## Abstract

A theoretical investigation was done concerning the resonant properties of a rotating Fabry-Perot cavity with dielectric media inside. The frequency splitting of orthogonal circularly-polarized modes had been predicted. Formula for the frequency splitting was obtained taking into account refractive index dispersion, dynamic-optical effect and the transversal structure of the electromagnetic field. We represent a simplified analysis, based on the geometrical optical theory, and we also present an accurate analysis on the basis of the general theory of relativity.

©1998 Optical Society of America

## 1. Introduction

The resonant properties of ring cavities in a noninertial frame of reference, (particularly, in rotating) were studied in Refs. [1–7]. In this paper we focus on the resonant frequencies of a Fabry-Perot (FP) cavity rotating around an optical axis.

A qualitative explanation of an observable phenomenon consists in considering the electromagnetic field structure as rest in an inertial frame of reference. An observer moving together with the cavity, crosses the field’s nodes and antinodes. (See Ref. [8] and literature quoted therein.) Thus, the observer in the cavity frame needs to be able to detect the motion of the frame either in the nonuniformity or the anisotropy of the field. There are no restrictions on the cavity geometry. Such resonant frequencies of the cylindrical microwave cavity rotating around the axis of symmetry were shown in Ref. [1] to be dependent on the angular rotation speed Ω. The frequency splitting of TE_{rmq} modes with azimuth indexes ±*m* is equal to 2*m*Ω. Because of the boundary conditions on the cavity’s cylindrical surface, the polarization and transversal structure of the modes are not independent. It is therefore impossible to detect the rotation when only rotational symmetrical modes (with m=0) are excited.

In optical range, open cavities are used (for example an FP-cavity) and the fundamental mode with rotational symmetry may have a linear polarization. As mentioned in Ref. [8], when the cavity rotates along the axis of symmetry, the linearly polarized electromagnetic field must maintain the same oscillation direction during rotation. The observer in the cavity frame may consider this as a presence of two orthogonal circularly-polarized modes with different frequencies in the cavity. In the present work we derived formula for the frequency splitting in the rotating FP-cavity taking into account the optical properties of the medium and the transversal structure of the electromagnetic field.

An accurate analysis of the phenomenon arising in a rotating cavity should be made on the basis of the general theory of relativity because of the noninertiality of the reference frame. An important point in such an analysis is the “4-tensor formulation” of the constitutive equations for the medium [1–6]. The constitutive equations used in Refs. [2,3] lead to the frequency splitting in rotating ring cavity in agreement with experimental measurements [7]. In the present work we are unable to directly use these equations because the frequency splitting arising in an FP-cavity is much lower than the corresponding value in a ring cavity. We need to take into account the dynamic-optical effect [10]. For this reason we must derive the constitutive equations for the medium in the rotating frame to include the birefringence induced by the Coriolis force. Therefore microscopic Maxwell’s equations in a covariant notation were chosen as the starting point. Then the equations were averaged and the polarization **P** was introduced. Next, the amendment to a dielectric permeability ε of a medium resting in noninertial frame was derived from the covariant equation of a charged particle movement in gravitational and electromagnetic fields. Finally the wave equation was obtained and the formula for the frequency splitting was obtained.

We suggest that the analysis of this phenomenon based on the geometrical optical theory will also be useful for readers.

## 2. A simplified analysis of the phenomenon

Consider an empty FP-cavity of length *L*, formed by two isotropic mirrors. There exists a polarization degeneracy of eigen mode frequencies in such a cavity. If a cavity is rotated with angular speed Ω around an optical axis, the angle of rotation for cavity round-trip time is 2Ω*L*/*c*. An observer moving with the cavity, finds that the linearly polarized wave returns after a round-trip of the cavity with polarization revolving at the angle -2Ω*L*/*c*. Thus, the rotation of the cavity is equivalent to the insertion of a nonreciprocal rotator (or circular birefringent wave plate) with a rotation angle of a polarization plane -Ω*L*/*c*. This results in the removal of the degeneracy: the frequency splitting ∆ω of two orthogonal circularly-polarized modes becomes 2Ω.

When a dielectric with refractive index *n* fills a cavity, it is necessary to take into account polarization dragging (the Fermi effect) within the rotating medium. For the observer in the inertial reference frame, the rotation angle of the polarization plane through a single trip between mirrors is (*n*
^{2} - 1)Ω*L*/*cn* [10], while the cavity rotation angle is Ω*Ln*/*c*. This results in the reduction of the frequency splitting:

Note, that the splitting is independent of the geometrical sizes of the cavity.

We shall now carry out an accurate analysis, based on the covariant form of Maxwell’s equations.

## 3. The microscopic equations of an electromagnetic field

For an accurate analysis of the frequency splitting phenomenon arising in rotating FP-cavity we will use the reference frame moving with the cavity. This reference frame is noninertial and the metric of space-time is different from the Galilean metric. The metric tensor *g*_{ik}
has the following nonzero components in the first approximation on Ω*r*/*c*:

In this paper we use the following notation: a component of a tensor with index 0 corresponds to the temporary coordinate (*dx*
^{0} = *cdt*) and with indexes 1,2,3- to the spatial coordinates (*dx*
^{1} = *dx* , *dx*
^{2} = *dy* , *dx*
^{3} = *dz*); Greek indexes accept meanings 1,2,3, and Latin indexes -0,1,2,3; twice repeated Greek or Latin identical indexes indicate the summation over only the space indexes or over all four indexes, correspondingly; an index separated by a semicolon designates the covariant derivative on the appropriate coordinate.

Maxwell’s equations in a covariant form are [9]:

$${{F}^{\mathit{ik}}}_{;k}=\frac{1}{\sqrt{-g}}\frac{\partial \sqrt{-g}{F}^{\mathit{ik}}}{\partial {x}^{k}}=-\frac{4\pi}{c}{j}^{i}$$

where *g* = det(*g*_{ik}
) , *F*_{ik}
and *F*^{ik}
are covariant and contravariant antisymmetric tensors of a microscopic electromagnetic field and *j*^{i}
is the 4-vector of a microscopic current. The tensors *F*_{ik}
and *F*_{ik}
are associated in a standard manner:

Now we represent components of antisymmetric 4-tensor through the components of polar and axial 3-vectors. Due to the difference of the space-time metric *g*_{ik}
from Galilean, the vectors associated with *F*_{ik}
do not coincide with the ones for *F*^{ik}
. The first equation in (3) does not change in the presence of matter (microscopic charges), therefore the microscopic electrical and magnetic fields **e** and **h** are components of *F*_{i}*k* . We also enter auxiliary 3-vectors **ẽ** and **h̃** associated with *F*^{ik}
in accordance to the scheme from Ref. [9]:

$${\tilde{e}}^{\alpha}=-\sqrt{{g}_{00}}{F}^{0\alpha}\phantom{\rule{1em}{0ex}}{\tilde{h}}_{\alpha}=-\frac{1}{2}\sqrt{\gamma}{e}_{\mathrm{\alpha \beta \gamma}}\sqrt{{g}_{00}}{F}^{\mathrm{\beta \gamma}}$$

where *e*
^{αβγ} and *e*
_{αβγ} are the contravariant and covariant fully antisymmetric unit 3-tensor, γ_{αβ} is the three-dimensional metric tensor, responsible for the space’s geometrical properties, γ = det(γ_{αβ}) , γ*e*
_{αβγ} = *e*
^{uvσ}γ_{αμ}γ_{βv}γ_{γσ} . The space metric tensor γ_{αβ} and space-time metric tensor *g*_{ik}
are related in the following way:

The relations (5) lead to the coincidence of **ẽ** and **h̃** with **e** and **h**, correspondingly, in empty space in an inertial frame of reference, and to the following vectorial form of microscopic Maxwell’s equations :

$$\mathrm{div}\phantom{\rule{.2em}{0ex}}\tilde{\mathbf{e}}=4\mathrm{\pi \rho}\phantom{\rule{2em}{0ex}}\mathrm{rot}\phantom{\rule{.2em}{0ex}}\tilde{\mathbf{h}}=\frac{1}{c\sqrt{\gamma}}\frac{\partial}{\partial t}\left(\sqrt{\gamma}\tilde{\mathbf{e}}\right)+\frac{4\pi}{c}\mathbf{s}$$

where s is a vector with components *s*
^{α} = ρ*dx*
^{α}/*dt*, ρ is the density of microscopic charges, and vectorial operators of divergence and curl appear with the appropriate space metric γ_{αβ}.

In a three-dimensional notation the relationships (4) take the form:

where we introduce a vector **g** with components *g*
_{α} = -*g*
_{0α}/*g*
_{00} . In the first approximation on Ω*r*/*c*, the vector **g** is rewritten as:

One could obtain from Eqs. (6) that in this approximation, the 3-dimensional space is Euclidean: γ_{αβ} = δ_{αβ}, γ = 1, where δ_{αβ} stands for the usual Kronecker symbol.

## 4. The macroscopic equations of a field

Now we average Eqs. (7) and (8) on 4-volume to derive macroscopic Maxwell’s equations and constitutive equations. In the first approximation on Ω*r*/*c*, the averaging is equivalent to averaging in an inertial system (*g*
_{00} = 1 and γ = 1). We introduce the macroscopic electrical
field **E** = 〈**e**〉 and the magnetic induction **B** = 〈**h**〉, the polarization **P** and the magnetization **M** as electrical and magnetic dipole moments of unit volume to obtain macroscopic Maxwell’s equations:

$$\mathrm{div}\mathbf{D}=4\mathrm{\pi \rho}\phantom{\rule{.9em}{0ex}}\mathrm{rot}\phantom{\rule{.2em}{0ex}}\mathbf{H}=\frac{1}{c}\frac{\partial \mathbf{D}}{\partial t}+\frac{4\pi}{c}\mathbf{j}$$

where we introduce **D** = 〈**ẽ**〉 + 4π**P** and **H** = 〈*h̃*〉-4π**M**. Averaging Eqs. (8), we obtain the constitutive equations:

Note that the above equations will coincide with the constitutive equations from Refs. [2,3] if we put **P** = (ε - 1)**E**/4π and **M** = (μ - 1)**H**/4π with ε and μ measured for the matter resting in the inertial frame. In our case the frequency splitting (1) is a small value, and we should take into account the influence of the Coriolis force.

In an optical range, the displacement current prevails over the conductivity current (ρ = 0, **j** = 0), the magnetization is absent (**M** = 0), therefore it is only necessary to find out the relation between **E** and **P** .

## 5. Dielectric permeability in a rotating medium

To find the response of the medium on the electromagnetic field in the same strict manner as was done in previous sections, we should start from the covariant equation of a charged particle (electron) movement in gravitational and electromagnetic fields [9]:

where the left-hand side of Eq. (12) is the covariant derivative of a speed 4-vector *u*^{i}
on an interval *ds*; ${\mathrm{\Gamma}}_{\mathit{\text{kl}}}^{i}$ is the Cristoffel symbol and *e* is the charge of electron. Expressing ${\mathrm{\Gamma}}_{\mathit{\text{kl}}}^{i}$ through the metric tensor components *g*_{ik}
(see ref. [9]), we find the following nonzero components of ${\mathrm{\Gamma}}_{\mathit{\text{kl}}}^{i}$ in the first approximation on Ω*r*/*c*:

Disclosing Eq. (12) for the spatial components (i=1,2,3), we obtain:

where **v** is the 3-dimensional speed. So, in this approximation, the Coriolis force term appears in Eq. (14) in comparison with the equation in the inertial reference frame. Its action on a free electron is equivalent to an influence of a permanent magnetic field with induction **B** = -2/*mc*
**Ω**/|*e*|.

The equation of a harmonic oscillator (bound electron) differs from Eq.(12) through the presence of the covariant derivative of the potential energy on the right-hand side. Since the covariant derivative of a scalar coincides with a common derivative, there are no changes in potential force in an noninertial reference frame. Therefore the circular birefringence induced by a Coriolis force could be represented by a vector of gyration:

where V is the Verdet constant, μ_{B} =|*e*|ħ/2*mc* is the Bohr magneton, ω is the optical frequency. The expression obtained for **g**
_{C} is equivalent to that in Ref. [10]. Using Eq. (15) we find the ratio of the polarization plane rotation angle caused by the Coriolis force to that caused by a real rotation of reference frame in the presence of polarization dragging. This ratio is equal to *Vħcn*/μ_{B} and it’s value for Nd:YAG (n=1.82, V=1.8∙10^{-6} rad/Gs∙cm at the 1.06 μm wavelength) is an order of 0.01. The exact expression for **g**
_{C} can only be obtained through
quantum mechanics.

Thus, the relation between **P** and **E** in a rotating medium could be written in the following form:

where ε is a dielectric permeability of a medium, at rest in an inertial system of reference, and the vector of gyration **g**
_{C} is determined by an angular velocity of the system:

with λ^{(C)} > 0. Note that the sign of an imaginary part in Eq. (16) depends on the selected convention of a sign for a wave phase. In our analysis, we have written the phase as *i*(ω*t* - **kr**).

## 6. The wave equation

From Maxwell’s equations (10) and constitutive equations (11) and (16), we obtain the following relationships:

$$\mathrm{div}\phantom{\rule{.2em}{0ex}}\mathbf{E}=-\frac{i}{\mathit{\epsilon c}}{\mathbf{g}}_{C}\frac{\partial \mathbf{B}}{\partial t}+\frac{2\mathbf{B\Omega}}{\mathit{\epsilon c}}-\frac{1}{c}g\frac{\partial \mathbf{E}}{\partial t}$$

Note that in the considered approximation div **E** ≠ 0. Disclosing a curl of a vector product rot [**E** × **g**] and expressing [rot **E** × **g**] through grad(**Eg**), we derive the following wave equation:

The first term on the right side determines the Sagnac effect in ring lasers. The following two terms are responsible for the investigated problem. The two final terms lead to deflection of an energy flux from the wave vector direction.

We consider a plane monochromatic **E** = **E**
_{0}exp{*i*(ω*t* - **kr**)} with a vector
**k** = τ*k* at small angle **θ** with respect to the rotation axis **Ω**. The substitution of **E** in Eq.(19) with the account of Eq. (17) and dispersion *∂n*/*∂*ω gives:

where *n*
^{2} = ε, ∆ω_{±} and *n*
_{±} are frequency shifts and refractive indexes for left and right circularly-polarized waves, respectively. Rewriting Eq. (20) in terms of wave amplitudes **E**
_{±} = (**E**
_{x} ± *i*
**E**
_{y})/2 we find refractive indexes:

Therefore, the phase difference between the left and right circularly-polarized waves for the layer of thickness *L* is:

For the cavity round-trip we shall consider propagation in forward and backward directions. In the backward direction (after reflection from the mirror) the left (right) polarized wave becomes the right (left) polarized. As in the backward direction the sign of a product **Ωτ** is inverse, the phase difference for a cavity round-trip is 2∆φ. With dispersion taken into account, the frequency splitting of left and right polarized waves is as follows:

where θ is the angle between the wave vector **k** of the outgoing waves and the axis of rotation **Ω** . If the rotation axis coincides with the optical axis of the cavity and the dispersion is absent, and if the Coriolis force is neglected, the frequency splitting (23) coincides with the result of the simplified analysis (1).

## 7. Frequency splitting of transversal modes

The result, obtained in an approximation of plane waves, can only be applied to the fundamental mode of a cavity.

For the propagation of an electromagnetic wave along a rotating FP-cavity, not only does the rotation of a polarization plane occur, but there is an image rotation at an angle of -2Ω*Ln*cosθ/*c*. This angle is of the same module and the opposite sign to the cavity rotation angle during a round trip.

The spatial distribution of the transversal modes is expressed through the generalized Laguerre polynomials ${L}_{r}^{m}$ (ξ) [11]:

where ρ is a transversal coordinate, φ is an azimuth, *q* is a complex parameter of the Gaussian beam, *w* is a beam spot size, *r* and ±*m* are the radial and azimuth indexes (*r*,±*m* = 0,1,2…). Because of the image rotation, the additional phase shifts ±2*m*Ω*Ln*cosθ/*c* arise for modes with the azimuth indexes ±*m* after one round trip in the cavity. It leads to the following frequency splitting of orthogonal circularly-polarized modes with azimuth indexes ±*m*:

where the upper sign corresponds to the case when the mode with index +*m* (-*m*) has the left (right) circular polarization, the lower sign corresponds to the opposite case.

Thus, contrary to a cylindrical microwave cavity, in an FP-cavity, the frequency splitting arises even for modes with *m* = 0. This splitting is purely caused by circular anisotropy of the cavity.

## 8. Conclusion

In this paper we have shown that even in an empty rotating FP-cavity there is a circular anisotropy, arising from the noninertiality of a cavity frame of reference and resulting in the frequency splitting of modes with orthogonal polarization.

We would like to note the independence of the frequency splitting on geometrical parameters of an FP-cavity contrary to the dependence in the case of a ring cavity. In accordance with the results we obtained, the presence of a gyromagnetic medium inside a cavity, or excitation of transversal modes with high azimuth index, increases the frequency splitting. We would like to stress that, if the cavity contains an active medium with inverted populations of laser transition, (in this case *∂n*/*∂*ω > 0 in a center of a gain line) a reduction of frequency splitting arises. To avoid considerable reduction, it is necessary to have a small excess of pumping above the threshold.

## Acknowledgments

The author is grateful to professor N.V. Kravtsov and Dr. E.G. Lariontsev for valuable discussions about the problem investigated. This research was financially supported by the Russian Foundation for Basic Research (project No. 96-02-17326) and the “Fundamental Metrology” Program.

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