external potentials Sigurdsson et al. (2014) have been used to generate specific vortices by breaking the central symmetry and indirect vortex control using complex optical setups was proposed theoretically Ma et al. (2016); Ma and Schumacher (2017).

Here, we demonstrate experimentally and theoretically the direct optical control of an optically imprinted trapped vortex charge using an ultra-short ($120$ femtoseconds in our experiments) off-resonant optical control pulse that can reverse the topological charge of an already formed vortex. The basic principle is that an off-resonant optical pulse close to the vortex excites a potential barrier, which stops the rotating vortex and flips it to the opposite rotation direction. Our detailed analysis shows that with the perturbation of the initially formed vortex mode the nonlinear dynamical system evolves into an oscillating dipole mode. After switching off the perturbation the system returns to the vortex mode with a charge that depends on the duration and the power of the control pulse. We directly measure the topological charge of the vortex and its temporal evolution using an efficient OAM sorting approach. Compared with interferometric approaches for which a phase reference is needed, this simple and robust detection scheme is directly relevant to future applications as it only requires the bare sample emission as the input and works down to the level of individual photons Lavery et al. (2012). In the present work the vortex switching is demonstrated in a high-quality GaAs-based semiconductor sample, however, we find that the scheme is quite robust against sample disorder which will be important for the transfer to other materials, potentially allowing operation at up to room temperature.Kéna-Cohen and Forrest (2010); Plumhof et al. (2014)

II Results

II.1 Principle of optical vortex control

Before moving on to the experimental realization, we will first explain the basic principle of vortex switching supported by a detailed numerical analysis. Our calculations are based on a microscopic model for the coupled spatio-temporal light-field exciton dynamics inside the semiconductor microcavity. More details are given in the Methods section. Let us first consider excitation of the microcavity system with an off-resonant continuous wave (CW) ring-shaped pump beam. In the case of full rotational symmetry, the polariton condensate that forms in the annular potential will spontaneously form a vortex with clockwise or counter-clockwise rotation depending on initial fluctuations in the coherent polariton field acting as a seed in the condensation process Ma et al. (2016). We will address the influence of sample disorder and reduced rotational symmetry in detail in a separate section below. In Fig. 2(a) the example of a spontaneously formed vortex rotating counter-clockwise is shown. Now, in addition to the CW ring pump we also switch on a Gaussian pump spatially localized in the ring region. This perturbation intentionally breaks the rotational symmetry of the optically imprinted external potential seen by the condensate as sketched in Fig. 1(c) and the condensate assumes a dipole mode as shown in Figs. 2(b) and 2(c). This dipole mode, however, is not static, but persistently oscillates back and forth between the two orientations in Figs. 2(b) and 2(c) as long as the Gaussian perturbation is present. Once the Gaussian control beam is switched off and the height of the potential barrier is reduced, the perturbation potential acting on the dipole becomes weaker and eventually the dipole transforms back into a vortex mode which may lead to a vortex rotating in the direction opposite from the initial vortex as shown in Fig. 2(d). The final rotation direction depends on the rotation direction of the dipole at the time when the perturbation potential is sufficiently reduced in strength to allow for a stable vortex mode to form again and thus can be set by altering the duration of the perturbation potential.

II.2 Experimental realization of vortex switching

Here we closely follow the concept discussed in the previous section. The main experimental results are shown in Fig. 3. We first create a vortex by only applying a ring-shaped CW pump beam as sketched in Fig. 1(a). In order to optimize the visibility of the vortex switching process, we employ a pump beam with a small intentional asymmetry, such that a slightly preferred direction of rotation for the vortex exists. To identify the vortex switching process we use our direct access to the OAM of the light emitted from the condensate on a shot-by-shot basis with picosecond time resolution. We cannot apply more common interferometric techniques in this case as phase patterns are not stable over several pulses and there is no phase reference for non-resonant excitation. Here we employ OAM sorting Berkhout et al. (2010); Berger et al. (2018), which converts the phase gradient into a spatial displacement and is applicable down to the single photon level. Figure 3(a) shows the measured time-resolved photoluminescence from the polariton condensate resolved into its OAM components (denoted as topological charge, m). Optical setup and data analysisBerkhout et al. (2010); Berger et al. (2018) as well as sample details Schmutzler et al. (2015) are discussed in the Methods section. With excitation by a CW ring pump only, at time $t<0\,\mathrm{ps}$ the emitted signal is dominated by the $m=+1$ topological charge, evidencing that a counter-clockwise rotating vortex is created. At $t=0\,\mathrm{ps}$ an additional off-resonant pulse with duration of $120\,\mathrm{fs}$ is applied, creating spatially localized reservoir excitations such that the rotational symmetry of the effective external potential seen by the condensate is temporarily destroyed as discussed in the previous section. At low pulse power of $1\,\mathrm{mW}$ as shown in Fig. 3(a), shortly after application of the pulse, at $t\sim 100\,\mathrm{ps}$ the $m=+1$ state is suppressed and the $m=-1$ state dominates for a short period of time. Then the system reverts back to the $m=+1$ state at $t\sim 200\,\mathrm{ps}$. For a slightly increased pulse power of $4\,\mathrm{mW}$ as shown in Fig. 3(b), the system oscillates back and forth twice with the $m=-1$ state dominating at $t\sim 50\,\mathrm{ps}$ shortly after application of the pulsed perturbation and then persistently dominating the signal from $t\sim 300\,\mathrm{ps}$ as the system switches into a new state with the formation of a clockwise rotating $m=-1$ vortex. In this case the application of the additional pulse inverts the vortex charge. When the pulse power is increased further to $20\,\mathrm{mW}$ in Fig. 3(c), the system again undergoes an intermediate oscillation period with dominant $m=+1$ and $m=-1$ states but then at longer times returns to a state with predominantly counter-clockwise rotation of the vortex. Finally we would like to note that while the duration of the control pulse in the experiments reported in Fig. 3 is only $120\,\mathrm{fs}$, the effective duration of the perturbation is determined by rather complex relaxation processes. Accordingly, the switching dynamics reported here evolves on a much longer, few hundreds of picoseconds, timescale. After having demonstrated vortex switching and control by simple application of a short off-resonant spatially localized light pulse in the present section, in the following section we will theoretically investigate the general dependence of the vortex dynamics on the power and duration of the control pulse.

II.3 Vortex dynamics

We write the intensity profile of the laser beam used for excitation of the microcavity system in the calculations as

[TABLE]

with $r=\sqrt{x^{2}+y^{2}}$. Here, we chose $P_{0}=6$ ps*-1* $\mu$m*-2*. $w=5$ $\mu$m is the radius of the ring-shaped CW pump beam, $a$ scales the intensity of the Gaussian pulse with spatial width $w_{p}=3$ $\mu$m and peak intensity at $r_{0}=5$ $\mu$m, $w_{t}$ is the duration of the pulse.

In the calculations, we first create a vortex in the polariton condensate from initial noise by application of the off-resonant ring-shaped CW pump beam. After the condensate reaches stationary behavior in the vortex mode as in Fig. 2(a), the Gaussian perturbation pulse with magnitude $a=3$ is also applied. Figure 4(a) shows examples for the time evolution of the total topological charge $m$ of the condensate when the pulse is applied at $t_{0}=0\,\mathrm{ps}$. The topological charge $m$ is calculated as

[TABLE]

Results are shown for two different pulse lengths with $w_{t}=10\,\mathrm{ps}$ and $w_{t}=20\,\mathrm{ps}$, respectively. For $w_{t}=10\,\mathrm{ps}$, after application of the perturbation pulse the vortex with topological charge $m=+1$ is smoothly switched to the vortex state with opposite topological charge $m=-1$. For a longer pulse with $w_{t}=20\,\mathrm{ps}$, however, the vortex is hindered in its rotation and temporarily converted into an oscillating dipole mode with near zero total topological charge and then it is stopped again and switched back to the original state with $m=+1$.

In a more complete picture, whether successful vortex switching (inversion of the vortex charge) occurs after application of the control pulse depends on both pulse duration and intensity as shown in Fig. 4(b). For a fixed intensity of the Gaussian control pulse, the duration of the pulse directly affects the lifetime of the potential barrier induced. The lifetime of the potential barrier in turn determines how many times the intermediate dipole mode oscillates back and forth. If it is ‘flipped’ an odd (even) number of times the final switching result is on (off). For a very long control pulse or CW control, the system persistently oscillates between the two topological states, topologically forming an oscillating dipole mode as shown in Fig. 2. For fixed duration, the switching dynamics and overall outcome depends on the intensity of the control. When the intensity is increased, a shorter pulse duration is required for vortex charge inversion. We find that for a higher potential barrier the oscillation period of the dipole is decreased. Moving along the vertical dotted line in Fig. 4(b) one can see that for fixed pulse duration the switching status alternates with increasing pulse intensity. This observation coincides well with the experimental results shown in Fig. 3. We note that in the experiment reported in Fig. 3, switching is achieved with a pulse duration of only $120\,\mathrm{fs}$ whereas in the simulations for such short pulses no switching would occur. In our theoretical description the relaxation processes from the reservoir to lower energy states are only treated on a phenomenological level, in which the complicated electron-phonon interaction processes are not considered in detail. As a consequence, especially for short pulses, control pulse durations needed to achieve switching can not directly be compared between theory and experiment. The timescales of the switching dynamics (hundreds of picoseconds), however, in both experiments in Fig. 3 and theory in Fig. 4(a) agree well with each other.

II.4 Role of disorder

In this section we discuss the role that sample disorder plays for our results and analysis. In this context it is worth noting that in the measurements performed in the present work, for a fully rotationally invariant system, the vortex switching would not be evidenced. Data shown in Fig. 3 are obtained by averaging over many control pulses and switching events. If the rotation direction of the initial vortex state was only decided by random fluctuations triggering the condensation process, clockwise and counter-clockwise rotation would occur with equal probabilities. In that case the OAM sorted signal would initially show either one or the other of the $m=+1$ and $m=-1$ states. Averaging over many successful switching events, after application of the control pulse, the measured signal would be distributed with equal intensity to the $m=+1$ and $m=-1$ states again, so even if a switching process occurred, it would not be evidenced in the measured data. In the presence of a moderate disorder profile, however, rotational symmetry is intrinsically broken slightly, resulting in an initial imbalance of the m=+1 and m=-1 states and vortex switching can be clearly observed as shown in Fig. 3. Another benefit of the disorder leading to broken symmetry is that the system as shown in Fig. 3(b) will predominantly occupy the $m=+1$ state again on a long timescale (several nanoseconds after the application of the pulse). Otherwise, an equal mixture of both $m=\pm 1$ states would be observed in Fig. 3(b), as the vortex would get switched back and forth with every additional pulse. The built-in asymmetry enables us to observe the switching dynamics even averaging over many events.

In Fig. 5(a) we show the time-integrated spatially resolved photoluminescence before application of the control pulse at $t<0$ in Fig. 3(a). Figure 5(b) shows the corresponding ring-shaped intensity profile of the CW pump beam used to excite the sample that due to its imperfections also slightly breaks the system symmetry. Apparently, in the real system with its imperfections the condensate state formed in the system as shown in Fig. 3(a) does not show the perfect ring-shaped emission profile expected for an ideal individual vortex as assumed in the theoretical calculations, e.g., shown in Fig. 2(a). Not surprisingly, for each fixed point in time in Fig. 3 also the OAM-resolved photoluminescence does not show only a single OAM contribution of well defined topological charge but two dominant OAM states with different amplitudes and $m=+1$ and $m=-1$, respectively. The simultaneous initial presence of $m=+1$ and $m=-1$ components is a necessary prerequisite to observe the vortex switching process and also explains the dipole-like appearance of the spatial intensity profile in Fig. 5(a).

In our numerical simulations, we now include a finite sample disorder as a random external potential for the coherent polariton condensate with realistic magnitude of $0.2\,\mathrm{meV}$ and correlation length $1\,\mathrm{\mu m}$ Schmutzler et al. (2015). For one random disorder configuration, the resulting condensate density and phase are shown in Figs. 5(c) and 5(d) before and in Figs. 5(e) and 5(f) after application of a control pulse to switch the rotation direction of the vortex. We note that in the numerical simulations we find that if sample disorder gets too strong, a stable dipole mode is formed that can not serve as the initial state for switching. For comparison, in Fig. 5 we also show the (g) density and (h) phase profiles of a superposition state $\Psi_{\text{sp}}({\bf r})=C_{+}\Psi_{+}({\bf r})+C_{-}\Psi_{-}({\bf r})$ of two vortices $\Psi_{\pm}({\bf r})$ with opposite topological charges $m=\pm 1$ and different amplitudes with $C_{+}/C_{-}=3$. Here $\Psi_{+}({\bf r})=e^{-{\bf r}^{2}/W^{2}}e^{i\phi}$ and $\Psi_{-}({\bf r})=e^{-{\bf r}^{2}/W^{2}}e^{-i(\phi+\delta)}$, with in-plane angle $\phi$ and a phase difference $\delta=\pi/3$ and $W=4$ $\mu$m. The resulting density profile shown in Figs. 5(g) shows good qualitative agreement with the experimental and numerical results in Figs. 5(a) and 5(c) while the phase profile (h) illustrates the vorticity of the state. Overall, this discussion illustrates that in the experimental observations in Figs. 3 and 5(a) a vortex state with two OAM contributions but one dominant contribution is created and switched in the presence of sample disorder.

III Discussion

In the present work we have demonstrated the all-optical switching of the charge of a single localized vortex in a microcavity polariton condensate. The generation of the vortex and the switching is achieved using only off-resonant control pulses. In contrast to resonant excitation, our switching scheme is universal. With the same optical beam we can switch from a $m=-1$ state into a $m=+1$ state and vice versa, while for resonant excitation this is not as simple. In principle the same scheme also works for higher topological charges. Using only off-resonant excitation, our approach may also allow for an electrically pumped realization in the future and our experimental readout of the vortex charge by OAM sorting is exceptionally simple and practical in contrast to an interferometric approach and works down to the single-photon level. We further show that our scheme is remarkably robust against sample disorder and system imperfections and may pave the way for practical realizations of all-optical switching and information processing based on polariton vortices in semiconductor microcavities.

methods

Theory. The dynamics of the coherent polariton system can be analyzed in detail based on a driven-dissipative Gross-Pitaevskii (GP) type model:Wouters and Carusotto (2007)

[TABLE]

Here $\Psi(\mathbf{r},t)$ is the coherent complex-valued polariton field on the lower polariton branch in effective mass approximation, with $m_{\text{eff}}=10^{-4}m_{\text{e}}$ and $m_{\text{e}}$ the free electron mass. Loss of polaritons due to their finite lifetime is included through $\gamma_{\text{c}}=0.1$ ps*-1*. The repulsive polariton-polariton interaction and resulting cubic nonlinearity in the equation is included with $g_{\text{c}}=6\times 10^{-3}\text{meV}\mu\text{m}^{2}$. The external disorder potential is given by $V_{\text{ext}}(\mathbf{r})$. In order to include the relaxation and scattering dynamics for non-resonant optical excitation [cf. Fig. 1(b)], we assume coupling to two reservoirs, one active reservoir ($n_{\text{A}}$) and one inactive reservoir ($n_{\text{I}}$) Lagoudakis et al. (2011); Schmutzler et al. (2015). The condensate is replenished directly from the active reservoir through a stimulated in-scattering process with $R=0.02$ ps*-1* $\mu$m2. The repulsive Coulomb interaction between condensate and excitations in the active reservoir is represented by $g_{\text{r}}=2g_{\text{c}}$. The density of the active reservoir satisfies

[TABLE]

Here $\gamma_{\text{A}}=0.15$ ps*-1* is the polariton loss from the active reservoir. The active reservoir is replenished from the inactive reservoir with $\tau=0.1$ ps*-1*. The inactive reservoir contains hot excitons excited directly by the external non-resonant pump and obeys the following equation of motion:

[TABLE]

Here $\gamma_{\text{I}}=0.01$ ps*-1* is the loss from the inactive reservoir and $P(\mathbf{r},t)$ represents the non-resonant pump. As in the experiment, the pump has a frequency far above the excitonic resonance. Here we assume that the phase information of the optical pulse is lost through the complicated relaxation processes occurring in the real system. In this work, two different optical beams are used for excitation: one is a ring-shaped CW pump for exciting and sustaining a vortex and the other one is a Gaussian-shaped pulse for controlling the topological charge of the vortex [cf. Fig. 1(a)].

Microcavity sample. The sample we investigate is a MBE-grown planar microcavity based on GaAs with a quality factor of about 20.000 and a Rabi splitting of 9.5 meV. It consists of two distributed Bragg reflectors (DBR) made of 32 and 36 alternating layers of $\text{Al}_{0.2}\text{Ga}_{0.8}\text{As}$ and AlAs enclosing a $\lambda/2$ cavity. In the central antinode of the electric field four GaAs quantum wells are placed. The sample is mounted in a helium-flow cryostat to cool it down to the measurement temperature of 17 K. The exciton-cavity detuning is -4 meV for all experiments shown in this work.

Spectroscopy. For non-resonant optical excitation a CW laser with a wavelength of 735.5 nm (1686 meV) is used. The CW laser is shaped by using a spatial light modulator (SLM) to generate a nominally annular optical potential. For the switching pulses a pulsed titanium-sapphire laser (repetition rate 75.39 MHz) emitting pulses with a duration of approximately 120 fs with the same central wavelength as the CW laser is used. Both beams are focused onto the sample using a microscope objective (numerical aperture 0.4). For real space imaging a liquid nitrogen-cooled CCD camera placed behind a monochromator (operated in zeroth order) is used. For time resolved imaging of the OAM, the signal beam is guided through the OAM sorting process and imaged onto a streak camera.

Orbital angular momentum sorting. The OAM sorting process works as follows: First, the beam carrying OAM is imaged onto a SLM displaying the transformation phase pattern of the OAM sorter, which transforms the helical phase gradients of OAM states to linear phase gradients. An ideal transformation of the gradients cannot be achieved by a single phase pattern. Accordingly, usually another lens is placed behind the SLM, which performs an optical Fourier transformation of the beam. In the Fourier plane an additional phase correction pattern is placed, which corrects remaining deviations of the wave front from the linear phase gradient. Finally the output light beam is imaged onto a detection device such as a CCD camera using another lens. This lens then focuses different OAM-sorted states with different phase gradients to corresponding spots at different lateral positions on the CCD, where each OAM state is mapped to one detector position. In our experimental implementationBerger et al. (2018) the two phase patterns of the OAM sorting process are displayed on two halves of a single SLM and a concave mirror is used to guide the beam and perform the optical Fourier transformation like a lens would do. The output light beam is then imaged onto the entrance slit of a streak camera, which is opened by 100 $\mu$m and aligned along the axis of OAM deflection.