Orbital angular momentum coupling in elastic photon-photon scattering

TL;DR

This paper explores how orbital angular momentum influences elastic photon-photon scattering in vacuum, proposing a new experimental approach to detect OAM signatures in generated photons using high-power lasers.

Contribution

It introduces exact solutions carrying OAM to the vacuum electromagnetic wave equation and derives the coupling effects within an effective field theory framework.

Findings

OAM affects the coupling between initial photon waves

OAM introduces a detectable signature in generated photons

Proposes a laser experiment to observe OAM effects

Abstract

In this letter, we investigate the effect of orbital angular momentum (OAM) on elastic photon-photon scattering in vacuum for the first time. We define exact solutions to the vacuum electro-magnetic wave equation which carry OAM. Using those, the expected coupling between three initialwaves is derived in the framework of an effective field theory based on the Euler-Heisenberg La-grangian and shows that OAM adds a signature to the generated photons thereby greatly improvingthe signal-to-noise ratio. This forms the basis for a proposed high-power laser experiment utilizingquantum optics techniques to filter the generated photons based on their OAM state

Figures

Equations

$$\text{E}(\text{r},t)={\mathcal{E}}_0{e}^{i(\text{k}\cdot \text{r}-\omega t)}\hat{\text{n}}$$

$${\text{E}}_l(\text{r},t)=\frac{{\mathcal{E}}_0}{2\pi }{\int }_0^{2\pi }(i{)}^l{e}^{i(\text{k}({\varphi }_k)\cdot \text{r}-{\omega }_{0}t+l{\varphi }_k)}\hat{\text{n}}({\varphi }_k)d{\varphi }_k$$

$$\text{k}({\varphi }_k)=k\cos (\alpha )\hat{\text{z}}-k\sin (\alpha )(\cos ({\varphi }_k)\hat{\text{x}}+\sin ({\varphi }_k)\hat{\text{y}})$$

$$\hat{\text{n}}({\varphi }_k)=\sin ({\varphi }_k)\hat{\text{x}}-\cos ({\varphi }_k)\hat{\text{y}}$$

$$\begin{split}\textbf{E}_{l}(z,\rho,\phi,t)=&\mathcal{E}_{0}e^{i(\kappa z-\omega t+l\phi)}\Bigg{[}-\dfrac{lJ_{l}(\beta\rho)}{\beta\rho}\hat{\mathbf{e}}_{\rho}\\ &+i\dfrac{J_{l+1}(\beta\rho)-J_{l-1}(\beta\rho)}{2}\hat{\mathbf{e}}_{\phi}\Bigg{]}\end{split}$$

$$\begin{split}\langle\mathfrak{L}\rangle=&\dfrac{\epsilon_{0}\mathcal{E}_{0}^{2}}{c}\Bigg{(}\dfrac{l}{k_{0}}J^{2}_{l}(\beta\rho)\hat{\mathbf{z}}-\dfrac{lz\sin(\alpha)}{\beta\rho}J^{2}_{l}(\beta\rho)\hat{\mathbf{\rho}}\\ &-\rho\cos(\alpha)\dfrac{J^{2}_{l+1}(\beta\rho)-J^{2}_{l-1}(\beta\rho)}{2}\hat{\mathbf{\phi}}\Bigg{)}\end{split}$$

$$\mathcal{L}=\dfrac{1}{8\pi}\Big{(}(\mathbf{E}^{2}-\mathbf{B}^{2})+\xi[(\mathbf{E}^{2}-\mathbf{B}^{2})^{2}+7(\mathbf{E}\cdotp\mathbf{B})^{2}]\Big{)}$$

$${\partial }^{\mu }{\partial }_{\mu }\mathbf{E}=4\pi (c^2\nabla (\nabla \cdot \mathbf{P})-{\partial }_t({\partial }_t\mathbf{P}+c\nabla \times \mathbf{M}))$$

$$\mathbf{M}=(4\pi {)}^{-1}\xi (-2({\mathbf{E}}^2-{\mathbf{B}}^2)\mathbf{B}+7(\mathbf{E}\cdot \mathbf{B})\mathbf{E})$$

$$\mathbf{P}=(4\pi {)}^{-1}\xi (2({\mathbf{E}}^2-{\mathbf{B}}^2)\mathbf{E}+7(\mathbf{E}\cdot \mathbf{B})\mathbf{B})$$

$${\mathbf{k}}_1+{\mathbf{k}}_2={\mathbf{k}}_3+{\mathbf{k}}_4{\omega }_1+{\omega }_2={\omega }_3+{\omega }_4$$

$${\partial }^{\mu }{\partial }_{\mu }{\mathbf{E}}_4\approx (-i{)}^l{\omega }_0^2\xi \hat{\mathbf{v}}({\varphi }_k)E_1{E}_2{E}_3^{\ast }{e}^{i({\mathbf{k}}_4({\varphi }_k)\cdot \mathbf{r}-{\omega }_{0}t-l{\varphi }_k)}$$

$${\partial }^{\mu }{\partial }_{\mu }{\mathbf{E}}_4\approx {\omega }_0^2\xi E_1{E}_2{E}_3^{\ast }{\mathbf{J}}_l(\rho ,\varphi )e^{i(-\kappa z-{\omega }_{0}t-l\varphi )}$$

$$\begin{array}{rl}{\mathbf{J}}_l(\rho ,\varphi )=& \sin (\alpha )J_l(\beta \rho )\hat{\mathbf{z}}+i\cos (\alpha )\frac{J_{l+1}(\beta \rho )-J_{l-1}(\beta \rho )}{2}\hat{\rho }\\ & -\frac{l\cos (\alpha )J_l(\beta \rho )}{\beta \rho }\hat{\varphi }\end{array}$$

$${\mathbf{E}}_4=\frac{(i{)}^l}{4\pi r{c}^2}{\omega }_0^2\xi E_1{E}_2{E}_3^{\ast }{\mathbf{\Lambda }}_l(\theta )e^{i(k_{0}r-{\omega }_{0}t-l\varphi )}$$

$$\begin{array}{rl}{\Lambda }_l(\theta )=& 2\pi (\sin (\alpha )A_l\hat{\mathbf{z}}-\cos (\alpha )\frac{A_{l+1}+A_{l-1}}{2}\hat{\rho }\\ & -i\cos (\alpha )\frac{A_{l+1}-A_{l-1}}{2}\hat{\varphi })\end{array}$$

$$A_l=\int {\int }_{V^{\prime }}{e}^{-i(\kappa +k_0\cos (\theta ))}{J}_{-l}(k_0\sin (\alpha ){\rho }^{\prime })J_l(\beta {\rho }^{\prime }){\rho }^{\prime }d{\rho }^{\prime }d{z}^{\prime }$$

$$N_{\gamma }=\frac{{ϵ}_{0}c\tau }{2\hslash {\omega }_4}{\int }_0^{\pi }{\int }_0^{2\pi }|E_4(r,\theta ,\varphi ){|}^2{r}^2\sin (\theta )d\theta d\varphi$$

$${\mathbf{E}}_j(\mathbf{r},t)=E_j{e}^{i({\mathbf{k}}_j\cdot \mathbf{r}-{\omega }_{j}t)}{\mathbf{n}}_j({\mathbf{k}}_j)$$

$${\partial }^{\mu }{\partial }_{\mu }\mathbf{E}=4\pi (c^2\nabla (\nabla \cdot \mathbf{P})-{\partial }_t({\partial }_t\mathbf{P}+c\nabla \times \mathbf{M}))$$

$$\mathbf{M}=(4\pi {)}^{-1}\xi (-2({\mathbf{E}}^2-{\mathbf{B}}^2)\mathbf{B}+7(\mathbf{E}\cdot \mathbf{B})\mathbf{E})$$

$$\mathbf{P}=(4\pi {)}^{-1}\xi (2({\mathbf{E}}^2-{\mathbf{B}}^2)\mathbf{E}+7(\mathbf{E}\cdot \mathbf{B})\mathbf{B})$$

$$\begin{array}{r}{\partial }^{\mu }{\partial }_{\mu }\mathbf{E}=\sum {\mathbf{A}}_{ijk}{E}_i{E}_j{E}_k{e}^{i(({\mathbf{k}}_i+{\mathbf{k}}_j+{\mathbf{k}}_k)\cdot \mathbf{r}-({\omega }_i+{\omega }_j+{\omega }_k)t)}\\ +{\mathbf{B}}_{ijk}{E}_i{E}_j{E}_k^{\ast }{e}^{i(({\mathbf{k}}_i+{\mathbf{k}}_j-{\mathbf{k}}_k)\cdot \mathbf{r}-({\omega }_i+{\omega }_j-{\omega }_k)t)}\\ +{\mathbf{C}}_{ijk}{E}_i{E}_j^{\ast }{E}_k^{\ast }{e}^{i(({\mathbf{k}}_i-{\mathbf{k}}_j-{\mathbf{k}}_k)\cdot \mathbf{r}-({\omega }_i+{\omega }_j-{\omega }_k)t)}\end{array}$$

$${\mathbf{E}}_4(\mathbf{r},t)=E_4(\mathbf{r},t)e^{i({\mathbf{k}}_4\cdot r-{\omega }_{4}t)}{\mathbf{n}}_4({\mathbf{k}}_4)$$

$$\begin{array}{r}{\partial }^{\mu }{\partial }_{\mu }{\mathbf{E}}_4(\mathbf{r},t)\approx -2i({\omega }_4\frac{\partial }{\partial t}+c^2\mathbf{k}\cdot \nabla )E_4(\mathbf{r},t)\\ \times e^{i({\mathbf{k}}_4\cdot r-{\omega }_{4}t)}{\mathbf{n}}_4({\mathbf{k}}_4)\end{array}$$

$${\partial }^{\mu }{\partial }_{\mu }{\mathbf{E}}_4\approx {\mathbf{B}}_{123}{E}_1{E}_2{E}_3^{\ast }{e}^{i(({\mathbf{k}}_1+{\mathbf{k}}_2-{\mathbf{k}}_3)\cdot \mathbf{r}-({\omega }_1+{\omega }_2-{\omega }_3)t)}$$

Full text

††preprint: APS/123-QED

Photon-photon scattering in vacuum is an effect that has attracted considerable interest over the past few decades Brodin et al. (2001); Kaplan and Ding (2000). In classical electromagnetism, the linearity of Maxwell’s equations indicates that scattering between photons cannot happen in the vacuum. Nevertheless, results from quantum electrodynamics (QED) have shown that the vacuum is, in fact, non-linear. This means that the aforementioned scattering is actually possible, albeit very difficult to observe Berestetskii et al. (1982). Many different proposals have been made to detect such processes using microwave cavities Brodin et al. (2001), plasma channels Shen et al. (2003) and high-power lasers Lundström et al. (2006); King et al. (2018). The main obstacle to these schemes has been the relatively weak predicted signal compared to the noise inherent to any experiment. As such, scattering between real photons has yet to be observed in the vacuum.

In this letter, we investigate the effect of orbital angular momentum (OAM) coupling in elastic photon-photon scattering and we show, for the first time, that OAM contributes an additional unique signature that will help filter the background noise and is thus predicted to make the detection of these weak signals easier. OAM is a relatively recently understood degree of freedom of light that is distinct from the more widely understood spin angular momentum (SAM) which is related to light’s polarization Torres and Torner (2011). It appears in electromagnetic modes which have an azimuthal phase dependence, $E(\mathbf{r})\propto e^{il\phi}$, where conventionally $l$ denotes the OAM state of the mode. It should be noted that $l$ here represents the projection of OAM on the direction of propagation in contrast to its use in the quantum mechanical treatment of bound electronic states where it represents the absolute value of the “total” OAM. Interest in researching these modes has grown ever since they were shown to carry a quantized net angular momentum separate from their spin Allen et al. (1992). Since it is unbounded and can have any integer value, OAM has many potential applications in different fields including: enhanced imaging techniques, light manipulation and high density coding of information for communication Torres and Torner (2011); Lei et al. (2017); Ghaleh et al. (2018). Consequentially, over the course of the previous two decades, the techniques required to efficiently generate OAM states, and detect and filter photons according to their OAM state (right down to the single-photon level) have been honed to a great level of precision Hadfield (2009); Leach et al. (2002). For this reason, we are interested in investigating its impact on photon-photon scattering.

This letter will be organized as follows: first, exact solutions to the electromagnetic wave equation which carry OAM will be discussed. These modes will then be used to investigate the effect of OAM on photon-photon interaction within the framework of an effective field theory Heisenberg and Euler (1936). Finally, a proposed experimental configuration will be presented that is expected to validate the accompanying theoretical work.

The simplest solution to the linear electromagnetic wave equation is the plane wave decomposition, where a plane wave is described by:

[TABLE]

where $\mathbf{k}$ is the wavevector, $\omega$ is the angular frequency and $\hat{\mathbf{n}}$ is the polarization unit vector of the plane wave which necessarily satisfies the transversality condition ($\textbf{k}\cdotp\hat{\textbf{n}}=0$) arising from Maxwell’s equations. We then consider an infinite superposition of plane waves whose wave vectors lie on the surface of a cone with half-angle $\alpha$ around the z-axix:

[TABLE]

[TABLE]

[TABLE]

where $l\in\mathbb{Z}$ is the azimuthal mode number. $\mathbf{E}_{l}$ is obviously then an exact solution to the electromagnetic wave equation as it is a linear superposition of exact solutions. Its integral representation can be simplified by decomposing the trigonometric functions into their exponential form and using the fact that $(1/2\pi)\int_{0}^{2\pi}e^{i(l\phi-x\sin(\phi))}d\phi=J_{l}(x)$ Abramowitz (1974), where $J_{l}(x)$ is the $l$-th order Bessel function of the first kind. Thus, the electric field in Eq.(2) can be rewritten in cylindrical coordinates ($z,\rho,\phi$) as:

[TABLE]

Where $\kappa=k\cos(\alpha)$ and $\beta=k\sin(\alpha)$. The corresponding magnetic field can then be calculated from Eq.(5) using Maxwell’s equations ($\nabla\times\mathbf{E}_{l}=-\partial\mathbf{B}_{l}/\partial t$). This solution can be seen as an OAM-carrying extension of the electromagnetic Bessel modes. One of the interesting properties of Bessel modes is that they propagate without diffraction as can be seen by noting that the radial profile of the field is independent of the longitudinal position along the direction of propagation. True Bessel beams are unphysical as they carry infinite energy. However, Bessel-like modes, which retain some of the aforementioned interesting properties over a known coherence length, have been generated using axicon lenses McGloin and Dholakia (2005). When considering the case of $l\neq 0$, the field has an azimuthal phase dependence, and therefore as shown in Allen et al. (1992), it carries nonzero OAM. This can be confirmed by calculating the time-averaged angular momentum density of the field $\langle\mathfrak{L}\rangle=(1/2)\mathbf{r}\times\Re{(\textbf{E}^{*}\times\textbf{B})}$.

[TABLE]

If $\langle\mathfrak{L}\rangle$ is integrated over a finite circular transverse profile of the field, only the z-component of the angular momentum density contributes as the polar unit vectors integrate to 0. This results in a total angular momentum $\langle\mathbf{L}\rangle\propto l\hat{\mathbf{z}}$.

It should be noted that this method of defining light modes that carry OAM differs from the standard approach of considering Laguerre-Gaussian (LG) modes Allen et al. (1992). The method used here offers two advantages, the first being that the field defined in Eq.(5) is, by construction, divergence-free and hence an exact solution of Maxwell’s equations. Secondly, it offers a simple representation of OAM beams as a superposition of plane waves which proves highly advantageous when calculating the nonlinear interaction terms compared to the LG modes which contain a complicated envelope that makes the calculation intractable.

Classically, photon-photon scattering is physically impossible since Maxwell’s equations are linear Maxwell (1865), meaning that any superposition of solutions to the wave equation is itself a solution. However, the full quantum electrodynamic treatment of the vacuum field predicts the existence of vacuum polarization and the possibility of photon-photon scattering mediated by virtual electron-positron pairs Berestetskii et al. (1982). The exact treatment of this problem is quite challenging, however, the interaction can be approximated by the Euler-Heisenberg Lagrangian which accounts for this vacuum polarization up to one loop in the corresponding Feynman diagram, thereby accounting for the lowest order contribution to the photon-photon QED scattering amplitude. It can be written in Gaussian units, in the limit for fields oscillating slower than the Compton frequency, as Heisenberg and Euler (1936):

[TABLE]

Where $\xi=\dfrac{\hbar e^{4}}{45\pi^{2}m^{4}c^{7}}$. A modified electromagnetic wave equation can then be derived using the Euler-Lagrange equations. The quartic corrections to the Lagrangian lead to a non-linear source term for the wave equation, even in the vacuum.

[TABLE]

Where $\partial^{\mu}\partial_{\mu}=\partial^{2}_{t}-c^{2}\nabla^{2}$, and $\mathbf{M}$ and $\mathbf{P}$ are the effective vacuum magnetization and polarization respectively, described by:

[TABLE]

[TABLE]

These equations indicate that the source term in Eq.(8) has a non-linear cubic dependence on the electromagnetic fields. Therefore, if we start with three plane waves $(\mathbf{k}_{i},\omega_{i})_{i\in\{1,2,3\}}$, it is clear that the three modes will mix generating a fourth wave ($\mathbf{k}_{4},\omega_{4}$) that satisfies the following energy and momentum matching conditions:

[TABLE]

There are, of course, many other interaction terms that involve different combinations of the three initial waves. However, in the rest of this letter, only terms that correspond to the matching conditions in Eq.(11) will be considered. This is because it is the most convenient generated wave to measure experimentally and allows for flexibility in the geometrical setup of the incoming beams as will be seen in the following parts. This does not necessarily mean that the other interactions terms are particularly weaker. However, they are also not any stronger, and waves generated through them will have frequencies and directions of propagation (and different OAM state as will be seen later on) different from Eq.(11) meaning that they can safely be neglected when estimating the number of detected photons in the experimental geometry that will be considered later on.

Let the three initial plane waves be $\mathbf{E}_{1}=E_{1}e^{i(k_{0}x-\omega_{0}t)}\hat{\mathbf{y}}$, $\mathbf{E}_{2}=E_{2}e^{i(-k_{0}x-\omega_{0}t)}\hat{\mathbf{z}}$ and $\mathbf{E}_{3}=(i)^{l}E_{3}e^{i(\mathbf{k}_{3}(\phi_{k})\cdotp\mathbf{r}-\omega_{0}t)}e^{il\phi_{k}}\hat{\mathbf{n}}(\phi_{k})$ where $\mathbf{k}_{3}(\phi_{k})=k_{0}\cos(\alpha)\hat{\textbf{z}}-k_{0}\sin(\alpha)(\cos(\phi_{k})\hat{\textbf{x}}+\sin(\phi_{k})\hat{\textbf{y}})$ and $\hat{\textbf{n}}(\phi_{k})=\sin(\phi_{k})\hat{\textbf{x}}-\cos(\phi_{k})\hat{\textbf{y}}$. It is clear that $\mathbf{E}_{3}$ is a plane wave that lies on the surface of a cone with half-angle $\alpha$ similar to the aforementioned case. Further, $\alpha$ is considered to be very small, which is a reasonable assumption with Bessel beams as they are generally generated with axicon lenses which have a very shallow angle. Keeping only the interaction terms of Eq.(8) that are resonant for the generated fourth wave $\mathbf{E}_{4}$. The non-linear wave equation can then be written as supplementary_material :

[TABLE]

where $\hat{\mathbf{v}}(\phi_{k})=\cos(\alpha)\cos(\phi_{k})\hat{\mathbf{x}}+\cos(\alpha)\sin(\phi_{k})\hat{\mathbf{y}}+\sin(\alpha)\hat{\mathbf{z}}$ and $\mathbf{k}_{4}=-\mathbf{k}_{3}$.

Consider now the case where $\mathbf{E}_{3}$ is not a plane wave, but an $l$-th order OAM mode whose average wavevector is directed along the positive z-axis. As seen above, the OAM beam can be decomposed into an appropriate superposition of plane waves. It is only then necessary to perform an integration on Eq.(12), similar to the one presented above in Eq.(2), and find that the source term, in this case, can be rewritten as Bessel functions with an overall azimuthal phase dependence.

[TABLE]

where $\beta=k_{0}\cos(\alpha)$, $\kappa=k_{0}\sin(\alpha)$ and $\mathbf{J}$ is a vector containing the transverse radial dependence of the source.

[TABLE]

Eq.(13) can be solved using the standard Green’s function method, and the generated field, far from the source, can be written as:

[TABLE]

where:

[TABLE]

[TABLE]

and $r\in\mathbb{R}^{+}$ is the spherical radial distance from the origin of the finite interaction volume $V^{\prime}$ whose extent will depend on the size of the focal spots of the three beams being used in the experimental setup. The interaction volume is artificially limited in this calculation because the beams used in the analytical derivation have infinite extent and are non-integrable over all of space, while physical beams will always have an integrable intensity envelope.

The direction of propagation of this generated wave can be seen in the angular dependence of $\mathbf{\Lambda}_{l}(\theta)$ in Fig.(1). From it we can conclude that the generated wave propagates along the same axis but in the opposite direction as the initial OAM mode $\mathbf{E}_{3}$, which is intuitive as $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ are anti-parallel, and linear momentum is conserved. It can also be clearly seen in Eq.(15) that the generated field has the opposite azimuthal phase dependence of $\mathbf{E}_{3}$. This means that the OAM of the generated photons will be opposite in sign, which is to be expected as a consequence of conservation of angular momentum since the two other intial waves $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ were plane waves. We can then use Eq.(15) to calculate the generated intensity $I_{4}=(1/2)c\epsilon_{0}|E_{4}|^{2}$ which can be integrated over the surface of the interaction volume to calculate the number of generated photons for specific initial laser parameters.

Eq.(15) shows that for a specific set of three-beam geometries, if one of the lasers is carrying some OAM ($l$) then the photons generated due to photon-photon scattering will carry the opposite value $l_{4}=-l$ as can be seen from the azimuthal dependence in the phase of the generated field. We propose to use this additional signature to design a novel high-power laser experiment to verify this hitherto undetected phenomenon. In order to make the generated photons even easier to distinguish, the geometry will be modified slightly from the one considered previously. Consider instead three initial waves, where ($\mathbf{k}_{1}=-2(\omega_{0}/c)(1/2\hat{\mathbf{z}}+\sqrt{3}/2\hat{\mathbf{x}})$, $\hat{\textbf{n}}_{1}=\hat{\mathbf{y}}$) and ($\mathbf{k}_{2}=-2(\omega_{0}/c)(1/2\hat{\mathbf{z}}-\sqrt{3}/2\hat{\mathbf{x}})$, $\hat{\textbf{n}}_{2}=\hat{\mathbf{y}}$) are plane waves, and $\mathbf{E}_{3}$ is an OAM beam with $l_{3}=1$ and $\omega_{3}=\omega_{0}$ that is propagating along the positive z-axis. These kinds of modes can be efficiently generated in the laboratory using mode-conversion methods such as so-called spiral phase mirors (SPM) which have efficiently generated OAM-carrying high power lasers longman_aps . According to Eq.(11) and Eq.(15), we should then expect to detect photons generated at $\omega_{4}=\omega_{1}+\omega_{2}-\omega_{3}=3\omega_{0}$ with $l_{4}=-1$ propagating along the negative z direction. These generated photons are then distinct, from any of the initial radiation in the interaction, in frequency, direction of travel and OAM state. The expected number of photons generated by photon-photon scattering $N_{\gamma}$ can be calculated from Eq.(15) using:

[TABLE]

where $\tau$ is the duration of the interaction. In the case of a high-power laser of $10\text{\,}\mathrm{PW}$ at a wavelength of $800\text{\,}\mathrm{nm}$, where the laser pulse duration is $\tau=$30\text{,}\mathrm{fs}$$, the beam must be split into three, where two of the component parts are frequency doubled. Considering a second harmonic conversion efficiency of 30% and a spot size of $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, Eq.(18) predicts that $N_{\gamma}$ is expected to be on the order of 100 photons per shot. It should be noted that since unphysical light modes were used for this derivation, it can only provide an initial estimate of the number of generated photons through photon-photon scattering. This is, however, not a major concern as any additional realistic envelope (such as a Gaussian envelope) will vary slowly compared to the oscillations of the field and as such will have negligible contributions to the photon-photon scattering interaction. Fig.(3) shows the scaling of $N_{\gamma}$ with respect to the power of the initial laser, while keeping all other parameters the same as the above example. Comparing this scaling to the one in the case with no OAM Lundström et al. (2006), it is clear that the introduction of OAM does not change the coupling strength of the photon-photon interaction and the predicted number of photons generated from it is of the same order when lasers parameters are matched.

Although the number of photons may seem small, especially compared to the other 3 high-power beams (e.g. the OAM-carrying beam contains $\mathrm{O}(10^{20})$ photons), its multiple different signatures will allow for efficient filtering of any other background radiation. There have been many successful experiments where even single photons have been filtered by their OAM state and detected using highly efficient single-photon detectors nicolas_quantum_2014 ; Leach et al. (2002). Along with OAM filtering, frequency filtering with ultra narrow-band bandpass filters would block any photons whose frequency is sufficiently different from what is expected from the previously defined three-beam interaction. The single-photon detectors can also be gated to only be sensitive to scattered photons that are generated during the interaction time in order to further reduce the noise. Furthermore, performing this experiment on a high-power and high-repetition rate laser facility, such as the ELI facility et al. (2010); mourou_extreme_2011 or the Apollon facility papadopoulos_apollon_2016 , would allow for gathering the statistics essential for verification of the scattering event. The scattering signal would also be boosted by orders of magnitude by performing the experiment on the upcoming $100\text{\,}\mathrm{PW}$ Station of Extreme Light (SEL) laser facility as may be seen from the scaling in Fig.(3).

Of course, it is practically impossible to achieve a perfect vacuum even in a laboratory setting. Even within a vacuum chamber, there are still some particles that can interact with the lasers and generate some noise. The main interaction contributing to this concern is Compton scattering. However, as shown in Lundström et al. (2006), the number of Compton-scattered photons that would possibly be generated in the appropriate frequency band is much smaller than the photon-photon scattering signal we expect. Also, as proven in Jentschura and Serbo (2011), the OAM state of these Compton-scattered photons would either be 0 or the opposite of the state of interest. Thus, using a projective OAM filtering technique Nicolas et al. (2015), the already small amount of noise can be reduced by $25\text{\,}\mathrm{dB}$ compared to the case without the OAM signature while the signal remains as strong, thus making detection of the signal of interest feasible.

In this letter, we have investigated the effect of orbital angular momentum on elastic photon-photon scattering. Using effective field theory, we have derived an expression for the expected generated field. We have shown that the OAM coupling will provide an additional useful signature for the interaction and will allow for the use of quantum optics techniques to discern them from any background radiation. We have also proposed an experimental setup, which we plan to perform at the SEL laser facility, that would allow for the detection of this effect using high-power lasers. The experimental verification of the polarization of the vacuum will open a new era in strong field, low energy quantum electrodynamics for fundamental physics investigations.

.1 Analysis of the non-linear electromagnetic wave equation

The Euler-Heisenberg Lagrangian shows that even in the vacuum, there is a non-linear coupling between electromagnetic waves. If one again considers the case with three initial waves, then the electromagnetic fields for the waves can be written in complex notation as

[TABLE]

With $n_{j}(k)$ being the polarization of the field and $j\in\{1,2,3\}$ labeling each individual wave. The electromagnetic wave equation derived from the vacuum Euler-Heisenberg Lagrangian is

[TABLE]

[TABLE]

[TABLE]

Where $\mathbf{P}$ and $\mathbf{M}$ are the effective vacuum polarization and magnetization respectively, and $\mathbf{E}$ and $\mathbf{B}$ are the total sum of the electric and magnetic fields respectively. It should be noted that since the complex notation is being used for the fields, the squared fields are calculated from their real part (e.g. $\mathbf{E}^{2}=\Re(E)^{2}$). From Eq.(S3) and Eq.(S4), it is clear that the non-linearity is cubic, so the inhomogenous wave equation can rewritten to first order as

[TABLE]

It is possible to investigate this equation via harmonic analysis in order to simplify it to a more solvable form. The field of interest, considered in the paper for the convenience of the geometry of its interaction, is the fourth wave that is generated through the photon-photon interaction

[TABLE]

with $\mathbf{k}_{4}=\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{k}_{3}$, $\omega_{4}=c|\mathbf{k}_{4}|=\omega_{1}+\omega_{2}-\omega_{3}$ and $E_{4}(\mathbf{r},t)$ is a slowly varying function of space and time. For this generated field, the left-hand side of Eq.(S5) can be expanded to first order as

[TABLE]

Multiplying both sides of Eq.(S5) by $e^{i(\mathbf{k}_{4}\cdot r-\omega_{4}t)}$, it is then Fourier transformed in space, thereby leaving only the time derivative of $E(\mathbf{r},t)$. When integrating over time, all the non-resonant terms oscillating at $\omega\neq\omega_{4}$ vanish and the only relevant non-linear coupling term is the resonant $E_{1}E_{2}E_{3}^{*}$ term. Hence Eq.(S5) is simplified to

[TABLE]

which is then solved using the Green’s function method as described in the paper.

.2 Comparison between photon-photon scattering with and without OAM

The main advantage to using beams with orbital angular momentum (OAM) in photon-photon scattering is the additional OAM signature that is carried by the generated photons distinguishing them from the background radiation. This is predicted to improve the signal-to-noise ratio of interaction making it much more detectable. To see this, first consider the signal of the generated photons. In the paper, the non-linear equation for an incoming beam OAM was derived by first considering a superposition incident plane waves with an angle $\alpha$ with respect to z-axis and an azimuthal phase shift and then integrating them with respect to the phase. The strength of the coupling with no OAM can be derived by considering case with with $\alpha=0$ and no phase variance. By comparing the predicted number of generated photons through photon-photon scattering for each case (with and without OAM), it can be shown that the number of generated photons is practically unaffected when parameters of the incoming beams (power, duration and focal spot) are held constant. This is intuitive as the effect of OAM would only be to modify the OAM state of the generated photons and not the coupling strength of the photon-photon interaction.

Of course, high-power lasers typically do not carry OAM and it must be imparted onto them using mode converters such as so-called spiral phase plates (SPP). Even if the coupling strength is not affected, the conversion efficiency is critical as a poorly designed converter would weaken the incoming beam, potentially undermining the positive effect of the additional signature.

Efficiently generating OAM-carrying beams has been a very active topic of research due to their potential applications in quantum computation and communication, and high-density coding of information nicolas_quantum_2014 ; ganic_generation_2002 . Conversion methods adapted to high-power lasers have also been developed shvedov_efficient_2010 ; campbell_generation_2012 , with methods such as spiral phase optics having a theoretical efficiency of up to $93\%$ when the waist ratio is optimized longman_mode_2017 . Spiral phase mirrors are particularly well adapted for high-power lasers as they avoid transmission through dielectric media thereby avoiding any problems with group velocity dispersion (GVD) or B-integral private_comms . They can also be made using coatings with high-damage thresholds similar to the ones used on the rest of the optics used to manipulate these high-power beams where dielectric coatings can have fluence damage threshold of over $1\ \mathrm{J}\cdot\mathrm{cm^{-2}}$ damage_threshold . Additionally, high-power beams generally have a large diameter so that their fluence becomes well below the damage threshold for the optical coatings. Tests have already been done using spiral phase mirrors to efficiently generate OAM on 100 TW laser beams longman_aps . This means that the conversion of one of the input beams into an OAM-carrying one will not have a detrimental effect on the number of photons generated through photon-photon scattering.

Considering now the noise, the additional unique OAM signature carried by these photons allows for the use of filtering techniques developed in quantum optics research. In nicolas_quantum_2015 , a projective filtering method is used to efficiently detect photons carrying a particular OAM signature while other modes are suppressed by around 25 dB. The filtering involves using a highly-efficient blazed fork phase hologram that adds a specific amount of OAM. The goal is to choose the hologram so that the desired OAM mode is converted into a regular Gaussian while all other modes are converted to various other OAM modes. The output is then coupled into a single-mode fiber optic cable itself coupled to a single photon detector. These cables can only transmit Gaussian modes while all other modes decay while propagating. Therefore, at the final output, the specific OAM, corresponding to the photons generated through photon-photon scattering, will be detected while the background noise will be filtered by 25 dB. From all these considerations, the signal-to-noise ratio (SNR) of the interaction using an OAM-carrying initial beam is significantly enhanced compared to that of the regular plane wave.