The gravitational field of a laser beam beyond the short wavelength approximation

Light carries energy, and therefore, it is the source of a gravitational field. The gravitational field of a beam of light in the short wavelength approximation has been studied by several authors. In this article, we consider light of finite wavelengths by describing a laser beam as a solution of Maxwell's equations and taking diffraction into account. Then, novel features of the gravitational field of a laser beam become apparent, such as frame-dragging due to its spin angular momentum and the deflection of parallel co-propagating test beams that overlap with the source beam. Even though the effects are too small to be detected with current technology, they are of conceptual interest, revealing the gravitational properties of light.


INTRODUCTION
The gravitational field of a light beam was first studied by Tolman, Ehrenfest and Podolski in 1931 [37], who described the light beam as a one-dimensional "pencil of light". Later, a description for the gravitational field of a cylindrical beam of light of a finite radius was presented by Bonnor [4]. In this description, light has been modeled as a continuous fluid moving at the speed of light. A central feature of these two models is the lack of diffraction; the beams do not diverge. This corresponds to the short wavelength limit where all wavelike properties of light are neglected. Further studies of the gravitational field of light that share this feature include the investigation of two co-directed parallel cylindrical light beams of finite radius [3,26], spinning non-divergent light beams [25], non-divergent light beams in the framework of gravito-electrodynamics [13], and the gravitational field of a point like particle moving with the speed of light [1,39]. In contrast, the wavelike properties of light have been taken into account in [38], where the gravitational field of a plane electromagnetic wave has been investigated. An approach to take finite wavelengths into account for the case of a laser pulse has been given in [28,30], where, however, diffraction has been neglected. In this article we describe the laser beam as a solution to Maxwell's equations. This is done perturbatively by an expansion in the beam divergence, which is considered to be small. The zeroth order of the expansion corresponds to the paraxial approximation and coincides with the result of [4]. In the first order in the beam divergence, frame-dragging due to the internal angular momentum of circularly polarized beams occurs. In the forth order in the divergence angle, a parallel co-propagating test beam of light is found to be deflected by the gravitational field of the laser beam. The gravitational field of laser beams is a phenomenon on the interface of general relativity and quantum mechanics as laser beams can be brought into non-classical states. For the progress of modern physics it is of great importance to study such phenomena, as they may give some insight into quantum gravity, supporting or ruling out possible candidates for the theory. Laser beams are widely used in the laboratory, and are therefore potential candidates to experimentally measure both gravitational and quantum mechanical effects. To do so, it is necessary to know precisely the features of laser beams, including their gravitational properties. We proceed as follows: In Sec. 2, we describe a focused laser beam as a solution to Maxwell's equations. This is done perturbatively, as an expansion in the small beam divergence angle θ. Furthermore, we derive the energy momentum tensor for a circularly polarized laser beam. In Sec. 3, we introduce the framework of linearized gravity. The equations determining the metric perturbation and solutions with Green's functions are given in Sec. 4. Then we discuss the specific effects appearing in the different orders of the expansion in θ of the gravtational field: In Sec. 5, we discuss the zeroth order, which corresponds to the paraxial approximation. Frame-dragging happens in the first order of the metric perturbation and is explained in Sec. 6. That a co-propagating parallel light ray is deflected in the gravitational field of the laser beam is shown in Sec. 7. Some conclusions are given in Sec. 8. Throughout the article we use the following notation: For spacetime coordinates we use greek indices, like x α , and for spatial coordinates we use latin indices, like x k . For the Minkowski metric, we choose the convention η αβ = diag(−1, 1, 1, 1).

DESCRIBING THE LASER BEAM
In this section we describe the laser beam as a Gaussian beam, a perturbative solution to Maxwell's equation. The solution is expanded in the beam divergence, which is assumed to be small. Finding a solution for the vector potential, we calculate the energy-momentum tensor which will be used in the next section to determine the spacetime metric.

The field strength tensor
The laser beam is a monochromatic plane wave whose intensity distribution in the directions perpendicular to the direction of propagation decreases with a Gaussian factor. It is a perturbative solution of Maxwell's equations: an expansion in the beam divergence, the opening angle of the beam, which is assumed to be small. This solution is obtained by making the ansatz that the vector potential is a plane wave enveloped by a function depending on the position. More specifically, the vector potential of the Gaussian beam is obtained as follows: It has to satisfy Maxwell's equations in form of the wave equations, is the d'Alembert operator and we choose the Lorenz gauge condition η αβ ∂ α A β = 0. For convenience, we work in the dimensionless coordinates τ = ct w0 , ξ = x w0 , χ = y w0 , ζ = z w0 , where w 0 is the beam waist. Writing {x α } for the coordinates {ct, x, y, z} and {xᾱ} for the coordinates {τ, ξ, χ, ζ}, we obtain for the Minkowski metric ηᾱβ = dx α dxᾱ dx β dxβ η αβ = w 2 0 diag (−1, 1, 1, 1) .
We consider θ to be small, which implies that the envelope function changes much more slowly in z-direction than in x-direction or in y-direction. Then, we make the ansatz that vᾱ can be written as a power series of θ, 1 vᾱ(ξ, χ, θζ) = ∞ n=0 θ n v (n) α (ξ, χ, θζ) , where v (n) α are the coefficients in the power series. The Helmholtz equation (4) leads to the differential equations Note, that this set of equations couples components of vᾱ of odd n to other components of odd n and components with even n to other components of even n. Therefore, we obtain two independent hierarchies of components of vᾱ. We will couple odd and even components later when we introduce helicity. Eq. (6) is known as the paraxial Helmholtz equation. It can be interpreted as a Schrödinger equation in two spatial dimensions with m/ = 2 when θζ is seen as a time variable, i.e.
i∂ θζ v where ∆ 2d = ∂ 2 ξ + ∂ 2 χ is the two dimensional Laplace operator. A solution of Eq. (9) has to spread similar to the wave packet of a massive particle in quantum mechanics. Here, the spreading of the wave packet corresponds to the divergence of the beam. The solution of Eq. (9) that we are interested in is a Gaussian wave packet. Furthermore, we want that the wave packet is centered on the optical axis and that it is rotationally symmetric about the optical axis. With these conditions, we obtain for the lowest order v (0) where the function v 0 is given by and where ρ = ξ 2 + χ 2 , α is the constant polarization co-vector and µ(θζ) = 1/(1 + iθζ) relates the spread of the Gaussian wave packet and the divergence angle of the beam. Eq. (10) represents the Gaussian beam in lowest order in the divergence angle θ. A graphic representation can be found in Fig. 1.
FIG. 1. Schematic illustration of the Gaussian beam, the beam waist w0, the Rayleigh length zR and the beam divergence θ. More specifically, the figure illustrates the scalar envelope function v0 of the vector potential of the Gaussian beam in a plane that contains the optical axis (represented by the dashed horizontal line). Due to the rotational symmetry of the envelope function about the optical axis, the vertical axis can be any direction transversal to the optical axis. The thick curved lines mark the distance w(ζ) = 1/|µ(θζ)| from the optical axis at which the absolute value of the envelope function reaches 1/e times its maximum.
The first order solution fulfills the same paraxial Helmholtz equation as the zeroth order solution. Therefore, we set v (1) The equations for the higher order terms in Eq. (8) correspond to Schrödinger equations with an additional term proportional to the solution of the equation two orders lower, which has the effect of a source term, Finally, we have to specify the polarization co-vectors ᾱ and the terms in the expansion of the envelope function of even n. We will do so for a Gaussian beam of circular polarization in the following. First, note that the components of the vector potential are not independent; the Lorenz gauge condition we imposed leads to With this identity, A τ can be eliminated from the space-time components of the field strength tensor Fᾱβ = ∂ᾱAβ − ∂βAᾱ as where δbc is the Kronecker delta. As the vector potential, the field strength tensor can be expanded as where E 0 = √ 2A/(w 0 θ) and a direct relation between v (n) α and f (n) αβ can be established, which is given in Appendix A.

Circularly polarized beams
In the last step, we have to specify the polarization of the beam that we want to consider. In this article, we will focus on circularly polarized beams. We define a circularly polarized beam as a helicity state which is an eigenstate of the generator of the duality transformations F ᾱβ = Fᾱβ cos ϕ + Fᾱβ sin ϕ, where Fᾱβ = 1 2 −det(η) ᾱβγδ Fγδ is the Hodge dual of Fᾱβ and ᾱβγδ is the completely anti-symmetric Levi-Civita symbol with 0123 = −1. The invariance of Maxwell's equations under these duality transformations and the corresponding conservation laws were worked out in [8]. The generator of the duality transformation D θ = exp(iϕΛ) : The vector potentials of well-defined helicity are eigenstates of Λ with eigenvalues λ = ±1. There are two options to obtain these eigenstates. One option is to start with a helicity eigenstate of zeroth order in θ, construct the corresponding higher order terms of the expansion of the envelope function of even n with Eq. (13), obtain the odd terms in the expansion of the envelope function with the Lorenz gauge condition in Eq. (14), calculate the field strength tensor and project it with (1 + λΛ)/2. This option is presented in Appendix C. In the main text of this article, we follow the second option, where a vector potential is constructed order by order by taking into account the condition (1 − λΛ)F λ αβ = 0 and the expansion in Eq. (16) in each order separately. This construction is presented in Appendix A. Starting from v 2 andā ∈ {ξ, χ, ζ}, and taking the solutions of even orders from [32] into account, we obtain where (1) a = w 0 (0, 0, 1). The corresponding vector potential is given as A λ α = 4 n=0 θ n Av λ(n) α (ξ, χ, θζ)e i 2 θ (ζ−τ ) , where the component A λ τ is given through the Lorenz gauge condition in Eq. (14). Linearly polarized Gaussian beams are obtained as linear combinations of helicity eigenstates; for example, A ξ α := (A + α + A − α )/ √ 2 is the vector potential of a laser beam that is linearly polarized in the ξ-direction. Note that all terms of higher than leading order in Eq. (17) decay faster than v λ(0) a for θζ → ∞. Hence, vā ≈ v λ(0) a for large θζ.

Three distinct scenarios
The beam divergence θ, which is assumed to be small, is related to the wave vector k, the beam waist w 0 and the Rayleigh length z R through The beam waist w 0 describes the width of the beam at its focal point, i.e. at ζ = 0, and the Rayleigh length is the distance from the focal point along the direction of propagation such that the cross section of the beam is doubled, as illustrated in Fig. 1. There are basically three scenarios for which the condition that θ is small is satisfied: 1. k = constant: If the wave vector k is kept constant, the beam waist w 0 and the Rayleigh length z R have to be large, and z R w 0 has to hold. Keeping the wave vector constant is the characteristic feature of a plane wave. If the beam is very long, it may be compared to the infinitely extended plane waves, which are described by the pp-wave metrics [9].
2. w 0 = constant: Keeping the beam waist w 0 fixed, the wave vector k and the Rayleigh length z R have to be large, and in addition we find z R 1 k . This situation describes an almost parallel beam of a given waist. If the beam is very long and the beam waist is considered to be small, such that it is approximately a cylinder of light, it may be compared to the solution found by Bonnor [4] for an infinitely long cylinder of light.
3. z R = constant: Keeping the Rayleigh length fixed, the wave vector k has to be large and the beam waist w 0 has to be small. This case corresponds to a very thin and almost parallel beam along the z-axis, whose energy-density is accordingly high. This is the solution given by Tolman, Ehrenfest and Podolski [37].
In the following, we will keep the beam waist w 0 constant.

The energy-momentum tensor
To derive the gravitational field of the laser beam, we have to derive its energy-momentum tensor first. Let us define the real part of Fᾱβ as Re(F )ᾱβ. In terms of Re(F )ᾱβ, the energy-momentum tensor is defined as Tᾱβ = c 2 ε 0 (Re(F )ᾱσRe(F )βσ − 1 4 ηᾱβRe(F )δρRe(F )δρ). Therefore, the energy momentum tensor can be decomposed into the real termTᾱβ and its complex conjugateT * αβ . The termTᾱβ is highly oscillating with i(ζ − τ )/θ while these oscillations cancel inTᾱβ. For eigenstates of the helicity operator with eigenvalue λ = ±1, the highly oscillating terms inTᾱβ and its complex conjugate vanish and it remains Tᾱβ =Tᾱβ. Therefore, the highly oscillating parts of the energy momentum tensor can be interpreted as a result of the interference of contributions of different helicity in the field strength that come into play for linear or elliptical polarization. In the following, we will only consider circular polarization. The components of the energy-momentum tensor are directly related to the energy density E λ , the Poynting vector S λ and the Maxwell stress tensor σ λ ij of the electromagnetic field, For the field strength tensor F λ αβ = ∂ᾱA λ β − ∂βA λ α of a circularly polarized laser beam which we specified in Sec. 2, the energy density, the Poynting vector and the stress tensor components are given in Appendix B. The power transmitted in the direction of propagation is given by P = 2π 0 dφ ∞ 0 dρ ρS ζ . In the leading order in the expansion in θ, we obtain P 0 = πcε 0 E 2 0 w 2 0 /2, where E 0 is the amplitude of the electric field in the leading order at the beamline. We may then express the amplitude in terms of the power as E 0 = 2P0 πcε0w 2 0 . For a power of P 0 ∼ 10 15 W and a beam waist of w 0 ∼ 10 −3 m, the amplitude is E 0 ∼ 10 12 V m . As the field strength tensor, the energy momentum tensor can be expanded in orders of θ as T λ αβ = n θ n t λ(n) αβ . Then, the gravitational field of the laser beam can be calculated for each order and effects of different orders can be identified. We will present this analysis up to fourth order in θ in the following sections.

LINEARIZED GRAVITY
Assuming that the energy of the laser beam is sufficiently small, we use the linearized theory of general relativity [9] to describe its gravitational field. In Appendix D, we make a rough estimation to show that this is reasonable. The metric g αβ consists of the metric for flat spacetime η αβ plus a small perturbation h αβ with |h αβ | 1, Therefore one neglects terms quadratic in the metric perturbation. In this case, one sees that the inverse of the metric reads g αβ = η αβ −h αβ . The Einstein equations can be simplified to a set of linear equations in the metric perturbation.
As the full general relativity has an invariance under coordinate transformation, its linearized approximation is invariant under linear coordinate transformations x α →x α = x α + ξ α , where the metric perturbation transforms as h αβ →h αβ = h αβ − ∂ α ξ β − ∂ β ξ α 2 . Since curvature is described by the second derivatives of the metric, quantities depending on the curvature are invariant under linear coordinate transformations. To derive the linearized version of the Einstein equations, we assume the Lorenz gauge condition, ∂ α h αβ = ∂ β h α α /2. The energy-momentum tensor has to be conserved, η αβ ∂ α T βγ = 0, which implies that the continuity equation is satisfied [23,28] . The remaining gauge freedom is given by linear coordinate transformations ξ α that satisfy ξ α = 0. Taking into account that the trace of the energy momentum tensor T σ σ is identically zero for the electromagnetic field, we obtain the linearized Einstein equations [9] h αβ = −κT αβ , where κ = 16πG/c 4 and G is Newton's constant.
In general relativity, coordinates have no physical meaning. Since the values of the components of the metric tensor depend on the choice of coordinates, we cannot extract physical information directly from them. Therefore, we have to investigate effects on test particles to learn about the gravitational field. The motion of test particles is governed by the geodesic equation where, in linearized gravity, the Christoffel symbols are given as A more direct way to analyse gravitational effects is through the spread and the contraction of the trajectories of test particles. This way, the test particles serve as each others reference. The relative acceleration between two infinitesimally close geodesics γ( ) and γ ( ) parameterized by is given by the geodesic deviation equation where s is the separation vector between the geodesics, D/d =γ µ ∇ µ is the covariant derivative along the geodesic γ( ) and R µ ρσα is the Riemann curvature tensor. This is illustrated in Fig. 3. In the linearized theory, the pulled down Riemann curvature tensor is given by [9] R αβγδ = 1 2 Since the metric perturbation transforms as h αβ →h αβ = h αβ − ∂ α ξ β − ∂ β ξ α , we find that R αβγδ is invariant under a linearized coordinate transformation.

THE METRIC OF THE LASER BEAM
Solving Eq. (27) for the energy momentum tensor (25) with emitter and absorber at general positions can be quite cumbersome. In the following, we will consider two different limiting situations instead; we consider the case of the distance between emitter and absorber being very large and very small.
In the first situation, we can neglect the rapid change of the field strength at the emitter and the absorber. Then we can take into account that Tᾱβ is changing slowly in ζ. In particular, we have T λ αβ =T λ αβ (ξ, χ, θζ). Therefore, we can expand the metric perturbation similar to Eq. (5) as and the linearized Einstein equations (27) lead to the differential equations The solutions h λ(n) αβ of Eqs. (33), (34) and (35) can be given by using the free space Green's function for the Poisson equation in two dimensions as where Q λ(n) is the source term on the right hand side of Eqs. (33), (34) and (35), respectively. The form of the solutions in Eq. (36) was fixed by an additional condition that we did not discuss yet; we want the components of the Riemann curvature tensor to vanish at infinite distance from the beamline. As stated in Sec. 3, the Riemann curvature tensor governs the spread and the contraction of the trajectories of test particles. This means, if the Riemann tensor vanishes, parallel geodesics stay parallel and there is no physical effect as the only reference for a test particle in linearized gravity can be another test particle. We can assume that there is no gravitational effect for infinite spatial distances from the beamline. Therefore, we assume that the Riemann curvature tensor R µ ρσα vanishes for ρ → ∞. The full discussion of the curvature condition and its implications are given in Appendix F. Additionally, Appendix F contains expressions for the components of the metric perturbation up to third order in θ. Note that, as the vector potential, the field strength tensor, the energy momentum tensor and the metric perturbation before, we have to expand the Christoffel symbols and the Riemann tensor as well in orders of θ as and respectively. With Eqs. (31), (29) and (32), we can derive direct relations between the terms of the expansions r λ(n) αβγδ and (γ λ(n) )ᾱ βγ and terms in the expansion of the metric perturbation h λ(n) αβ . They are given in Appendix E.

Small distance between emitter and absorber
In the second situation, where we assume a short distance between emitter and absorber, the rapid change of the field strength at emitter and absorber cannot be neglected. Then, we solve the Einstein equations (27) by use of their retarded solution Furthermore, we can set θζ 1 and we can expand the function e −|µ(θζ)| 2 ρ 2 appearing in the energy momentum tensor in θ before the integration, which simplifies the calculations significantly. 3 Expressions for h λ αβ up to second order in θ for the case of small distances between emitter and absorber can be found in Appendix H. In the following, we discuss the metric perturbation in different orders in θ and present its physical effects.

ZEROTH/LEADING ORDER
The metric in the leading order corresponds to the full metric at θ = 0, and thus to the metric for the laser beam in the paraxial approximation. Then, the components of the Poynting vector transversal to the beamline vanish and the only non-zero component of the Maxwell stress tensor is σ λ ζζ . Furthermore, σ λ ζζ = E λ = −S λ ζ /c, which leads to where . Therefore, the metric perturbation is found as where, for the case that the emitter and absorber are far away from each other, we find from Eq. (36) where Ei(x) is the exponential integral function. The solution (42) can be compared with the exact solution derived by Bonnor for an infinitely extended beam of a light-like medium without divergence. The derivation of the metric for a Gaussian profile of the energy density of the medium is given in Appendix G. Bonnor's solution is split into an interior and an exterior solution that are matched at a finite transversal radius a. If the beam is infinitely extended in the transverse direction, we are left with an interior solution only which reads For θ = 0, we have µ(θζ) = 1, and the solution in Eq. (42) coincides with (43).

Small distance between emitter and absorber
For the case when the emitter and absorber are close to each other, we have to take the second approach described in Sec. (4). With θζ 1, the retarded potential (39) in leading order in θ becomes where J 0 is the Bessel function of the first kind. For small beam waists, w 0 1, the solution for the laser beam (44) approaches the solution for the infinitely thin beam (45), as shown in Appendix I. We obtain Thus, in the paraxial approximation, we may say that the solution for the laser beam approaches the solution for the infinitely thin beam of constant energy per length of [37] as the beam waist goes to zero. Note that the limit w 0 → 0 can only be considered for the leading order of the laser beam here. This is because θ = 0 implies that the condition θζ 1 can be satisfied for all w 0 . In contrast, for any non-vanishing θ, the conditions w 0 → 0 and θz/w 0 = θζ 1 imply z → 0. In Fig. 3, the function I (0) and its derivatives are illustrated for the three cases of the infinitely long Gaussian beam, the Gaussian beam with short distance between emitter and absorber with a Gaussian profile, and the infinitely thin beam.  3. These plots show the value of the leading order of the metric perturbation I (0) and its first derivatives for the Gaussian beam with infinite distance between emitter and absorber (plain, blue), the Gaussian beam with short distance between emitter and absorber (dashed, red), and the infinitely thin beam (dotted, purple) in units of κP0w 2 0 /(2πc). In the second and the third case, the distance between emitter and absorber is chosen to be 6. In the first row, the functions are plotted for ζ = 1 and in the second row for ρ = 1/2. The second row does not contain plots for the case of large distances between emitter and absorber as there is no dependence of I (0) on ζ in that case. We find that the values for I (0) and its first derivatives are usually larger for the infinitely thin beam than for the other two cases. This is due to the divergence at the beamline for the case of the infinitely thin beam. In the other two cases, the gravitational field is spread out as the sources are. In b), we see that the absolute value of the first ρ-derivative of I (0) reaches a maximum at a finite distance from the beamline. Note that ∂ρI (0) is proportional to the acceleration that a test particle experiences if it is initially at rest at a given distance ρ to the beamline. We see that acceleration is always directed towards the beamline. It is larger in the case of an infinite distance between emitter and absorber than in the case of a finite distance, which we can attribute to the larger extension of the source (and thus the larger amount of energy) in the former than in the latter. In e), which shows plots for the cases of finite distance between emitter and absorber, we see that ∂ρI (0) still is the largest at the center between emitter and absorber and decays quickly once their positions at ζ = ±3 are passed. ∂ ζ I (0) is proportional to the acceleration in the ζ-direction. As expected it vanishes for infinite distance between emitter and absorber. In f), we see that the acceleration is directed towards the center between emitter and absorber and its absolute values reaches its maximum at ζ = −3 and ζ = 3, the ζ-coordinates of emitter and absorber respectively.

Acceleration of a test particle at rest
Let us consider the acceleration a massive test particle would experience if it was initially at rest at given ρ and ζ. Then, the initial normalized tangent to its worldline γ( ) is given as τ τ /2, 0, 0, 0). From the geodesic equation (28) and the form of the metric in zeroth order, we find that d 2 γ µ /d 2 = ∂ µ I (0) /2 and therefore Plots of ∂ ρ I (0) and ∂ ζ I (0) for the three different cases above are given in Fig. 3.

Curvature
For the leading order, we can find the components of the curvature tensor using Eq. (E2) in Appendix E and Eq. (41). The only non-zero independent components of the Riemann curvature tensor for the metric perturbation given in Eq. (42) and Eq. (44) and the limit of an infinitely thin beam in Eq. (45) are For the case of a far extended beam neglecting emitter and absorber that was given in Eq. (42), we obtain Comparison to the inifinitely thin beam In the paraxial approximation (i.e. for θ = 0) and for small beam waists, the Riemann curvature tensor of the infinitely long laser beam approaches the Riemann curvature tensor of the infinitely thin beam, as the metric does so. It is also interesting to compare the curvature for the infinitely thin beam with that for the full solution given in [4] by Bonnor.
The analysis can be found in Appendix G for a beam with a Gaussian profile cut off at a radius a. The corresponding solution splits into an interior solution and an exterior solution. For a → ∞, we obtain the solution in Eq. (43) that we compared with our leading order metric perturbation already. In Appendix G, we give the components of the curvature tensor in the exterior region (r > a) in Eq. (G8). We show that it coincides with the components of the curvature tensor of an infinitely thin beam. In particular, the curvature is independent of the radial dependence of the beam intensity; only the total power of the beam is important.

FIRST ORDER AND FRAME DRAGGING
The metric perturbation for large distances between emitter and absorber in first order in θ is determined by the first order of the energy momentum tensor,t λ (1) αβ , which has the only independent non-zero components Note thatζ = θζ is the coordinate that is considered for the asymptotic expansion in Eqs. (6), (7) and (8). Therefore, where For small θζ, the terms proportional to θζ can be neglected in (53) such that we find It is interesting to note that our solution coincides with the exact solution of Einstein's equations presented in [5] by Bonnor for a rotating null fluid. In particular, we can identify our functions in the metric with those of [5] as α = θI . Our equation (34) corresponds to the equations (2.16) and (2.17) in [5]. Similar expressions for the metric of a circularly polarized light beam are presented in [16].

Small distance between emitter and absorber
For small distances between emitter and absorber, we find directly Eq. (55), where I (0) has to be taken from Eq. (44). In Fig. 4, the function I λ(1) ξ is illustrated as a function of ξ and ζ for χ = 1. The plots for I λ(1) χ would look similar when plotted as a function of χ and ζ for ξ = 1. for an infinite distance between emitter and absorber (plain, blue) and a short distance between emitter and absorber (dashed, red) as a function of ξ for ζ = 0.1 and χ = 0 (plot a) and as a function of ζ at ξ = 1/2 and χ = 1 (plot b). The functions are plotted in units of κw 2 0 P0/(2πc). In b) there is no plot for the case of infinite distance between emitter and absorber as the result does not depend on θζ. Plot c) shows the deflection in the χ-direction a light light test particle would experience if it would move radially outwards in the ξ-direction at χ = 0 for an infinite distance between emitter and absorber (plain, blue) and a short distance between emitter and absorber (dashed, red). This effect is induced by frame dragging. We see that the effect changes sign for the case of a short distance between emitter and absorber.

Curvature
It was shown in [5] that the rotation of the null fluid leads to frame dragging. This has been shown to be the case as well in [36] for a laser beam of light with angular momentum. Here, we obtain the frame dragging effect in the curvature tensor components. The only non-zero components of first order (see Eq. (E2)) are The non-zero curvature components r λ(1) ξτ ξχ and r λ (1) χτ χξ lead to the precession of gyroscopes, which can be seen most straight forward in the framework of gravitomagnetism [24]; they can be interpreted as gravitomagnetic fields that govern the motion of test particles in a gravitational Lorentz force law.

Deflection of test particles
The frame dragging effect can be studied alternatively using the geodesic equation (28) and the expressions for the Christoffel symbols in Eq. (E1). Let us consider a test particle moving radially outwards with velocity v. We will only consider terms liner in v in the following. Then, the initial tangentγ(0) = w −1 0 (1 − f, v/c, 0, 0) to the test particle's world line γ(ρ) at γ(0) = (0, ξ, 0, 0) and f (v, ξ, χ, θζ) is chosen such thatγ( ) fulfills the condition gμν(γ( ))γμ( )γν( ) = −1 at = 0. In first order in the metric perturbation, we find thaẗ We see that massive test particles do not propagate radially. Their trajectories are transversally bent, where the sign of the bending depends on the polarization of the laser beam. This is the effect of frame dragging. Let us consider the acceleration in natural coordinates (t, x, y, z). Application of the Leibniz rule gives d 2 γ a /dt 2 = w −1 0 (dτ /d ) −2 (γ a − v aγτ /c), which leads to in first order in v. The effect in Eq. (59) decreases exponentially with the distance to the beamline. The same is true for the curvature components in Eq. (57). The effect is due to the intrinsic angular momentum due to the helicity of the beam. In contrast, in [36], frame dragging effects for ρ 1 have been shown to arise from the orbital angular momentum of optical vortices. In Fig. 5, the above deflection is illustrated. It is interesting to note that, by direct calculation from the expressions for the metric perturbation up to third order in θ in Appendix F, we find for ρ 1 up to third order in θ. All other terms decay exponentially with ρ 2 . Therefore, far away from the beam and up to third order in θ, there are no effects beyond those that already exist in zeroth order. All additional effects appear only where the energy distribution of the source beam is non-vanishing; they are effects of a local gravitational coupling between the source and test particles. In the next section, we will discuss another such effect in fourth order in θ, the deflection of parallel co-propagating test rays.

FOURTH ORDER -THE DEFLECTION OF PARALLEL CO-PROPAGATING TEST RAYS
As discussed in [37] for a finitely long and infinitely thin light beam, a test ray propagating parallel to it is not deflected. It has also been shown [4] that the superposition of two exact solutions of the Einstein equations for pp-waves travelling in the same direction is again a solution, confirming the result of the linearized theory. In our description, there are two more important characteristics of the laser beam playing an important role, both of them coming from the wave-like nature of light: First, the laser beam is diverging. Second, in [15], it was argued that light in a laser beam does not move with the speed of light along the beamline, but with a slightly smaller velocity. The origin of the effect is the superposition of plane waves with different wave vectors, which leads to a reduced effective propagation speed. This was confirmed experimentally in [17]. In [15], the difference between the speed of light and the group velocity of light in a laser beam was found to be 4 δv θ = c − v θ = c/(k 2 w 2 0 ) = cθ 2 /4. It has been shown by Scully that two parallel co-propagating thin beams in a wave-guide, since they are propagating slower than the speed of light, do gravitationally interact with each other [34]. Therefore one may wonder whether the laser beam deflects an originally parallel co-propagating test ray. We will investigate this question in the following. The setup is illustrated in Fig. 6. A parallel co-propagating test light ray is described by the light-like tangent vectorγᾱ = w −1 0 c(1, 0, 0, 1 − f ), where f is determined by the null-condition and found to be of the same order of magnitude as the metric perturbation, and therefore does not contribute in the following. With the geodesic equation, we obtain From the expression for the components of the energy momentum tensor in Appendix B and Eqs. (35), we find which is solved by Eq. (36) as The components of the metric perturbation in third order in θ which appear in the second term in equation (62) can be found in Appendix F. We obtain forj ∈ {ξ, χ}, assuming that θζ 1, For large distances from the beamline (ρ 1) and j ∈ {x, y}, the acceleration becomes which is an apparent repulsion. This is due to the second term in Eq. (62). If we would have considered only the first term in Eq. (62), we would have obtained the same absolute acceleration as in Eq. (67), but with the opposite sign. Hence, the first term in Eq. (62) induces an attraction and the second term a repulsion. However, coordinate acceleration does not have any physical meaning in general relativity. Therefore, we have to investigate the geodesic deviation to learn about the meaning of the coordinate acceleration (67). With the separation vector sᾱ = (0, 1, 0, 0) and the tangentγᾱ = w −1 0 c(1, 0, 0, 1−f ), we obtain for the acceleration of the separation vector in ξ-direction from Eq. (30) With the expressions for the combinations of the metric perturbation given above and in appendix G, we obtain in the case of θζ 1 which vanishes far from the beamline. Therefore, we found that the deflection in Eq. (66) is a coordinate effect. More precisely, the geodesic deviation in Eq. (68) can be split into two parts. The first part is the ξ-derivative of the coordinate accelerationγj in Eq. (66). The second part is the second θζ-derivative of h (2) ζζ which corresponds to the change of the definition of length in the ξ-direction. The contributions of the two parts to the geodesic deviation cancel for large distances from the beamline.

Comparison to the boosted infinitely long massive cylinder
The reduced propagation speed argued for in [15] suggests that the result in Eq. (66) may be compared to the deflection of a parallel test ray by a cylindrically symmetric mass distribution moving with v = c − δv θ along the cylinder axis (see Fig. 7). That is the content of this subsection. The exterior gravitational field of a cylindrically symmetric mass distribution of rest of mass per unit length m (dimensionless units) is described by the Levi-Civita metric [20], in the cyllindrical coordinates (ct, ρ, φ, z), where ρ = x 2 + y 2 and we set P = 1. The parameter m can be considered to be a dimensionless quantity representing the mass or energy per unit length for 0 < m < 1 2 [6]. Now, we let the cylinder move in positive ζ-direction with normalized velocity β = v/c, and thus make the coordinate transformation where γ = (1 − β 2 ) −1/2 and β = v/c. The line density of energy m is a quotient of an energy scale E and a length scale L. The energy seen by an observer in the rest frame is E = γE. Due to Lorentz contraction, the length scale seen in the rest frame becomes L = L/γ. Therefore, the line density of energy seen in the rest frame becomes m = γ 2 m. Then, the metric becomes Transforming to cylindrical coordinates according to ρ = x 2 + y 2 and dρ = 1 ρ 2 (xdx + ydy), as well as dφ = 1 ρ 2 (−ydx + xdy), and assuming γ −2 m to be small and expanding the terms ρ γ −2 m as ρ m = 1 + γ −2 m log(ρ) and neglecting quadratic terms in γ −2 m , we obtain This metric can be decomposed into the Minkowski metric plus the small perturbation We can identify the line density of energy with that of a beam of light as m c 4 /G = P 0 /c. Then, the metric η αβ + h αβ coincides with the metric of an infinitely long beam of light with constant energy density P 0 /(π(w 0 /2) 2 c) confined to a cross section of π(w 0 /2) 2 for β = 1 given in [4], up to constants. From the metric (74), we find that the parallel test ray with tangentγ µ = c(1, 0, 0, 1) is deflected in x-direction according toγ Assuming 1 − β = δv/c = θ 2 /4, we find that the result in Eq. (75) differs from Eq. (67) by its sign and a factor 1/2. Considering the geodesic deviation with the separation vector s α = (0, 1, 0, 0), we obtain and, inserting the expressions for the metric, In contrast, for the gravitational field of the focused laser beam, we did not find a deflection for large distances. From this result we see that the gravitational field of light in a Gaussian beam does not simply behave as massive matter moving with the velocity derived in [15] along the beamline. The reason is that the divergence of the laser beam does not only lead to a reduced group velocity, but also to a change of the metric along the beamline. This leads to the appearance of the second and third term in Eq. (68), which cancel the effect of the first term for large distances from the beamline. In particular, we mentioned above that the first term in Eq. (62) induces an attraction with the same absolute value as the acceleration in Eq. (66). Accordingly, if we had considered the first term in Eq. (62) only, we would have obtained an expression that would coincide with that for the geodesic deviation induced by the boosted rod given in Eq. (77) up to a factor 2. Therefore, we can conclude that the additional effects due to the divergence of the light beam cancel the attraction due to the reduced propagation speed of the light in the beam.

CONCLUSION
We analyzed the gravitational field of a focused laser beam, describing the laser beam as a solution to Maxwell's equations. We calculated the five leading orders of the metric perturbation expanded in the divergence angle θ of the beam explicitly and discussed the difference to the solutions when the laser beam is treated in the paraxial approximation. For small values of the beam waist and for θ = 0, which corresponds to the paraxial approximation in our case, our solution for the laser beam corresponds to the solution for the infinitely thin beam [37]. If in addition we consider the laser beam to be infinitely long, we recover the solution for an infinitely long cylinder [4]. In first order in the divergence angle, we found frame dragging due to intrinsic angular momentum of the circular polarized laser beam. This is similar to the result of [36] for beams with orbital angular momentum. In contrast to frame dragging induced by orbital angular moment, the effect we find decays exponentially with the distance square from the beamline divided by the beam waist parameter w 0 . This property coincides with the decay of the energy density of the beam. Hence, frame dragging due to the intrinsic angular momentum of the beam is proportional to the local energy density of the beam. The statement of [37] by Tolman et al. that a non-divergent light beam does not deflect gravitationally a co-directed parallel light beam has been recovered in different contexts: two co-directed parallel cylindrical light beams of finite radius [3,4,26], spinning non-divergent light beams [25], non-divergent light beams in the framework of gravitoelectrodynamics [13], parallel co-propagating light-like test particles in the gravitational field of a one-dimensional light pulse [28]. In fourth order in the divergence angle, we found a deflection of parallel co-propagating test beams. This shows that the result of [37] and [4] only holds up to the third order in the divergence angle. This could have been expected from the fact that the group velocity of light in a Gaussian beam along the beamline is not the speed of light [11,14]. However, the deflection of parallel co-propagating light beams by light in a focused laser beam decays like the distribution of energy of the source beam with the distance from the beamline. This means that the effect does not persist outside of the distribution of energy given by the laser beam like the frame dragging effect due to intrinsic angular momentum. This is in contrast to the deflection that one obtains from a rod of matter boosted to a speed close to the speed of light. Therefore, we conclude that focused light does not simply behave like massive matter moving with the reduced velocity identified in [28,36]. This is due to the divergence of the laser beam along the beamline which leads to additional contributions to the metric perturbations which do not appear in the case of the boosted rod. These additional contributions cancel the effect induced by the reduced propagation speed of light in the focused beam.

OUTLOOK
As an extension of the research presented in this article, it would be interesting to study the gravitational interaction of two parallel co-propagating focused laser beams in the description presented here. The result could be compared to the corresponding results presented in [3,4,26]. In particular, it would be interesting to see if a contribution to the gravitational interaction of the two beams exists that does not decay exponentially with the square of the distance between the beamlines of the beams. It would be interesting to know if the solutions to Maxwell's equations developed in this article can be used as a basis for a quantum field theoretical description of the gravitational interaction of two laser beams in the framework of perturbative quantum gravity (PQG). Then, the effect of localization on light-light interactions could be considered for light with quantum properties. For example, in [27,29] it is shown that the differential cross section for gravitational photon scattering can be amplified or suppressed when the scattering photons are in specific polarization entangled states initially. It would be interesting to see how this effect depends on the distance between the beams. Furthermore, in [7], the effect of entanglement in the position of a source of a gravitational field was investigated in the framework of semiclassical gravity. Similar questions could be considered in the framework of PQG using focused laser beams in spatial superposition states or with squeezed light as sources.
Another step from the results presented in this article into a different direction could be the consideration of a pulse of light in a focused laser mode. The framework used in this article would need to be extended to envelope functions that depend on time and the position along the beamline. Approaches for the description of such beams are given for example in [2,21,33,40]. An expression for the gravitational field of a focused laser pulse could be used to have a closer look at the implications of focusing for possible experiments trying to detect the gravitational field of light.
In particular, the pulsed beams would produce a pulsed gravitational signal that could be detected with resonator systems like small scale gravitational wave detectors (for example [18,31,35]) or quantum optomechanical systems.
The gravitational field of a focused laser pulse could also be used to check the results of [28] where the laser pulse is modeled as a one-dimensional rod of light with an energy density that is modulated as that of a plane wave. In particular, for the model used in [28], all gravitational effects are induced by the emission and the absorption of the light pulse alone; there is no gravitational effect related to the propagation of the pulse. This situation may change once divergence of the beam is taken into account. It could be worthwhile to see whether a similar solution for the gravitational field of a focused laser beam as we derived in this article could be derived considering the full coupled set of the Einstein-Maxwell equations. The resulting metric could be compared to the one in [22] and it could be investigated if the results of [22] about the effective gravitating mass and angular momentum can be reproduced when divergence of the beam is taken into account. It would also be interesting to consider the gravitational field of the electromagnetic field distribution used in this article to model a focused laser beam in dynamical spacetime theories with spacetime torsion like Einstein-Cartan-Theory and the Poincar-Gauge-Theory of gravity [19]. In particular, we found that frame dragging due to the intrinsic angular momentum of light is proportional to the local energy density of the beam. This is similar to the effect of intrinsic angular momentum on test particles or fields via spacetime torsion as torsion is not a propagating degree of freedom in Einstein-Cartan-Theory and Poincar-Gauge-Theory.
we obtain a direct relation between v λ(n) α and f λ(n) αβ (where λ refers to the polarization state) as Since the vector potential fulfills the wave equation (1), we have that Fᾱβ = 0. In particular, The components of the dual Hodge dual field strength tensor are given as and we obtain that a helicity eigenstate has to fulfill the conditions where the last three conditions are fulfilled if the first three conditions are fulfilled. The remaining conditions can be rewritten as The sum and the difference of the second and third equation lead to and respectively. For the leading/zeroth order envelope function, we find from Eq. (A20) that v . For the first order envelope function, we obtain from Eq. (A17) the condition . Furthermore from Eq. (A20), we find the condition For the second order, we obtain from Eq. (A17) which is always fulfilled since v λ(0) ζ fulfills Eq. (6). Additionally from Eq. (A20) and with v λ(0) ξ = iλv λ(0) χ , we find the condition Assuming v λ(2) ξ = iλv λ(2) χ , we find that the first term in the condition vanishes and we can solve for v The condition in Eq. (A21) is automatically fulfilled in second order due to v . For the third order, we find from Eq. (A17) which is just the θζ-derivative of Eq. (A24). From Eq. (A20) follows that where we used Eq. (A22). The last condition of third order comes from Eq. (A21) as which is fulfilled since Eq. (9) has to hold. In fourth order, we find from (A17) which is satisfied due to Eqs. (7) and (8). From Eq. (A20), we obtain in fourth order Assuming v λ(4) ξ = iλv λ(4) χ and taking into account v , which we assumed before, we obtain Again with Eq. (A24), we can check that the higher order Helmholtz equation (8) is fulfilled by v given in (A31). The last condition that we have to check is the fourth order condition in Eq. (A21), which becomes which may be written as, using v and is fulfilled due to Eqs. (A24) and (6). We conclude that a vector potential for a circularly polarized laser beam up to fourth order in the divergence angle θ is given by Eqs. (6), (7) and (8) Starting from v (0) α = ᾱ v 0 , where ᾱ = w 0 (0, 1, −λi, 0)/ √ 2 and the solutions of even orders that can be found in [32], where v 0 (ξ, χ, θζ) = µ(θζ)e −µ(θζ)ρ 2 . This leads to the expressions for the odd orders v (1) Appendix C: The projected solution Following the second option to construct the field strength tensor for a circularly polarized beam described in Sec. 2, we start from the zeroth order envelope function v We define cylindrical coordinates (ρ, φ, ζ) such that ξ = ρ cos φ and χ = ρ sin φ. Then, the components of the field strength tensor of the helicity eigenfunction F λ,prō Since ΛFᾱβ = −i Fᾱβ, the projection (1 + λΛ)/2 is equivalent to adding the dual field of Fᾱβ. In the approach of complex source points presented in [10], adding the dual corresponds to adding a magnetic dipole to the electric dipole that would create Fᾱβ. In contrast to [10], we add the dual with a phase shift of −π/2 to add −i Fᾱβ and not just Fᾱβ 5 .

Poynting vector, Maxwell stress tensor and energy
For the field strength tensor F λ,prō αβ of a circularly polarized laser beam given in Eq. (C1), the energy density, the Poynting vector and the stress tensor components are given as where E (0) = c 2 ε0w 2 0 E 2 0 |v0| 2 .

Appendix D: Validity of the Linear Approximation of General Relativity
In the linearized version of general relativity, we decompose the metric into the Minkowski metric plus a perturbation, which is assumed to be small, Eq. (26). In this section, we make a rough calculation -just considering orders of magnitude -to verify that the linear approximation is justified, i.e. that it is possible to neglect terms quadratic in the metric perturbation. From the Einstein equations it follows that the second derivative of the metric perturbation is proportional to 8πG c 4 times the energy-momentum tensor, When considering spatial components (the other components can be considered to be of the same order of magnitude), we integrate to obtain an area A on the right hand side, Identifying T Ac as the Power P , we obtain h ∼ 8πG In our calculation, we wrote the metric perturbation in the form (where we write for all expressions of order O(1) in θ) The linearized theory is valid if one can neglect terms of the order O(h 2 ), i.e. if h 2 h. In our case, this condition translates to θ 2 . From the above equations, we see that ∼ 8πG c 5 P . The condition then becomes 8πG For a power of the order of magnitude P ∼ 10 15 W, we thus have to require θ 10 −18 . If we consider θ to be equal to zero, the condition becomes 2 , which is also satisfied.

Appendix E: Expansion of Christoffel symbols and curvature tensor
With the Eqs. (29) and (32), we can derive a direct relation between the terms of the expansions (γ λ(n) )ᾱ βγ and h λ(n) αβ . We obtain forī,j,k ∈ {ξ, χ} where h n αβ = 0 if n < 0. With the Eqs. (31) and (32), we can derive a direct relation between the terms of the expansions r λ(n) αβγδ and h λ(n) αβ . Withj,k ∈ {ξ, χ}, we obtain r λ(n) where h λ(n) αβ (ξ, χ, θζ) is given in Eq. (36). Since the Riemann curvature tensor is linear in the metric perturbation, it consists of a term induced by h λ(n) αβ and a term induced by h λ(n)rest αβ . The term induced by h λ(n)rest αβ (θζ) does not depend on the distance to the beamline. Let us assume that the term in the curvature tensor induced by the first term in Eq. (31) vanishes for ρ → ∞. Then, the term in the curvature tensor induced by h λ(n)rest αβ (θζ) has to vanish everywhere for the curvature to vanish for ρ → ∞.
Therefore, h λ(n)rest αβ (θζ) cannot contribute to the curvature tensor and can be set to zero in Eq. (F1). It turns out that the contribution of the first term in Eq. (F1) to the curvature tensor vanishes at infinity, indeed, up to the fourth order in θ, as we will show in the following. Therefore, we assume h λ(n)rest αβ (θζ) = 0 in this article. In the following, we give expressions for h λ(n) αβ (ξ, χ, θζ) up to third order in θ.
In zeroth order, we have (see Sec. 5 for comparison) In first order, we have In second order, we find for the only non-vanishing independent components of the metric perturbation In third order, we obtain the only non-vanishing independent components Now, with the expressions for the terms in the expansion of the curvature tensor given in Appendix E, we can show that the contribution of h λ(n) αβ (ξ, χ, θζ) to the curvature vanishes for ρ → ∞. From Eq. (F1), we obtain that where xj ∈ {ξ, χ}. From the expressions in Appendix B, we see that all terms in the energy momentum tensor decay like exp(−2|µ| 2 ρ 2 ) for ρ → ∞. From the expressions above, we find that this is true for ∂ 2 θζ h λ(0) αβ and ∂ 2 θζ h λ(1) αβ as well. Furthermore, αβ decays at least as ρ −2 for ρ → ∞. Hence, for n ≤ 4 we find that the sources Q λ(n) αβ -the terms on the right hand side of the differential equations in Eqs. (33), (34) and (35) -are falling off at least as ρ −2 for ρ → ∞. Therefore, the first derivatives of h λ(n) αβ (ξ, χ, θζ) in the directions ξ and χ will go to zero for ρ → ∞ for n ≤ 4. From the expressions above, we find that ∂ 2 θζ h λ(n−2) τ τ and ∂ 2 θζ h λ(n−2) τk decay like exp(−2|µ| 2 ρ 2 ) for ρ → ∞ for n ≤ 4. Therefore, we find that the contribution of h λ(n) αβ (ξ, χ, θζ) to the curvature vanishes for ρ → ∞ and n ≤ 4. Hence, the term h λ(n)rest αβ (θζ) can be set to zero as argued above.

Appendix G: An exact solution for the infinitely long laser beam with boundary in the paraxial approximation
An exact solution for the infinitely long laser beam in the paraxial approximation, i.e. for θ = 0, is constructed as follows: We make the ansatz of a plane wave metric [4], in the dimensionless coordinates (τ, ξ, χ, ζ) = (ct, x, y, z)/w0. The radius of the beam is supposed to be a, such that the energy density is given by w 2 0 = Tττ = T ζζ = −T τ ζ within this radius, and vanishes outside of it. Then the function K = K(ξ, η) in the interior region, for ρ ≤ a, and in the exterior region, for a < ρ, is determined by For the laser beam for θ = 0, the energy density is given by w 2 0 = E (0) = 2P 0 πc e −2ρ 2 . Writing equation (G2) in cylindrical coordinates, we obtain Integrating twice over ρ leads to where Ei(x) = γ + log(|x|) + iarg(x) + x + x 2 4 + x 3 18 + ... is the exponential integral. For the metric to be finite at r = 0, we set C1 = −κE 2 0 w 2 0 /(2πc), and for the interior solution to match the exterior solution at r = a, i.e. to be continuous and differentiable, we choose D2 = 0 and C2 = κP0w 2 0 (2πc) −1 e −2a 2 log(a) − 1 2 Ei(−2a 2 ) , s.t. the final solution reads If the beam is infinitely extended in the transverse direction, we are left with an interior solution only which reads The metric may be written as the Minkowski metric plus a small perturbation hµν = K(ρ)M0, s.t. the only non-vanishing independent components of the Riemann curvature tensor Rτiτj = R ζiζj = −R τ iζj = − 1 2 ∂i∂jK(ρ) (for i, j ∈ {ξ, η}) are given by R int τ ητ η = R int ζηζη = −R int τ ηζη = is the total power in the circular region with radius a that contains the source of the gravitation field seen in the exterior region. Therefore, expressing the curvature in the exterior region through the total power P0, we obtain the same result as for the infinitely thin beam. This coincides with the result from Newtonian gravity that the gravitational field outside of a spherical symmetric source distribution does not depend on the radial dependence of its density.
Appendix H: Metric perturbation for small distances between emitter and absorber In this appendix we provide the calculations for the metric perturbation for the case of a small distance between the emitter and absorber up to the second order in more detail. In the beginning we calculate the integrals we would need to calculate if the mirrors at ζ = α and ζ = β were not curved. In a next step we will include the correction for the case when they are curved. The beam is assumed to be emitted at the location of the wavefront for which ζ = α on the ζ-axis, propagate along the ζ-axis and be absorbed at the location of the wavefront for which ζ = β on the ζ-axis. The mirrors at the emission and absorption are curved such that the phase along them is constant. The phase of the Gaussian beam (without the term including the time) is given by ϕ(ρ, ζ) = θζρ 2 1 + θ 2 ζ 2 + 2 θ ζ .
For the ζ-coordinate of the mirror at the emission at ζ = α, which we callζα, setting ϕ(0, α) = ϕ ρ,ζα(ρ) , and for thē ζ β -coordinate of the mirror at the absorption, setting ϕ(0, β) = ϕ ρ,ζ β (ρ) , we obtain We start by calculating two integrals that will be useful in the following. The first one is Introducing for any quantity u a shifted quantity u by u = u − u, and changing to cylindrical coordinates (ξ , χ , ζ ) = (ρ cos(φ ), ρ sin(φ ), z ), we obtain Using the Bessel function of the first kind, J0(ix) = 1 π π 0 dφe x cos(φ) , leads to The second integral we calculate is the same as before but with a factor ζ in the nominator, In the same way as before, we obtain Every other integral we need to solve to calculate the metric perturbation can be expressed through derivatives of these integrals, using the following identities (and equivalently for I b if there is an additional factor ζ in the numerator): e −2(ξ 2 +χ 2 ) = 1 16 Including the correction of the boundaries of the integral due to the curvature of the mirrors, we obtain for the first integral Insertingζ β (ρ ) = β 1 − θ 2 2 (ξ 2 + χ 2 ) andζα(ρ ) = α 1 − θ 2 2 (ξ 2 + χ 2 ) and expanding to the second order of θ leads to where we defined Changing the coordinates as done previously and using equation (H8), we obtain where we express again the integration over the angle through the Bessel function of the first kind. Adjusting the boundaries in the second integral, we obtain Since the integral IB only appears in the second order of the metric perturbation, it is enough to expand it to the first order in θ, The metric perturbation, which is given by integrating over the retarded energy momentum tensor divided by the distance from the observer to the source point, is then found to be, expressed in terms of the integrals calculated above,