# Light by light diffraction in vacuum

###### Abstract

We show that a laser beam can be diffracted by a more concentrated light pulse due to quantum vacuum effects. We compute analytically the intensity pattern in a realistic experimental configuration, and discuss how it can be used to measure for the first time the parameters describing photon-photon scattering in vacuum. In particular, we show that the Quantum Electrodynamics prediction can be detected in a single-shot experiment at future 100 petawatt lasers such as ELI or HIPER. On the other hand, if carried out at one of the present high power facilities, such as OMEGA EP, this proposal can lead either to the discovery of non-standard physics, or to substantially improve the current PVLAS limits on the photon-photon cross section at optical wavelengths. This new example of manipulation of light by light is simpler to realize and more sensitive than existing, alternative proposals, and can also be used to test Born-Infeld theory or to search for axion-like or minicharged particles.

The linear propagation of light in vacuum, as described by Maxwell equations, is a basic assumption underlying our communication system, allowing e.g. that different electromagnetic waves do not keep memory of their possible crossing in the way to their reception points. However, this superposition principle is expected to be violated by quantum effects. In fact, Quantum Electrodynamics (QED) predicts the existence of Photon-Photon Scattering in Vacuum (PPSV) mediated by virtual charged particles running in loop diagrams Costantini1971 (), although the rate is negligible in all the experiments that have been performed up to now. On the other hand, additional, possibly larger contributions to the process may appear in non-standard models such as Born Infeld theory BornInfeld (); Denisov (); Gaete () or in new physics scenarios involving minicharged MCP () or axion-like ALP () particles. Therefore, the search for PPSV is important not only to demonstrate a still unconfirmed, fundamental quantum property of light, but also to either discover or constrain these kinds of new physics.

In the last few years, there has been an increasing interest in the quest for PPSV PPSsearch (); PVLAS (); JHEP09 (); NaturePhotonics2010 (). Here, we present a new scenario to search for this phenomenon using ultra-high power lasers mourou06 (). In our proposal, two almost contra-propagating laser pulses cross each other. Due to PPSV, the more concentrated pulse behaves like a phase object diffracting the wider beam. The resulting intensity pattern can then be observed on a screen, and will correspond to a direct detection of scattered photons.

The effective Lagrangian for photon-photon scattering. Following Ref. JHEP09 (), we will assume that for optical wavelengths the electromagnetic fields and are described by an effective Lagrangian of the form

(1) |

being the Lagrangian density of the linear theory and .

In Quantum Electrodynamics, would be the Euler-Heisenberg effective Lagrangian Euler-Heisenberg (), that coincides with Eq. (1) with the identification , being

(2) |

However, in Born-Infeld theory BornInfeld (); Denisov (); Gaete (), or in models involving a new minicharged (or milli-charged) MCP () or axion-like ALP () particle, and will have different values, as computed in Ref. JHEP09 ().

On the other hand, the current 95% C.L. limit on PPSV at optical wavelengths has been obtained by the PVLAS collaboration PVLAS (). As shown in Ref. JHEP09 (), it can be written as

(3) |

Assuming as in QED, this can be translated in the limit , which is times higher than the QED value of Eq. (2).

Proposal of an experiment: analytical computations. In our present proposal, illustrated in Fig. 1, a polarized ultrahigh power Gaussian pulse A of transverse width crosses an almost contra-propagating polarized ‘probe’ laser pulse B of width . For simplicity, we assume that the two beams have the same mean wavelength and frequency , although in principle they may have different durations and . We also suppose that the uncertainty in frequency is much smaller than , in such a way that we can consider the pulses as being monochromatic with a good approximation. Similarly, we assume that the uncertainty in the components of the wave vector are much smaller than .

From Ref. JHEP09 (), we learn that the central part of the probe B, after crossing the pulse A, acquires a phase shift

(4) |

where is the peak intensity of the high power beam at the crossing point, the indexes and refer to the two beams having parallel or orthogonal linear polarizations, respectively, and we have defined and .

Let and describe the dependences of the non-vanishing components of the two waves on the radial coordinate orthogonal to the direction of the motion, chosen in the -axis. The intensity of the pulse A in the colliding region will then have the transverse distribution . As a consequence, the space-dependent phase shift of the wave B just after the collision with the beam A is

(5) |

where we understand one of the sub-indexes or .

Due to this phase shift, the shape of the pulse B becomes . As discussed in Ref. JHEP09 (), is expected to be very small at all the facilities that will be available in the near future. Therefore with a very good approximation, and we obtain

(6) |

where we have defined .

After the collision, the field propagates linearly, so that we can just sum the free evolution of each term in Eq. (6), that can be computed with the approximation for the angular frequency, where , assuming that . As a result, the linear evolution of on the screen-detector plane produces an intensity pattern , where and correspond to the undiffracted and diffracted waves respectively, and represents the interference term. The result is

(7) |

where we have defined the widths of the undiffracted and diffracted patterns, and , and is the peak intensity of the wave B at the collision point, that can be related to the total power of the pulse B as . The interference term can be evaluated by multiplying by the factor

and turns out to be numerically negligible, as compared to , in all the configurations that we will discuss below.

On the other hand, the total power of the diffracted pulse can be obtained by integrating Eq. (7) in the screen plane, so that , which is much smaller than . However, an interesting feature of Eq. (7) is that the diffracted wave is distributed in an area of width (for ), so that it can be separated from the undiffracted wave e.g. by making a hole in the screen. Let be the radius of the central region that is eliminated from the screen. We require that the total power due to the diffracted wave for is much larger than the the power of the undiffracted wave in the same region, . A safe choice can be , that implies

(8) |

Finally, using Eqs. (4) and (7), we can compute the number of diffracted photons that will be detected after repetitions of the experiment in the ring region of the screen, being its external radius. We obtain

(9) |

where is the efficiency of the detector, and and are the total energies of the two pulses.

Angular constraints and Optimization of the sensitivity. Eq. (9) shows that the number of scattered photons is proportional to the product of the energies of the two laser beams. It would then be convenient to use a ultrahigh power pulse also for the probe B. This can be done economically by producing both beams simultaneously, e.g. by dividing a single pulse of energy before the last focalizations. The maximum value for is then obtained by taking and .

The other parameters that can be adjusted in order to maximize are the widths and of the two colliding beams. The choice of is constrained by the requirement that the pulse A must not spread in a significant way during the crossing, so that . However, a more stringent constraint, involving also the angle , originates from the condition that the center of pulse A has to remain close to the central part of beam B during the interaction. This implies that . A safe choice can then be . On the other hand, the angle has to be such that, out of the collision point, the trajectories of the two beams are separated by a distance sufficiently larger than their width. We conservatively ask that such a distance is 6 times the width of the beam A at the distance , although one can keep in mind that smaller separations, if they turned out to be experimentally viable, would allow for better sensitivities. For small , we then have , and we can solve for ,

(10) |

On the other hand, the value of that maximizes will be computed numerically, and the outer radius will be chosen slightly larger than , by requiring that only a few percent of the diffracted wave is lost.

Finally, the measurement of the number of diffracted photons can be used to determine the values of the parameters and . To evaluate the best possible sensitivity, we will suppose that the background of thermal photons and the dark count of the detectors can be made much smaller than the signal. Although this goal may be difficult in practice, in principle it can be achieved by cooling the ring detector and covering it with a filter that selects a tiny window of wavelengths around , and by optically isolating the experimental area within the time of response of the detector, which should be as small as possible (the ultrashort time of propagation of the beams can be neglected in comparison). Of course, the actual background should be measured by performing the control experiments in the absence of the beams, and with only one pulse at a time.

Under these assumptions, the best sensitivity would correspond to the detection of, say, 10 diffracted photons, so that the zero result could be excluded within three standard deviations. The ideal, minimum values of and that could be measured would then be given by Eq. (9), taking and all the choices reviewed above. (In the numerical computations that we present below, we also include a small correction that appears in the expression of as shown in Ref. JHEP09 ().)

Sensitivity at future 100 Petawatt laser facilities. Let us now study the possibility of performing our proposed experiment, in its economical version discussed above, using a 100 Petawatt laser such as ELI ELI () or HIPER HIPER (), that are expected to become operative in few years. In this case, we can use the following values: total power , duration , energy and wavelength . With these data, using Eq. (10), we can compute the suggested value for the width of the spot to which the pulse A should be focalized at the collision point. Taking e.g. , we find numerically that the best choice for is . We then obtain the value for the central hole in the screen, for its outer radius, and and for the widths of undiffracted and diffracted waves. The focused intensities of the two beams are and , and the angle . Even if we assume an efficiency as small as , which is a realistic value today for , we obtain that our proposed experiment can resolve and as small as and , values that are well below the QED prediction even for . Therefore, ELI and HIPER will be able to detect PPSV at the level predicted by QED in a single shot experiment. Two single-shot experiments, using parallel and orthogonal polarizations of the colliding waves respectively, would allow to measure both and .

We can restate this result in terms of the number of diffracted photons per pulse that will be be scattered in the ring detector. In the experiment with orthogonal polarizations, this number is , which is two orders of magnitude higher that the value obtained in the scenario of Ref. NaturePhotonics2010 (). This indicates that our present proposal is by far more sensitive than that of Ref. NaturePhotonics2010 (), in spite of the fact that we have applied much more realistic assumptions on the width of the higher power pulse in order to clearly separate the beams out of the crossing region. Moreover, the configuration of Ref. NaturePhotonics2010 () is less economical, since it requires an additional high power laser, and it will also present a greater experimental difficulty, as far as it needs the alignment of three ultrashort pulses (two of them having a spot radius as small as ).

Let us now compare our present proposal with that of Ref. JHEP09 (). The main differences between the two scenarios, both based in the crossing of two contra-propagating beams, are the following: 1) in Ref. JHEP09 (), the probe pulse could have much smaller power; 2) in Ref. JHEP09 (), the two beams had the same width, and the effect of PPSV was observed by measuring the phase shift of the probe by comparing it with a third beam that was originally in phase with it. Actually, the sensitivity calculated in Ref. JHEP09 () corresponded to a greater power, , and shorter , giving for . To compare that result with the present proposal, we have to use the same power, , and restate the sensitivity obtained in a single shot in Ref. JHEP09 () as and , where and are the total energy and focused width of the high power beam. It can be seen that a safe choice for the angle in that configuration would have needed , so that even at the setup of Ref. JHEP09 () would have preferred to use instead of . This would lead to and . These results would be significantly worse than those that we have obtained above in the present proposal. Moreover, an additional advantage of our new configuration is the fact that it can be systematically improved by increasing the number of crossing events in Eq. (9).

Sensitivity at present Petawatt lasers. Facilities such as OMEGA EP OMEGA () can already provide , and at . From Eq. (10), we obtain . Taking , we find numerically the optimal choice , leading to , , , , , and . Unfortunately, at present the efficiency of photon detectors for is just . Nevertheless, we obtain that our proposed experiment can resolve and as small as and . Even for the single shot experiment, , this result is at least an order of magnitude below the current limit of Eq. (3). As a result, this experiment at present facilities can already either detect PPSV of non-standard origin, or substantially improve the limits on and . In the former case, the measurement of both and can be used to discriminate between different kind of new physics, such as Born Infeld theory BornInfeld (); Denisov (); Gaete () or models involving minicharged MCP () or axion-like ALP () particles, using the expressions for the corresponding contributions that have been computed in Ref. JHEP09 ().

Finally, we note that in the minimal realization of our proposal, using only one high energy laser pulse divided in two parts of the same time duration , the optimal turns out to be proportional to (usually by a factor ). Taking into account Eqs. (9) and (10), this implies that the discovery potential for PPSV is proportional to . One could then study the possibility of using a laser having a wavelength in the visible window, in order to compensate in part the lower power (as compared with the case discussed above) with the higher and factors. In this case, assuming , our computations show that PPSV at the QED rate can be detected by accumulating a number of repetitions .

We thank D. Novoa and F. Tommasini for useful discussions and help. This work was supported by the Government of Spain (project FIS2008-01001) and the University of Vigo (project 08VIA09).

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