Gravitational limits on the frequency stabilityof hemispherical cavities

S. Ulbricht,1, 2, ∗ J. Dickmann,1, 2, 3, † R. A. Muller,1 S. Kroker,1, 2, 3 and A. Surzhykov1, 2, 3

1Physikalisch–Technische Bundesanstalt, D–38116 Braunschweig, Germany2Technische Universitat Braunschweig, D–38106 Braunschweig, Germany

3LENA Laboratory for Emerging Nanometrology, D–38106 Braunschweig, Germany(Dated: October 20, 2020)

We theoretically investigate the influence of gravity on the output frequency of a hemisphericalcavity, operating on Earth. The propagation of light in such a cavity is modeled by a Gaussianbeam, affected by the Earth’s gravitational field. On laboratory scale, this field is described bythe spacetime of homogeneous gravity, known as Rindler spacetime. In that spacetime, the beamis bent downwards and acquires a height dependent phase shift, while it propagates. We found,that the gravitationally induced phase shift limits the stability of the cavity’s output frequency.Detailed calculations are performed to investigate how this gravitational limit on frequency stabilitydepends on the beam parameters and the cavity setup. The results of our calculations indicate, thatgravitational effects can limit the precision of next generation laser stabilization cavities and haveto be considered in future cavity designs.

I. INTRODUCTION

Laser interferometry and laser spectroscopy are today’smost precise techniques to perform measurements at thefrontiers of metrology [1]. Due to that fact, in modernscience they are indispensable to gain a better under-standing of our world and to investigate nature. Theunprecendented precision of these techniques, would nothave been possible without major advances in laser sta-bilization technology [2, 3]. Therefore, the optical ref-erence cavities used for this purpose are fundamental tothe world’s most precise measuring instruments like op-tical atomic clocks [2, 4, 5], atom interferometers [6] andnext generation gravitational wave detectors [7, 8]. Be-ing essential to such a variety of devices, optical cavitieshelp to answer open questions of physics, like the time-variation of fundamental constants [9], the structure ofthe early universe [10, 11] and the nature of dark matter[12, 13].

The precision of cavity-based frequency measurementsis limited by a multitude of influences. The biggest ofthese influences is the stability of the optical referencecavity itself. Since the frequency of light in such a cavityis stabilized to the length of the resonator [3, 14], smalllength variations directly translate into an uncertainty ofthe output frequency. Many sources of length variations,such as seismic vibrations and temperature fluctuations,can be suppressed in modern state-of-the-art cavities. Inthis case, the stability of the cavity is limited by the fun-damental Brownian noise of the mirror coatings [3, 14].Using a cavity, that is fabricated from single-crystal sil-icon and cooled down to cryogenic temperatures, a rel-ative uncertainty of the resonator length and, thus, theoutput frequency, of 10−17 can be archived [14]. Beyond

∗ sebastian.ulbricht@ptb.de† johannes.dickmann@ptb.de

that, future experiments at lower temperatures, utiliz-ing crystalline coatings [15] or meta mirrors [16] are verypromising to enhance the frequency stability of opticalcavities by more than one order of magnitude.

With further improvements of the frequency stability,additional physical effects become relevant to a cavitysetup. Beside the well elaborated effects of quantumnoise [17] and thermo-optic noise [18], one can expect,that also the influence of gravity can not be neglectedanymore. To discuss the gravitational influences on thecavity, we have to bear in mind that in most experimentsthis device is operated in a laboratory on Earth. FromEinstein’s theory of general relativity it is known that thepropagation of light is affected by gravity in the presenceof heavy masses [19, 20]. Thus, also the light in a cav-ity is slightly deflected by the Earth’s gravitational field[21, 22]. In this work we, therefore, investigate theoret-ically, how the relative frequency stability, as an impor-tant characteristic of an optical cavity, is affected by grav-itational light deflection. Our analysis starts in Sec. II A,where we motivate Rindler spacetime as a model of theEarth’s gravitational field on laboratory scale. The co-variant Maxwell equations in this spacetime are used inSec. II B to obtain a wave equation, which accounts forthe leading order gravitational effects on light propaga-tion. In Sec. II C and Sec. II D, this wave equation isutilized to derive the vector potential for a gravitation-ally modified Gaussian beam. The obtained result is em-ployed to study the propagation and reflection of light ina hemispherical cavity, as used to stabilize the frequencyin recent laser experiments. In order to describe, howlight evolves between the plane and the spherical mirrorof such a cavity, in Sec. III A we present a method tocalculate the round trips of the gravitationally modifiedGaussian beam iteratively. The round trip calculationmethod then is used to study the phase of the beam atthe cavity output and to estimate its uncertainty, causedby gravitational effects. In Sec. III C, we discuss how thisuncertainty can be translated into a limit on the stability

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of the cavity output frequency. The gravity induced fre-quency instability in Sec. IV is discussed for a wide rangeof cavity setups. In particular, we argue that for certaincavity parameters, the gravitational effect can overcomeknown contributions to the frequency uncertainty bud-get. The summary of the results and our conclusions aregiven in Sec. V.

II. GAUSSIAN BEAMS IN A HOMOGENEOUSGRAVITATIONAL FIELD

A. The spacetime of homogeneous gravity

In the present work, we want to describe experimentswith optical cavities in a laboratory on Earth. In conse-quence, the equipment in such an experiment is affectedby the Earth’s gravitational field. On usual laboratoryscales, this gravitational field can be considered as ho-mogeneous. Additional effects, accounting for the Earthas a spherical body, can be neglected in a small regionaround the position of an observer. In the theory of gen-eral relativity, the co-moving frame of this observer isdescribed by the spacetime of homogeneous acceleration,i.e. Rindler spacetime [23, 24] with the line element

ds2 = gµνdxµdxν =(

1− gz

c2

)2d(ct)2 − dr2 , (1)

where g is the module of g = −9.81 m/s2 × ez, whichpoints into negative z-direction in the coordinate system(ct, x, y, z). Here, c is speed of light and gµν is the metrictensor with the sign-convention (1,−1,−1,−1). More-over, we use the Einstein notation, which means a sum isperformed from 0 to 3 when paired Greek letters appear.

The line element (1) describes merely flat Minkowskispacetime, but seen by an accelerated observer. Due tothe non-geodesic motion of the observer it contains a z-dependent factor (1−gz/c2)2, that accounts for the grav-itational redshift [23, 24]. Apparently, for vanishing ac-celeration the line element of Rindler spacetime reducesto the Minkowski line element. The same holds in theplane z = 0, where flat Minkowski spacetime is reachedasymptotically.

B. Light propagation in Rindler spacetime

As we know from general relativity, the properties ofspacetime not only affect the motion of matter, but alsothe propagation of light. This propagation mathemati-cally is described by the wave equation. In what follows,we will motivate the gravitational modifications of thiswave equation in Rindler spacetime from first principlesin a general relativistic framework [25, 26].

As usual in electrodynamics, the wave equation is ob-tained from the vacuum Maxwell equations, which inRindler spacetime can be written in the covariant form

∇µFµν = 0 . (2)

Those differential equations for the electromagnetic fieldstrength tensor Fµν = ∂µAν−∂νAµ are constructed frompartial derivatives of the four-potential Aµ = ( Φ/c , A ),which contains the vector and scalar potentials A andΦ, respectively. Moreover, in Eq. (2) the covariantderivative ∇µ = ∂µ + Γρµρ also carries information aboutRindler spacetime, encoded in the Christoffel symbol

Γρµρ = −g/c2(1− gz/c2

)−1δ3µ. By inserting the four-

potential in Eq. (2), the Maxwell equations in Rindlerspacetime can be written as

∇µ∇µAν = 0 , (3)

where we assumed the Lorentz gauge condition∇µAµ = 0. In Eq. (3) the equations of motion for thepotentials Φ and A decouple. Since the scalar potentialis fully determined by the Lorentz gauge condition [27],we can restrict our discussion to the wave equation forthe vector potential, which is given by

1

c2∂2

∂t2A−D2A = 0 +O(ε2) . (4)

Here D =(1− gz/c2

)∇ differs from the usual nabla

operator ∇ = (∂x, ∂y, ∂z) by the factor(1− gz/c2

), that

now accounts for the gravitational redshift and light de-flection. Moreover, we have considered effects of grav-ity only to linear order in the dimensionless parameterε = gL/c2, which is in the range of ε ∼ 10−18 . . . 10−13

for typical experiments. In these experiments the lengthscale L ranges from 1 cm to 1 km and is small in compar-ison to the Earth radius, such that the gravitational fieldcan be considered homogeneous.

In order to solve Eq. (4), we assume paraxial lightpropagation. Restricting our discussion to the paraxialregime, the polarization vector e of the vector potentialclose to the axis of light propagation can be assumed tobe coordinate independent [28]. Under this assumptionwe make the ansatz

A(t, r) = T (t)X(x)Y (y)Z(z) e , (5)

where the functions T , X, Y and Z obey the linear ordi-nary differential equations

T ′′ + ω20T = 0 , (6a)

X ′′ + k2xX = 0 , (6b)

Y ′′ + k2yY = 0 , (6c)

Z ′′ − 2γZ ′ + (k2z + δk3zz)Z = 0 . (6d)

In Eqs. (6a) - (6c), the constants ω0, kx and ky arearbitrary constants of separation, that determine k2z inEq. (6d) by

k2z = ω20/c

2 − k2x − k2y , (7)

The triplet k = (kx, ky, kz) can be seen as a general-ized wave vector and Eq. (7) as the corresponding dis-persion relation. This picture is justified in the case of

3

vanishing gravity, where k is a real valued vector. In con-trast to that, in the presence of gravity, k2z from Eq. (7)can be also negative, to account for gravitational damp-ing in z-direction. Moreover, an additional damping fac-tor γ = g/2c2 and the height dependent deviation of thewave vector δkz = (2gω2

0/c4)1/3 appears in Eq. (6d). As

we will see in the next section, the latter gives rise to theAiry-kind nature of light propagation in z-direction.

C. Solution to the wave equation up to O(ε2)

Above, we derived the wave equation for the elec-tromagnetic vector potential, that contains the influ-ence of gravity on light propagation to leading order inε = gL/c2. In what follows, we will solve this equation,using the ansatz (5).

We find, that Eqs. (6a)-(6c) are solved by plane wavesin the (x, y)-plane, while the solution to Eq. (6d) is givenby a damped Airy Ai-function. Therefore, and due to thelinearity of the wave equation (4), the vector potential Acan be expressed as a linear combination of the basisfunctions

ψkδkz (t, r) =

1

2π√δkz

eγz Ai

[−(kzδkz

)2

− δkzz

](8)

× eikxxeikyye−iω0t ,

which are defined for particular values of kx, ky and k2z .The normalization of these basis functions is chosen suchthat the integral over the full parameter space with theinfinitesimal volume dκ4 = dkxdkyd(k2z) gives

∞∫−∞

∞∫−∞

∞∫−∞

ψk∗δkz (t, r′)ψk

δkz (t, r) dkxdkyd(k2z)

= [1 + γ(z + z′)] δ(3)(r − r′) +O(ε2) (9)

= (1 + 2γz) δ(3)(r − r′) +O(ε2)

and produces a spatial delta function up to a prefactor(1 + 2γz) (cf. Ref. [29]). This prefactor, arising from thedamping terms eγz = 1 + γz+O(ε2) cancels the redshiftfactor in the coordinate invariant infinitesimal space vol-ume (1− 2γz)dxdydz, such that the spatial integral overEq. (9) is one. The relation (9) implies, that the set ofbasis functions (8) is complete and can be used to expressthe vector potential A as superposition of the ψk

δkz(t, r).

In the limit of vanishing gravitation g = 0, the set ofbasis functions {ψk

δkz→0(t, r)} becomes a plane wave ba-

sis {ei(k·r−ω0t)}, which describes light propagation in thespacetime of an inertial observer. This can be retracedto the asymptotic behavior of the Airy Ai-function [30].In this limit, moreover, the normalization integral (9)becomes an integration over the standard Fourier spacedkxdkydkz, which produces a spatial delta function.

D. Propagation of a Gaussian beam in Rindlerspacetime

In what follows, we want to discuss the findings of theprevious section for the particular case of a horizontallypropagating Gaussian beam. This beam is a suitablemodel for laser light, as it is used in many experimentaldevices, such as laser stabilization cavities [31] and grav-itational wave detectors [32]. Choosing the x-axis as thepropagation axis of the beam, we define the boundarycondition of the Gaussian beam at x = 0 by

A(t, 0, y, z) =(

1 +gz

2c2

) A0

2πb20e− z2+y2

2b20 e−iω0t , (10)

where b0 is the beam waist. Moreover, we requirethe boundary condition to share the redshift behavioreγz = (1 + gz/2c2) +O(ε2) of the basis functions (8) asthe fundamental solution to the wave equation (4). Inorder to express the boundary condition in terms of thebasis functions, we make use of the delta function andthe relation (9) to obtain

A(t, 0, y, z)

=

∫A(t, 0, y′, z′)δ(3)(r − r′)dr′

3(11)

=

∫∫A(t, 0, y′, z′)ψk∗

δkz (t, r′)ψkδkz (t, r) (1− 2γz′) dκ4dr′

3

=

∫Ab

k(t)ψkδkz (t, r) dκ4 .

Here the transformation of the vector potential at theboundary

Abk(t) = A0 exp

[−1

2(k2z + k2y)b20

]δ(kx) (12)

× 1√δkz

Ai

[−(kzδkz

)2

+1

4(δkzb0)4

].

is obtained by evaluating the spatial integral in Eq. (11),[33]. The first line of expression (12) is the Fourier trans-formation of a spatial Gaussian profile, as it is expectedin the case of vanishing gravity g = 0. The gravitationalcorrections in Eq. (12) are introduced by the Airy Ai-function in the second line.

Eqs. (11) and (12) only describe the vector potentialA(t, 0, y, z) at the boundary x = 0. However, we aim tofind the vector potential of the Gaussian beam A(r, t)in the entire coordinate space and for all times. There-fore, we recall, that the vector potential has to obey thewave equation (4) and the generalized dispersion relation(7), everywhere in space. This requirement can be fullyfulfilled by the replacement

δ(kx) → δ(kx −

√ω20/c

2 − k2y − k2z)

(13)

in the Eqs. (11) and (12). Moreover, the resulting re-placement in the transformation of the vector potential

4

Abk(t) → Ak(t) still has the proper boundary condition

(10). Since we know, that the solution to the wave equa-tion is uniquely determined by the boundary condition(e.g. [34]), we obtain the vector potential in the full co-ordinate space by

A(t, r) =

∫Ak(t)ψk

δkz (t, r) dκ4 . (14)

Having made the substitution (13) in Eqs. (11) and(12) and making use of the paraxial approxima-

tion√ω20/c

2 − k2y − k2z ≈ ω0/c − c(k2z + k2y)/2ω0 for

kx � ky, kz, we can perform the kx-integral over the deltafunction in Eq. (14) to obtain

A(t, r) = A0eγz eiω0c (x−ct)

× 1√2π

∞∫−∞

e−12k

2yB(µ)eikyy

′dky (15)

× 1√2π δkz

∞∫−∞

e−12k

2zB(µ) Ai

[−(kzδkz

)2

+1

4(δkzb0)4

]

×Ai

[−(kzδkz

)2

− δkzz′]

d(k2z) .

Here we introduced the complex quantityB(µ) = b20(1 + iµ), with µ = x/xR being the propagationdistance in units of the Rayleigh length xR = ω0 b

20/c

[35]. In order to further investigate the vector potentialA(t, r), we perform the Fourier transformation in thesecond line of Eq. (15) and employ Eq. (9) to calculatethe k2z-integral (cf. Ref. [30]). We obtain

A(t, r) =(

1 +gz

2c2

)AG(t, r)eSg(r) +O(ε2) , (16)

as a product of an unperturbed Gaussian beam

AG(t, r) =A0

2πb20(1 + µ2)exp

[− y2 + z2

2b20(1 + µ2)

](17)

× exp

[iω0x/c− iω0t+

i(y2 + z2)µ

2b20(1 + µ2)− i arctan(µ)

]and an exponential function with the complex argument

Sg(r) =gω2

0b20z

2c4(1 + µ2)

[µ2 + i(2µ+ µ3)

]. (18)

This complex exponent accounts for the leading ordereffects of a homogeneous gravitational field and resultsin two major modifications of the Gaussian beam: Onthe one hand, the real part of Sg(r) leads to a bendingdownwards of the intensity profile, so that the intensitymaximum of the beam follows the line z(x) = −gx2/2c2,see Fig.1. On the other hand, the imaginary part ofSg(r) gives rise to a z-dependent gravitational phase shiftφg(x � xR) ∼ gω0/c

3 z x, that grows while the beampropagates along the x-axis.

While in [22] we concentrated on the implicationsof the gravitational modifications (18) for the intensity

FIG. 1. Schematic picture of the Gaussian beam propagationin a hemispherical cavity in the presence of a homogeneousgravitational field (a) and without gravity (b). In the presenceof gravity, the wave fronts do not coincide with the surface ofa spherical mirror. Moreover, the intensity maximum followsthe line z(x) = −gx2/2c2 in the gravitational affected case.

profile of the Gaussian beam, in the present work wewill focus on their consequences for its phase. In or-der to do this, we formally rewrite Eq. (16) in the formA(r, t) = |A(r, t)|eiφ(r,t) e, where the phase of the grav-itationally modified Gaussian beam is given by

φ(r, t) =ω0

cx+

(y2 + z2)µ

2b20(1 + µ2)(19)

+gω2

0b20z

2c4(1 + µ2)(2µ+ µ3)− ω0t .

The gravitational effect on the phase becomes more pro-nounced for large propagation distances µ � 1. There-fore, far away from the focus, the phase (19) becomes

φ(r, t) ≈ ω0

c

(x+

(y2 + z2)

2x+

g

2c2z x− ct

)(20a)

≈ ω0

c

(√x2 + y2 + z2 +

g

2c2z x− ct

), (20b)

where in the last step we recognize, that Eq. (20a) isthe paraxial approximation of Eq. (20b). The behaviorof the phase (20b) can be illustrated in terms of phasefronts φ(r, t) = const. These phase fronts, up to theorder O(ε2) are given by the implicit equation

x2 + y2 + z2 +gL

c2z x = L2 (21)

for some L = const. As we can see from this expres-sion, in the case of vanishing gravitational accelerationg = 0, the phase fronts become spherical, as expectedfor a Gaussian beam. However, when gravity is takeninto account, the phase fronts described by Eq. (21) aredistorted and deviate from spherical shapes. Since theshape of the phase fronts has to coincide with the posi-tion of a mirror to construct a phase matched resonator,

5

the gravitational distortion can lead to a lack of preci-sion in an optical cavity. In the next section, we willconsider this effect in more detail for the particular caseof hemispherical cavities.

III. HEMISPHERICAL CAVITIES

In the previous section we developed a formalism todescribe a Gaussian beam, that is affected by a homo-geneous gravitational field. The results of this analysiscan be used to discuss a wide range of applications inEarth-based optical experiments. In what follows, weconsider the particular example of beam propagation ina hemispherical cavity. Specifically, we want to inves-tigate, how the frequency stability of such a device isaffected by gravity.

A hemispherical cavity consists of a plane mirror,which in our model is located at x = 0 and a spheri-cal mirror at distance x = L, which is referred to as thecavity length. In the absence of gravity, the propagationof light in such a cavity would be described by the un-perturbed Gaussian beam (17). Therefore, the beam isan eigen-mode of this particular cavity system, such thatthe phase fronts of the beam are transposed into them-selves by the mirrors. However, this will not hold, whenthe propagation of light is affected by the Earth’s gravi-tational field as described by Eq. (16). In this scenario,the properties of the beam slightly deviate between tworound trips in the resonator. In what follows, we will dis-cuss, how the cumulative effect of these deviations willaffect the phase properties of the cavity output signaland, hence, the frequency stability of the device.

A. Round trip calculation method

In order to consider gravitational effects on the cavityoutput signal, we first have to describe, how the beamproperties evolve with every reflection in the cavity. Forthat purpose, let us follow the beam on its first roundtrip, as displayed in Fig. 2:

After entering the cavity at x = 0, the gravitationalmodified Gaussian beam propagates towards the spher-ical mirror. As discussed in Sec. II D, on this way, itsbehavior deviates from common Gaussian beam propa-gation. That discrepancy becomes crucial, when the lightreaches the spherical mirror, which is designed to reflectthe unperturbed beam (17). In result, it is not trans-posed into itself, but reflected under a non-zero angle. Af-ter reflection, the beam again is affected by gravity and,therefore, has to obey the wave equation (4). Followingthese considerations, we can express the reflected light,utilizing the solution (16), but with a slightly changed di-rection of propagation. Thus, we assume, that the vectorpotential after reflection can be written as

A(1)(r, t) = A(0)(r1, t) , (22)

FIG. 2. First reflection of the gravitational modified Gaussianbeam in a hemispherical cavity. Due to the homogeneousgravitational field, the modified Gaussian beam is no eigen-mode of the cavity. In result the beam axis of the initialbeam are not reflected into itself by the spherical mirror. Thisdeviation of the reflected beam axis from the initial one ismodeled by the angle θ1 and the shifting parameter a1.

where A(0)(r, t) = A(r, t) denotes the initial beam,which coincides with the modified Gaussian beam (16).The change of the propagation direction

r1 = R(θ1) · r + a1ez (23)

is determined by the angle θ1 and position a1 under whichthe reflected beam hits the plane mirror at x = 0 in theend of the first round trip. Since these orientation coef-ficients are of linear order in ε = gL/c2, the propagationaxis of the beam after reflection keeps horizontal and,hence, remains perpendicular to g to the desired orderO(ε2).

The parameters θ1 and a1 can be found in an elegantway by comparing the phase fronts of the initial beamA(0)(r, t) and the reflected beam A(1)(r, t) at the posi-tion of the spherical mirror. To do that, we calculatethe wave vectors k(0) = ∇φ(r, t) and k(1) = −∇φ(r1, t),which at the spherical mirror have to obey the relation

k(1) = k(0) − 2(k(0) · n)n . (24)

Here n ∼ ∇φ(r, t)|g=0 is the normal vector of the spher-

ical mirror, which is normalized to one (cf. Ref. [28]).

In the example above, we have considered a singleround trip of light in the cavity. In order to start the nextround trip we only have to change x → −x in A(1)(r, t)due to the reflection at the plane mirror and repeat theprocedure explained above. That way, we obtain a seriesrn = R(θn) · r + anez, that characterizes the orientationof the beam A(n)(r, t) = A(rn, t) in every round trip.

6

B. Properties of the beam orientation coefficients

In the previous section, we developed a method to ob-tain the orientation of the gravitational modified Gaus-sian beam for every round trip in the hemispherical cav-ity. While the presented method can be used to analyze awide range of specific cavity setups, characterized by thebeam waist b0, cavity lengths L and wavelengths λ of theused laser light, in this section we restrict our discussionto two limiting cases. First, we consider the case of smallbeam waists b0 �

√Lλ, in which the phase properties of

the Gaussian beam resemble the properties of a spheri-cal wave. In this regime, the orientation coefficients aregiven by

an = −gL2

2c2

[1 + (−1)n+1 cos

(4πb20Lλ

n

)](25a)

θn =gL

c2(−1)n+1 Lλ

4πb20sin

(4πb20Lλ

n

), (25b)

as displayed in Fig. 3 for a beam with b0 = 50µm andλ = 1064 nm in a L = 21 cm cavity. As seen from thefigure and Eqs. (25a) and (25b), the evolution of bothparameters is described by an alternating series, whichis modulated by a periodic envelope with the frequencyπb0/Lλ. The parameters an and θn oscillate around the

10 20 30 40 50 60

-4

-2

0

n

an/10-18m

-1

0

1

10 20 30 40 50 60

θn/10-17

FIG. 3. Oscillations of the beam orientation coefficients anand θn in the parameter regime b0 �

√Lλ for a 21 cm cavity

with a beam waist of b0 = 50µm for light with λ = 1064 nm.Due to the strong focusing effect of the spherical mirror, theseries alternates with (−1)n.

mean values 〈θ〉 = 0 and 〈a〉 = −gL2/2c2, respectively.The periodicity of the parameters can be explained dueto the interplay of two effects: Since the beam is bentdownwards by gravity, it tends to leave its initial axisof propagation. However, the further the beam removesfrom its initial propagation axis, the more it is reflectedback into the direction of this axis by the focusing ef-fect of the spherical mirror. In result, for small, butfinite beam waists b0, the hemispherical cavity acts asa stable resonator. However, that is not the case fora vanishing beam waist b0 → 0, where the orientation

coefficients become an(b0 → 0) = gL2

2c2 {(−1)n − 1} and

θn(b0 → 0) = gLc2 (−1)n+1 n. As the second limiting case,

we consider a gravitational modified Gaussian beam witha large beam waist b0 �

√Lλ, where the orientation co-

efficients with every round trip are given by

an =gL2

c2

(2πb20Lλ

)2 [cos

(Lλ

πb20n

)− 1

](26a)

θn =gL

c22πb20Lλ

sin

(Lλ

πb20n

). (26b)

In Fig. 4 this is shown for a beam with b0 = 900µm andλ = 1064 nm in a L = 21 cm cavity. From the figure andthe Eqs. (26a) and (26b) it can be seen, that both pa-

10 20 30 40 50 60 70

-4

-2

0

n

an/10-15m

-4

-2

0

2

4

10 20 30 40 50 60 70

θn/10-16

FIG. 4. Oscillations of the beam orientation coefficients anand θn in the parameter regime b0 �

√Lλ for a 21 cm cavity

with a beam waist of b0 = 900µm for light with λ = 1064nm. The shift an initially follows the trajectory of free fall− g

2(2Ln)2 (dashed), before the weak focusing effect of the

spherical mirror leads the beam back to the initial axis ofpropagation.

7

rameters again oscillate around their mean values, which

now read 〈θ〉 = 0 and 〈a〉 = −gL2/c2 ×(2πb20/Lλ

)2, re-

spectively. The frequency of these oscillations is givenby Lλ/πb20, what, in contrast to the case of small b0, be-comes smaller with increasing beam waists. Therefore,also in the regime of large, but finite b0 the cavity actsas a stable resonator. However, the cavity becomes un-stable in the ultimate case b0 → ∞, when the orienta-

tion coefficients reduce to an(b0 → ∞) = − g2(2Ln)2

c2 and

θn(b0 →∞) = 2 gLc2 n. In that limit, the phase propertiesof the Gaussian beam approach those of a plane wave.Hence, the resonator becomes a Fabry-Perot cavity witha plane mirror at x = L. This mirror has no focusingeffect and does not prevent the beam from falling down.Therefore, after n round trips and a propagation distanceof 2Ln in the cavity, the beam finds itself at a heightan = − g

2c2 (2nL)2, as expected for a light ray, bent byEarth’s gravity in the limit of geometrical optics [21, 25].

C. Phase and frequency stability of hemisphericalcavities

Previously, we laid down a theory to describe the prop-agation of light in a hemispherical cavity, that is affectedby the Earth’s gravitational field. Below, we apply thistheory to cavities with the particular resonator lengthsof L = 21 cm, 30 cm and 50 cm, which currently repre-sent the world’s most stable optical frequency references[3, 36–38] and, therefore, are indispensable for the im-provement of high precision instruments, such as opticalatomic clocks [2, 4, 5] and gravitational wave detectors[7, 8]. In what follows, we will focus on the gravitationaleffects on the output frequency of these devices. Onecan expect that the stability of the output frequencyis affected by the gravitational distortion of the lightpropagation, discussed in Sec. II. Due to this distortion,the beam hits the plane mirror at the cavity output atslightly different positions in each round trip. Hence, alsothe phase of the vector potential A(n)(r, t) slightly varieswith every reflection in the cavity. Therefore, the super-position of the A(n)(r, t) at the output mirror allows usto analyze the accumulated effect of gravity on the phasestability. We can calculate that effect by using the the-ory, developed in Sec. III A and the summation of thecontributions from all round trips [39], which are charac-terized by the orientation coefficients an and θn. Since inreal experiments the mirrors have a finite transmittanceT , a fraction of light leaves the cavity with every reflec-tion and only an effective number of round trips N hasto be considered. This number is in the range of the cav-ity finesse F ≈ π/T 2 [40], which is above 105 for recentexperiments [3, 14, 41].

Having discussed qualitatively, how the gravitationaldistortion of the light propagation affects the phase sta-bility at the cavity output, we are ready to perform thisanalysis quantitatively. We start with Eq. (19), whichdirectly relates the trajectory of the beam and its phase.

50 100 200 500

b0

μm

0.5

1

5

10

δθ2 / 10-16

FIG. 5. Averaged deviation of the angle√〈δθ2〉 for light with

a wave length of λ = 1064 nm as a function of the beam waistb0 for 21 cm cavities (solid), 30 cm cavities (dashed) and 50cm cavities (dot-dashed).

From that expression we can define the phase variationat the output mirror

δφ(r) =

√⟨(φ(r, t)− φ(rn, t))

2⟩∣∣∣∣∣x=0

(27)

as the averaged difference between the phase of the initialGaussian beam and the phase of the beam on every roundtrip. Since the beams orientation rn only slightly differsfrom r, we can formally rewrite rn = r + δrn + O(ε2),where the structure of δrn = ( θnz , 0 , an− θnx ) followsfrom Eq. (23). That way, we can expand Eq. (27) aroundr, to obtain

δφ(r) =

√⟨(∇φ(r, t) · δrn)

2⟩∣∣∣∣∣x=0

. (28)

Due to the fact, that rn already is of linear order inε = gL/c2, we can evaluate the phase gradient of theunperturbed Gaussian beam (17) at x = 0. At this po-sition, the phase fronts of the beam are those of planewaves, such that ∇φ(r, t)|x=0 = (ω0/c , 0 , 0 ) + O(ε).This simplifies Eq. (28), which now reads

δφ(r) =zω0

c

√〈δθ2〉+O(ε2) . (29)

As seen from the above expression, δφ(r) becomes zero inthe case of vanishing gravity, since the phase of the beamremains unchanged with every round trip. In contrast,the presence of the gravitational field leads to a non-zero phase variation, that is proportional to the light fre-quency ω0 and the variation of the tilting angle

√〈δθ2〉,

which can be calculated as the root mean square value ofthe series θn:

√〈δθ2〉 =

√√√√ 1

N

N∑n=1

(θn)2 . (30)

8

In this formula, we took into account, that the periodsof oscillations in θn are much smaller, than the effectivenumber of round trips N ∼ F , such that no relativeweighting of the contributions from every round trip hasto be considered. Eq. (30) was used to calculate

√〈δθ2〉

for light with a wave length of λ = 1064 nm as a functionof b0 for cavities with a resonator length of L = 21 cm,30 cm and 50 cm, see Fig. 5. As can be seen there, theaveraged deviation of the angle increases with L and israther sensitive to the beam waist b0.

Eq. (29) describes the phase variations at some par-ticular point at the cavity output mirror. In order tocalculate the variation of the phase for the entire outputsignal, we have to weight it with the intensity distribu-tion of the Gaussian beam at the position x = 0 of themirror and obtain

√〈δφ2〉 =

√1

πb20

∫e− z2+y2

b20 δφ(r)2 dy dz

=b0ω0

2c

√〈δθ2〉. (31)

The resulting averaged phase instability of the cavity out-put signal directly translates into a relative frequency in-stability via the relation

δν/ν =λ

2πL

√〈δφ2〉 =

b02L

√〈δθ2〉 (32)

(c.f. [40]). Making use of this relation and Eq. (30), weobtain the gravitational limit on the frequency stabilityof a hemispherical cavity operating on Earth. In whatfollows, we will discuss the implications of this limit forrecent and future cavity designs.

IV. DISCUSSION

Above, we developed a theory to analyze the propaga-tion of a Gaussian beam in a hemispherical laser cavity,affected by the Earth’s gravitational field. This theorywas used to investigate, how gravity influences the prop-agation direction of the beam on every round trip and,hence, produces phase deviations at the cavity output.The resulting uncertainty of the phase, via Eq. (32) isrelated to the relative frequency instability, which is oneof the most important characteristics of a cavity device.Applying the methods, presented in Sec. III, for a widerange of cavity settings, we find, that the contributions tothe frequency instability, caused by the Earth’s gravity,are given by

δν/ν =g√2c2

(Lλ

8πb0+πb30Lλ

), (33)

depending on the characteristic lengths scales L, λ andb0 of the cavity system. In order to illustrate the de-pendence of δν/ν on theses parameters, in Fig. 6 the fre-quency instability is displayed for light with λ = 1064 nm

50 100 200 500

b0

μm0.05

0.10

0.20

0.50δν/ν / 10-19 m

FIG. 6. Relative frequency stability δν/ν of a hemisphericalcavity for light with λ = 1064 nm, depending on the beamwaist b0, for 21 cm cavities (solid), 30 cm cavities (dashed)and 50 cm cavities (dotted). In the logarithmic axis scales ofthis plot all three curves have the same shape. They are onlyshifted along the dashed straight line of optimum beam waist(dot-dashed).

and cavity lengths of 21 cm, 30 cm and 50 cm. As seenfrom that figure, the gravitational limit on frequency sta-bility strongly depends on the beams waist. For smallb0 < 100µm, for example, δν/ν is proportional to 1/b0.In contrast, for large values of b0 > 300µm, it growswith b30. This behavior can be traced back to the b0-dependence of the orientation coefficients (25b) and (26b)and the definitions (30) and (32).

The increase of the relative frequency instability forsmall and large beam waists, implies the existence of aminimum in the intermediate region. The position ofthat minimum

bgrav0 =

√Lλ

(24π2)1/4, (34)

can be obtained from the analysis of Eq. (33). At thisvalue, the best achievable frequency stability with respectto gravitational perturbations is(

δν

ν

)min

=g

c2

√Lλ

(216π2)1/4=

g√3c2

bgrav0 . (35)

As seen from this expression, (δν/ν)min linearly dependson the optimum beam waist. In order to visualize thisdependence, it is shown as a straight line in Fig. 6 by tak-ing b0 = bgrav0 . For the sake of completeness, we presentthe values for bgrav0 and (δν/ν)min for L = 20 cm, 30cm and 50 cm, in Tab. I. As can be seen there and inFig. 6, the optimum beam waist ranges between 100 µmand 300 µm, which fits very well to present experimentalconfigurations [3, 14, 37].

So far, we discussed the gravity-induced frequency in-stability of a hemispherical cavity. However, it lends itselfto compare these gravitational contributions to the fre-quency uncertainty budget with contributions of other

9

well elaborated effects. In particular, state-of-the-artlaser stabilization cavities are limited by the fundamen-tal Brownian noise of the mirror coatings [3, 14]. For ahypothetical 21 cm sapphire cavity formed by two siliconon sapphire meta-etalon mirrors at 4 K temperature, thefundamental Brownian noise limits the frequency stabil-ity as (δν/ν)Brownian = 2.41×10−18 (b0/µm)−1/2 [16, 40].In Fig. 7 it can be seen, that this frequency limit scales

with b−1/20 and, therefore, has a qualitatively different

behavior comparing to Eq. (33). Owing to that, in oursample case, the gravitational limit on frequency stabilitycan be even more dominant for beam waists grater than450µm. Around that value, a combined limit on the rel-ative frequency stability would be in the range of 10−19,which is aimed to be achieved in future laser cavities[13, 37, 42]. A common way to improve the frequency sta-bility of these devices, is to increase the resonator lengthof the cavity. For larger cavities, larger beam waists areexperimentally realizable. As seen from our studies, how-ever, the approach of increasing L and b0 has its limits,due to gravitational effects. The fact that the gravita-tional limit on frequency stability grows rapidly withincreasing beam waist can be seen as a game changerfor future designs of highly stable laser cavities. There-fore, future cavity designs, which are supposed to reacha frequency stability in the range of 10−20 are obligedto consider the influences of gravity on the experimentalsetting.

V. SUMMARY AND CONCLUSION

In this article, we present a theoretical framework todescribe the propagation of light in the presence of a ho-mogeneous gravitational field. In particular, we derivedthe wave equation for the vector potential of the elec-tromagnetic field in Rindler spacetime, that accounts forleading order gravitational corrections. The wave equa-tion is used to obtain a gravitationally modified Gaus-sian beam, that can be utilized to model the propagationof light in Earth-based laser experiments. As a specificsetup of such an experiment, a hemispherical cavity isconsidered in detail. Our theory is applied to describe the

TABLE I. Optimum beam waist bgrav0 and corresponding bestachievable relative frequency stability (δν/ν)min for the threedifferent cavity lengths 21 cm, 30 cm and 50 cm operated ata laser wavelength of λ = 1064 nm.

L (cm) bgrav0 (µm) (δν/ν)min

21 120 7.58× 10−21

30 144 6.06× 10−21

50 185 1.17× 10−20

50 100 200 500 1000

b0

μm

0.1

0.5

1

5

10

50δν/ν / 10-19 m

FIG. 7. Relative frequency stability δν/ν caused by Earth’sgravity (solid) compared to the relative frequency stabilitylimit induced by Brownian noise (δν/ν)Brownian (dashed) as afunction of b0 for a hypothetical sapphire 21 cm cavity formedby two silicon on sapphire meta-etalon mirrors at 4 K temper-ature and a laser wavelength of λ = 1064 nm.

round trips of light in the cavity and their gravitationalperturbations in such a device. In particular we found,that these perturbations lead to a frequency instability atthe cavity output. Detailed calculations where performedto analyze the gravitational limit on frequency stabilityfor a wide range of cavity settings, characterized by thebeam waists b0, the cavity lengths L and wavelengths λ.Special attention was paid to the effect on the currentlymost stable cavities with typical resonator lengths of L =21 cm, 30 cm and 50 cm, which are used to stabilize highprecision instruments like optical atomic clocks and grav-itational wave detectors. Based on our calculations, weconclude, that future laser cavity designs can not gen-erally neglect the contributions of gravitational effects,when frequency stabilities in the range of 10−20 are pur-sued. Moreover, we emphasize, that the state-of-the-artapproach of increasing the cavity length and the beamwaist in order to reduce the limiting Brownian noise goesalong with an increasing gravitational contribution to thefrequency uncertainty budget. This underpins our con-clusion, that future cavity concepts need to be optimizedby taking gravitational effects into account as well.

ACKNOWLEDGMENTS

The authors would like to thank Marcel Reginattofor helpful discussions. We acknowledge the supportby the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) under Germanys Excellence Strat-egy - EXC-2123 QuantumFrontiers - 390837967.

10

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