1 Introduction
Long-baseline interferometry can be carried out in tabletop experiments using optical fibers, thus enabling the measurement of small perturbations in the dispersion relation of light (even at the single-photon level). Such phase shifts arising from the Sagnac effect have already been measured1,2, and an experiment is being built to measure such phase shifts arising from vertical displacements of one arm of the interferometer in the gravitational field of the Earth3,4. Compared to experiments involving light propagation in vacuum, such experiments face the challenge of minimizing, controlling, and compensating for noise arising inside the fiber, or arising from the response of fibers to changes, drifts, and fluctuations of ambient conditions5. In particular, for experiments that modulate their signal by moving one or both interferometer arms (as, e.g., in Refs. 3,4), local variations in pressure, temperature, and local gravity lead to elastic deformations of the fibers that influence light propagation through photoelastic effects. These effects need to be modeled accurately in order to extract the effect which one aims to measure. The aim of this work is to carry this out, with focus on the effects relevant for the planned GRAVITES experiment3.
A simplified schematic diagram of this last experiment is shown in Figure 1.1. The interferometer aims to measure phase shifts induced by Earth’s gravity on entangled multi-photon states propagating in optical fibers. To modulate this effect, one interferometer arm is gradually lifted by 1 m relative to a stationary reference arm. If the fiber length is held constant in the process, the gravitational redshift implies a difference in phase for the two interferometer arms, which results in a height-dependent change in detection statistics of individual photons at the two output ports of the final beam-splitter. Now, the gravitational field in the laboratory changes with height. Further, in general, the pressures and temperatures at the two interferometer arms are not equal. The differences in gravitational acceleration, ambient pressure, and ambient temperature, will be denoted by δg, δ𝖕, and δT. We wish to determine here the changes of the Maxwell field in a cylindrical waveguide as functions of these variables.
Figure 1.1. Schematic representation of the GRAVITES experiment3.
Light propagates through optical fibers in two vertically separated arms of an interferometer and traverses spools with a total fiber length of the order of 100km, accumulating different phases due to the gravitational redshift. We consider a simplified setup as in previous work6, where the spools are unwound and treated as straight fiber sections. We allow for a difference in gravitational acceleration δg, ambient pressure δ𝖕 and ambient temperature δT at the spool locations, leading to elastic effects on the photon propagation. This figure has been reproduced with permission6.
As such, the elastic response of (straight) optical fibers to variations in gravity, pressure, andtemperature, was accurately modeled in a previous work6. We build upon these results by solving Maxwell’s equations in so-deformed fibers to determine the effect of variations of ambient parameters on light propagation in single-mode fibers. This is done using the perturbative scheme developed in Refs. 7–9.
This paper is organized as follows. In Section 2 we review the formulation of Maxwell’s equations in linear dielectrics in terms of gauge-fixed wave equations for the electromagnetic potential. These equations can be solved explicitly for the case of undeformed and unstrained step-index fibers, yielding the standard mode solutions. Following the approach of Refs. 7,8 and restricting to single-mode fibers, we extend the calculations there to include generic perturbation terms, and derive the general solution to first order in perturbation theory. This result is then used in the following sections to describe the phase shift and birefringence induced in the fiber by elastic deformations of any origin.
In Section 3 we review the elastic deformations of optical fibers derived in Ref. 6, and derive the induced correction terms in the electromagnetic field equations describing light propagation in such fibers. These corrections arise directly from the geometric fiber deformation (that alters the shape of the core-cladding interface), and indirectly through the induced anisotropy (arising from the strain due to photoelasticity). To first order in perturbation theory these effects add and can thus be computed separately: In Section 3.1 we describe the purely geometric deformation effects, and in Section 3.2 we discuss the photoelastic effects. The resulting expressions for the phase shifts and birefringence contain integrals that need to be evaluated numerically.
Figure 1.2 and Figure 1.3 summarize the results of our analysis and can be viewed as the main result of this paper.
Figure 1.2.
Varying the wavelength (left column) or the fiber core radius (right column) changes the birefringence induced by the elastically deformed core-cladding interface (1.2a and 1.2b) and the birefringence caused by the photoelastic material response (1.2c and 1.2d). Fiber 1 describes a fiber with parameters as listed in Table 3.1–Table 3.2, and Fiber 2 has refractive indices n1 = 1.4715 and n2 = 1.4648 instead, with the fibers identical otherwise. The parameter b2 quantifies the strain in the waveguide as defined in (3.11). In Figure 1.2c, and in the physically relevant region of Figure 1.2d, both fiber’s responses are identical for all practical purposes.
Figure 1.3.
Figure 1.3a and Figure 1.3b show the dependence of the phase shift on the elastic deformation of the core-cladding interface as functions of the optical wavelength and the fiber’s core radius. Figure 1.3c and Figure 1.3d, on the other hand, show the dependence of the photoelastically induced phase shift on these parameters. The material parameters are the same as in Figure 1.2. In Figure 1.3c, and in the physically relevant region of Figure 1.3d, the responses of both fibers are identical for all practical purposes.
Table 3.1. Properties of silica oxide glass from Ref. 17.
| Poisson ratio ν | 0.17 |
| Young modulus E | 73.1 GPa |
| Mass density ρ | 2.2 g cm–3 |
| Thermal expansion coefficient α | 5.5 × 10–7 K–1 |
Table 3.2. Parameters for the GRAVITES experiment3.
| Fiber outer radius a | 62.5 µm |
| Fiber core radius ρ | 4.1 µm |
| Core refractive index n1 | 1.4712 |
| Cladding refractive index n2 | 1.4659 |
| Variation in grav. acceleration, δg | 3 µm s–2 |
| Variation in pressure, δ𝖕 | 10 Pa |
| Variation in temperature, δT | 10–2 K |
| Fiber length L | 105 m |
| Propagation constant β | 6 × 106 m–1 |
| Wavelength λ | 1.55 µm |
In Section 3.3 we provide numerical results with parameters relevant for the GRAVITES experiment.
Our calculations are presented in units in which the vacuum permeability µ0, the vacuum permittivity ϵ0, and hence also the speed of light in vacuum c, are set to unity. Note that the final formulæ do not involve these constants. Coordinate indices are denoted by µ, ν, . . ., and range from 0 to 3, while the frame indices are denoted by a, b, . . ., and range over the set {t, z, ♯, ♭}, compare Equation (2.10) below. Implicit summation over repeated indices is employed throughout.
2 Perturbative fiber optics
Let us start by introducing the formalism used to describe electromagnetic modes in deformed single-mode fibers. To solve Maxwell’s equations in such fibers it is useful to formulate the equations in curvilinear coordinates adapted to the deformation of the fiber. For this in Section 2.1 we recall a generally covariant formulation of Maxwell’s equations in linear dielectrics. As these equations are generally not solvable explicitly, perturbation techniques are required. In our case, the unperturbed reference problem is that of an undeformed fiber, the modes of which are described in Section 2.2. Based on this, in Section 2.3 we present an approximation scheme that allows to compute perturbations of such modes. This scheme is general in that it applies to arbitrary small deviations from unperturbed field equations, regardless of their origin, as long as the perturbations are invariant under translations along the fiber. The results in this section will be used in Section 3 to determine the effects of internal deformations and of stress-induced anisotropy.
2.1 Maxwell’s equations in linear media
Maxwell’s equations without sources can be written in generally covariant form as
where Fµν and Gµν are antisymmetric tensors describing the electromagnetic field strength and excitation, where ∇µ is the covariant derivative associated to the space-time metric tensor gµν, and where square brackets indicate antisymmetrization (cf. e.g. 10). To obtain a closed system it is necessary to formulate a constitutive law that relates Gµν and Fµν. For linear and non-dispersive media, the standard form of this law takes the form
with the constitutive tensor χµνρσ exhibiting the symmetries
For isotropic dielectrics with four-velocity uµ, permittivity ϵ, and permeability µ, it can be shown11 that the constitutive tensor takes the form
where γµν is Gordon’s optical metric
in which n = is the refractive index.
Optical fibers are often modeled as linear isotropic dielectrics with µ = 112. However, to describe photoelastic effects in fibers, we will allow more general constitutive tensors that reduce to with µ = 1 in the absence of deformations and stresses.
The following calculations are carried out in terms of an electromagnetic potential Aµ such that
In this formulation Equation (2.1a) is identically satisfied, and Equation (2.1b) does not directly provide well-posed evolution equations due to the gauge redundancy Aµ → Aµ + ∂µλ. Following Refs. 7,8,13 we therefore consider the gauge-fixed Lagrangian
where γµν is the Gordon metric with µ = 1 and ϵ = n2 being the permittivity of the undeformed reference configuration, without deformations and stresses. The corresponding Euler–Lagrange equations
reduce to Equation (2.1b) whenever χgauge vanishes, and are well-posed irrespective of this gauge condition7.
In everything that follows the metric is taken to be the Minkowski metric, which we denote by ηµν, and the four-velocity vector is constant throughout the medium. Hence, in the case of an idealized step-index fiber, where = with µ = 1 and ϵ = n2 being locally constant (n takes different values in the fiber’s core and cladding), in Minkowski spacetime and in inertial coordinates Equation (2.8) reduces to
2.2 Solutions for ideal optical fibers
In this section we review the mode solutions to Equation (2.9) for straight step-index fibers with circular cross-sections. These serve as unperturbed reference solutions to the perturbative scheme described in Section 2.3. Whereas the description of such modes at the level of the field strength is described in textbooks12, the following calculation is carried out using the electromagnetic potential following Ref. 13. To distinguish the reference solution derived here from the actual solution for deformed fibers, the solutions to the unperturbed problem will be marked by a ring diacritic .
The unperturbed optical fiber is modelled as a cylindrical dielectric fiber with a core of radius ρ and refractive index n1 that is surrounded by an annular dielectric cladding of outer radius a and refractive index n2< n1. While a is finite for the purpose of the elasticity calculations, because of the exponential decay of the electromagnetic field in the cladding we use the approximation a = ∞ when solving Maxwell’s equations.
An efficient approach to this problem proceeds by solving Equation (2.9) in regions of constant refractive index independently and applying matching conditions at the interface. Due to the symmetry of the problem, the calculations to follow are carried out using cylindrical coordinates (t, r, θ, z). Following Ref. 13, mode solutions are obtained by decomposing the potential Aµ using the complex frame
and using the ansatz
so that the wave equations for the frame components åb decouple. Here, ω is the angular frequency of the mode, β its propagation constant, and m is the azimuthal mode index. Writing
the system (2.9) is equivalent to
where Hm is the Helmholtz operator
The requirements of regularity at the origin r = 0 and of exponential fall-off as r → ∞ imply that solutions of (2.13) can be expressed in terms of the functions
where c1 and c2 are arbitrary constants, Jm are Bessel functions of first kind, Km are modified Bessel functions of second kind, and the parameters U and W are defined as
Using this notation, the relevant solutions to Equation (2.13) take the form
with
denoting a set of constants that needs to be determined from the matching conditions at the core-cladding interface located at r = ρ. Specifically, we require continuity of the following field components, as discussed in Section II.D. of Ref. 13
in which ⟦f⟧ = (fcore – fcladding)|r=ρ denotes the jump of a function at the interface and nµ denotes any normal to that interface (due to linearity neither the sign nor the normalization are relevant). For the ansatz (2.11) these conditions can be written concisely as = 0, where is the eight-component vector
where we kept some of the continuous terms as this simplifies the algebra later.
For solutions of the form (2.17), the interface conditions can be written in terms of matrices acting on the vector :
where
with
and
For conciseness, we suppress here the arguments of the Bessel functions, which are evaluated at the interface and read Jm(Uρ) and Km(Wρ), respectively.
Non-trivial solutions to = 0 exist if and only if det M = 0. One can show that this determinant factorizes into a term whose roots yield physical solutions and a term whose roots correspond to gauge and ghost solutions that have either vanishing field strength Fµν or violate the gauge condition χgauge = 013. The equation leading to physical solutions reads
where individual roots of this transcendental equation yield propagating modes. Defining the normalized frequency
and the normalized guide index b implicitly by
The numerical solution to Equation (2.25) with n1 = 1.4712 and n2 = 1.4659 is presented in Figure 2.1, compare Ref. 12. Note in particular the shaded area in this diagram, where, for a given normalized frequency, there exist only roots for (2.25) with mode numbers m = ±1 (the curves are indifferent to the sign of m). Typical applications of single-mode fibers take place in this regime.
Figure 2.1. Mode diagram for an optical fiber with refractive indices n1 = 1.4712 and n2 = 1.4659.
The parameter κ enumerates distinct roots for a fixed azimuthal mode number m. Note that in the shaded region only a single propagating mode exists for a given normalized frequency. This region corresponds to single mode optical fibers (SMF).
For a given simple root of the determinant, the kernel spans a one-dimensional subspace, leading to solutions being determined up to an overall constant. This becomes important when one tries to define the polarization state of a superposition of simultaneously propagating solutions, as the solution vectors spanning the m = ±1-subspaces can be scaled independently8. In the case of single-mode fibers the m = ±1 solutions share the same root, and inspection of (2.22) shows that a change of sign for the mode number m corresponds to some sign changes and re-ordering of the entries, namely
up to a multiplicative factor that we chose to be equal to one.
2.3 Multiple-scales analysis
Basing on the above description of light propagation in ideal optical fibers, in this section we derive a general formula that describes the polarization dynamics in deformed optical fibers that are invariant under translations along the fiber. This result is applied in Section 3 to compute elastically induced phase and polarization dynamics resulting from stress-induced optical inhomogeneity and anisotropy, as well as from deformations of the core-cladding interface. Denoting by ε ≪ 1 a dimensionless expansion parameter associated to such perturbations, corrections to the field Equations (2.8) can generally be written as
where Σ is an operator that describes all first-order perturbations of the wave equation that arise from elasticity, and we neglect second order ε-terms throughout. For our purposes there is no loss of generality in working with coordinates such that the core-cladding interface lies at r = ρ, as perturbed fiber geometries can be transformed to this setting by a change of coordinates (see Section 3.1 for details). This method facilitates the implementation of junction conditions at the core-cladding interface.
Restricting to the single-mode fiber solutions of Section 2.2, the O(ε0)-modes have azimuthal mode number m = ±1. The remaining modes are a priori O(ε). As shown in Section III.C. of Ref. 8, the propagation of these leading modes in (2.29) is governed by those Σ-terms that are isotropic (i.e., independent of θ) and those having a relative azimuthal mode number of ∆m = ±2. Higher-order perturbation theory would allow a wider range of relative azimuthal mode numbers, since multiple combinations can again result in contributions to these leading order modes, see Refs. 8,9 for such calculations.
Consider the unperturbed single-mode solution consisting of a superposition of m = ±1 modes of the form (2.11):
where we factored-out an overall amplitude 𝒜, so that the coefficients 𝒥± are normalized to |𝒥+|2 + |𝒥–|2 = 1. As shown in Ref. 13, the coefficients 𝒥± can be interpreted as projections of a complex Jones vector 𝒥 = 𝒥xex + 𝒥yey, describing the polarization of the solution, onto the complex basis . This translates into , and hence
so that 𝒥+ and 𝒥− describe, respectively, the contributions of right-handed and left-handed circular polarizations. While the fields 𝒜 and 𝒥m are constant in the unperturbed case, they satisfy propagation equations along the fiber in the perturbed case, and in fact the main aim of this work is to derive these equations in the current setting.
Following Ref. 9 we describe gradual changes in the electromagnetic field along the fiber using the multiple-scales method. This proceeds by introducing an additional length variable ζ = εz. Using z and ζ simultaneously allows separating short-distance dynamics, involving distances of the order of the optical wavelength, from that on larger length scales which are determined by the deformations of the fiber. We thus consider the following ansatz for the perturbed wave equation
where tilde diacritics denote first-order perturbation terms. Note that the expansion above is to be understood in terms of the multiple-scales method, and not as a Taylor series. We will thus not approximate functions of ζ by f(ζ) = f(0) + εzf′(0) + O(ε2), see Chapter 11 of 14 for a textbook exposition.
The wave equation splits then into terms proportional to ei(±1±∆m)θ, where ∆m is the relative azimuthal mode number induced by the perturbations. To describe the evolution of the polarization along the fiber it suffices to consider the terms involving e±iθ, leading to a coupled set of two equations
where are source terms introduced by the perturbation. The current formalism allows for arbitrary such explicit expressions for concrete models will be considered in Section 3 below.
According to the multiple-scales method, both the leading-order terms and the ε-terms in (2.33) must vanish. It follows from the analysis of the unperturbed problem that the zero-order solutions take the form
which generalize the unperturbed solutions in (2.17) by allowing for a Jones vector 𝒥 that depends on ζ. Its components 𝒥±(ζ) will be determined by the equations, as follows: The ε-part of (2.33) is a Helmholtz equation for with known source terms
whose solutions are given by
where 𝒢± = 𝒢±1 is a multi-component operator9
in which Gm(f) is the Green’s operator for the Helmholtz equation
Here the Ym’s are Bessel functions of the second kind, and Im’s are modified Bessel functions of the first kind.
Equation (2.36) solves the field equations in the core and cladding separately. To obtain a solution that is valid throughout the waveguide the field must satisfy the interface conditions given in Equation (2.19). In the perturbed case these can now contain functions of order ε which depend upon the radial and angular coordinates and the unperturbed solutions . Following the reasoning leading to (2.33) we again restrict to isotropic and quadrupole perturbations, i.e., angular dependence ∆m = {0, ±2}, yielding the perturbed interface conditions
Here Γ± is given in Equation (2.20), while are the correction terms specific to the perturbations under consideration and will be determined in Section 3.
These can be written using (2.36) as
where M± = M±1 and N± = N±1 are given by Equation (2.22) and Equation (2.24). Since M± is singular due to the dispersion relation, the vectors given by the right-hand side of Equation (2.40) have to lie in the image of M±. An explicit calculation shows that the cokernel of M± is one-dimensional. Let us denote by ξ± some basis vectors of the cokernel of M±:
Then Equation (2.40) admits solutions if and only if
Since and depend on the coefficients 𝒥+ and 𝒥–, respectively, this equation provides the desired propagation law for the Jones vector. Indeed, inserting (2.34) one obtains
where
where and . The Cartesian components of the Jones vector thus satisfy
where can be decomposed in terms of the four Pauli matrices as
In all cases below one has and , so that takes to the form
In this case, the diagonal components of ℳ (which arise from perturbations with ∆m = 0) thus describe phase shifts, as they produce the same effect on 𝒥x and 𝒥y, while off-diagonal components of ℳ (corresponding to ∆m = ±2) describe birefringence, producing opposite phase shifts on 𝒥x and 𝒥y.
3 Elasticity
To continue, we wish to determine the source terms Σ(±,0) and Σ(±,2) coming from elastic perturbations of the fiber, after which (2.44) can be evaluated numerically.
For this we consider elastic deformations of the optical fibers resulting from ambient pressure, temperature, and gravity acting as a body force orthogonal to the principal axis of the fiber. Considering a circular optical fiber as described in Section 2.2, we take the elastic parameters of the core and cladding to be identical and constant throughout the medium, resulting in a simple model of a homogeneous and isotropic cylinder of length L and radius a.
In the following, we restrict the analysis to linear elasticity and introduce the main equations, referring for a more detailed treatment to Ref. 6. The displacement field ui describes small deformations of the material and determines the strain tensor uij via
The strain is related to the material stress σij via the generalized Hooke law for isotropic media15,16, as extended to account for changes of temperature T – T0. Assuming the material to be thermally-linear and thermally-isotropic one has16
where λ and µ are Lamé’s first and second parameters of the material, and where α is the coefficient of thermal expansion, all taken to be constant throughout the waveguide. The second Lamé parameter µ is also referred to as the shear modulus, ν = λ/[2(λ + µ)] is known as the Poisson ratio, and the Young modulus is given by E = 2µ(1 + ν). Material parameters for fused silica, which is widely used in optical fibers, are listed in Table 3.1.
Static configurations are described by solutions to the equilibrium equations16
with Fi = −∂iV denoting the gravitational body force, where V = gρy is the gravitational potential, with ρ being the mass density of the material and g the gravitational acceleration of Earth. At the object’s surface, one has the traction boundary conditions15
where nj is the outward-pointing unit normal vector of the material surface and Pi is the external pressure.
Taking into account the translational symmetry along the fiber, as well as the fact that the radius a of the fiber is significantly smaller than its length L, a ≪ L, the calculations can be reduced to those on two-dimensional cross-sections orthogonal to the main axis. The textbook approach does so by setting the out-of-plane strain components to zero [16, Chapter 7] leading to algebraic expressions for the out-of-plane stress and two-dimensional equilibrium Equations (3.3). This can be generalized by allowing for a linear displacement along the fiber (cf. 18)
where κ is a constant, implying
The equilibrium equations can now be solved by introducing a stress function ϕ, related to the stress tensor in cylindrical frame components via
The general solution to these equations has been written down by Michell19. Assuming regularity, vertical reflection symmetry, and absence of friction at the fiber’s boundary one is led to6
where the angle θ used here has been shifted by π/2 relative to that in 6 to measure the usual angle from the horizontal. The parameters d0 and bℓ are determined by the boundary condition (3.4). Combining (3.2) and (3.7) yields a parameterized expression for the strain tensor,
This can be integrated using (3.1), leading to the following form of the frame-components of the displacement vector:
The parameter Ξ describes rigid vertical displacements of the fiber and can thus be determined, for example, by prescribing the displacement at a point of the fiber’s boundary, e.g., by setting
Anticipating, the only expansion coefficients of the Michell solution that couple to the optical phase and polarization are d0 and b2. It is worth noting that these two coefficients are completely determined by the strain tensor at the axis of the fiber. Indeed, in Cartesian coordinates we have
In Ref. 6 a number of physically motivated boundary conditions were considered. For the numerics of Section 3.3 we restrict ourselves to the model discussed in Ref. [6, Section 4.2.1], namely the waveguide resting on a rigid plane with an extended contact region. In this case one has
where 𝖕 is an ambient isotropic pressure.
The following Section 3.1 and Section 3.2 describe the influence of such fiber deformations on the electromagnetic modes propagating therein.
3.1 Displacement
Transverse deformations of an optical fiber which are invariant under translation along its axis are described by displacements of the form
which is assumed to be sufficiently small so that nonlinear terms in ur and uθ are negligible.
The deformation implies that the core-cladding interface is displaced from its reference location, while the calculations in Section 2.3 require the core-cladding interface to lie at a constant radial coordinate distance from the fiber’s axis. This can be resolved by performing a coordinate transformation, namely the inverse of (3.13), so that the core-cladding interface is again located at r = ρ in the new coordinates. This coordinate transformation results in additional source terms in the field equations, similarly for the interface equations.
The details are as follows.
The equations. For definiteness let
denote the coordinates before the waveguide has been deformed, and let describe the deformed configuration:
The interface between the core and the cladding is located at x2 + y2 = ρ2.
We assume that ui and its derivatives are of order ε, where ε is small compared to the remaining scales involved. In particular the map is invertible with
Denoting by the vector potential in the barred coordinate system,
one has
Away from the interface the vector potential satisfies the wave equation
In the xµ-coordinates this becomes
where gij is the space-part of the transformed contravariant metric
with ∇µ denoting the covariant derivative of the transformed metric. In particular, keeping in mind that the ’s are Euclidean coordinates in which = δij = δij where is the Kronecker delta,
The Christoffel symbols are
so that
Hence, up to terms of order ε2, the system (3.20) can be rewritten as
3.1.1 Boundary conditions. Let ni denote the field of normals to the interface between the core and the cladding. In the unbarred coordinate system the interface is located at where denotes the position vector in the plane perpendicular to the axis of the fiber. The covariant components of the normal nA, A = 1, 2, are proportional to xA/r, say nA = αxA/r. As already mentioned above, the normalization of nA is irrelevant for the equations, and thus we take
As in the unperturbed case, the matching conditions at the core-cladding interface take the general form (2.19), reproduced here for the convenience of the reader
with
Here, as above, ⟦f⟧ = (fcore − fcladding)|r=ρ denotes the jump of a function f at the surface where r2 = x2 + y2 equals ρ2.
To continue, we need explicit formulae for Gµνnν. Keeping in mind our assumption that µ = 1, we have
Thus, assuming that
we obtain
and
3.1.2 Explicit expressions. As already pointed-out in Section 2.3, the results obtained from first-order perturbation theory only depend on perturbation terms with relative azimuthal mode number ∆m ∈ {0, +2, −2}, i.e., only terms without angular dependence or ∝ e±2iθ. We treat these cases independently in the following paragraphs.
Radial perturbations In the simplest case ∆m = 0 the perturbative terms are independent of the angular coordinate θ, and we can write the wave equation again in terms of Helmholtz operators
Explicitly, for the displacements given in (3.13) together with (3.10) one finds
This is equivalent to the unperturbed Equation (2.13) with rescaled frequency and wave vector. Consequently, the solutions are given as in (2.15)–(2.17) with the substitutions
This, in turn, introduces a shift in the roots of (2.25) and thus a perturbation of β, namely
where is the effective refractive index, and db/dV is the derivative of the normalized guide index with respect to the normalized frequency, which can be read from Figure 2.1.
Alternatively, in the current simple case one can directly work in the original coordinates, with δℋm = 0 in (3.32). The interface region is then displaced as ρ → ρ + ur(ρ). Such a perturbation in Equation (2.16) is equivalent to (3.34).
Perturbations with angular dependence Apart from the angle-independent terms described above, the displacements given in Equation (3.10) lead to correction terms to the field equations with angular dependence e±2iθ that can be written in the form of Equation (2.33) with
For the correction terms arising for the interface conditions (2.39) we find explicitly
These expressions can be used in Equation (2.44) to compute the matrix ℳ to obtain an explicit form of the propagation law given by Equation (2.43)–Equation (2.44); equivalently (2.45)-(2.46). Since the dispersion relation can only be solved numerically and the integrals arising here admit no concise explicit form, we evaluate ℳ numerically. Concrete numerical results for a range of fiber parameters, including those relevant for the GRAVITES experiment22, are presented in Section 3.3 and Figure 1.2–Figure 1.3.
3.2 Photoelasticity
We introduced the constitutive tensor for linear dielectric media in (2.2), specializing so far to isotropic media. Now, we consider small deviations from an ideal, isotropic, non-magnetic dielectric, writing the constitutive tensor as
where ε is the same small expansion parameter introduced in (2.29), with a tensor field δχµνρσ arising from the change in permittivity introduced by elastic deformations. Taking a perturbative expansion for the field strength tensor Fµν as well, we can write (2.2) as
so that, using (2.6), the gauge-fixed Equation (2.8) becomes
We assume that the medium remains non-magnetic and induces no magneto-electric coupling, so that δχµνρσ takes the form
where δϵij is the elastically induced change in the permittivity tensor. In the linear regime, δϵij is related to the material strain ukl via the photoelasticity tensor pijkl [20, Appendix D]:
The components pijkl generally need to be determined empirically, though in the case of an isotropic solid they simplify to 21
in Voigt notation, where 11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5, 12 → 6. Alternatively, employing tensor notation, we introduce material constants p and q through the formula
where gij are the components of the spatial metric (equal to the Kronecker δij in Cartesian coordinates). For the case of fused silica glass, as used in optical fibers, the photoelasticity tensor is characterized by 22,23
Note that p1212 is measured independently, but in good agreement with the isotropy assumption made in (3.43). The induced source terms for the perturbative calculation as determined via (3.40) and the strain tensor (3.9) are given by
as well as
Further, for the correction terms arising for the interface conditions (2.39) we find explicitly
and
with the remaining components of vanishing identically.
Based on these source terms, Equation (2.44) determining the transport matrix for the Jones vector can be evaluated numerically by inserting the relevant Bessel functions and integrating. We find that the matrix ℳ in (2.44) has the form
The corresponding matrix entering the propagation law (2.45) in Cartesian coordinates thus takes the form
Here, ξp, ξb are coefficients that depend only on the fiber core radius, its unperturbed refractive indices, and the optical frequency of the fiber mode. Numerical results are presented in Section 3.3.
Since the unperturbed solutions described in Section 2.2 are confined to a small region around the fiber’s axis, with the modes decaying exponentially in the cladding, it is natural to consider an approximate form of the above expressions that is valid for r ≪ a. Using this confinement approximation, the strain components take the form
Thus, the source terms for ∆m = 0 are identical to (3.46) and (3.48), while and simplify significantly to
and
The numerical results produced in Table 3.3 show good agreement between the confinement approximation and the full solution.
Table 3.3. Estimates for the phases, in radians, arising in the GRAVITES experiment described in 3.
The phase shifts arising from the longitudinal fiber expansion were computed in Ref. 6 The first column differs due to a correction of the linear thermal expansion coefficient α; the remaining shifts are determined in this work. Temperature and pressure differences over a vertical separation of 1m give rise to phase shifts via elastic deformations of the fiber and induced photoelastic effects; gravity gradients induce in-homogeneities and thus give rise to both phase perturbations and birefringence effects. Confinement approximation refers to (3.52) wherein the material strain components are expanded for r ≪ a.
| Gravitational phase shift | −6.52 × 10−5 |
|---|
| Systematic effects | Temperature | Pressure | Gravity gradient |
|---|
| Phase | Phase | Phase | Birefringence |
|---|
| Longitudinal expansion6 | 3240 | 28 | 7.0 × 10−7 | |
| Geometric deformation | 8.7 | −0.15 | −3.0 × 10−9 | 2.42×10−11 |
| Photoelasticity | 3190 | −53 | −1.1 × 10−6 | −1.27×10−6 |
| Photoelasticity confinement approx. | 3190 | −53 | −1.1 × 10−6 | −1.28×10−6 |
3.3 Numerical results
The Section 3.1 and Section 3.2 provide explicit expressions for the Σ- and -terms that enter the matrix ℳ, defined in Equation (2.44), which determines the propagation law of light polarization according to Equation (2.43) and Equation (2.45). Here we present numerical values based on the material properties of silica oxide glass given in Table 3.1 and the experimental parameters of GRAVITES given in Table 3.2. The values for δg, δ𝖕 and δT correspond to a height difference of 1m for an experiment without thermal insulation and pressure control. Since the response of the system is linear in the environmental parameters, these numbers can directly be used to determine the constraints on the experimental setup so that environmental fluctuations do not overwhelm the signal of interest.
The numerical values determined in this work are juxtaposed in Table 3.3 with the results of Ref. 6, where the change of length of the optical fiber has been determined.
In Figure 1.2 and Figure 1.3 we provide plots showing how a change in fiber parameters affects these values. Both for phase and birefringence the effect is essentially due to the photoelastic material response, as the contribution arising from the core-cladding interface deformation is smaller by four orders of magnitude. Additionally, this effect is practically independent of the fiber’s material indices n1 and n2.
