## Circular anisotropy of a rotating Fabry-Perot cavity

Optics Express, Vol. 2, Issue 9, pp. 397-403 (1998)

http://dx.doi.org/10.1364/OE.2.000397

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### 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.

© Optical Society of America

## 1. Introduction

1. C.V. Heer, “Resonant Frequencies of an Electromagnetic Cavity in an Accelerated System of Reference,” Phys. Rev. **134**, A799–804 (1964) [CrossRef]

1. C.V. Heer, “Resonant Frequencies of an Electromagnetic Cavity in an Accelerated System of Reference,” Phys. Rev. **134**, A799–804 (1964) [CrossRef]

_{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.

1. C.V. Heer, “Resonant Frequencies of an Electromagnetic Cavity in an Accelerated System of Reference,” Phys. Rev. **134**, A799–804 (1964) [CrossRef]

2. E.J. Post, “Sagnac Effect,” Rev. Mod. Phys. **39**, 475–493 (1967) [CrossRef]

**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.

## 2. A simplified analysis of the phenomenon

*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.

*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:

## 3. The microscopic equations of an electromagnetic field

*g*

_{ik}has the following nonzero components in the first approximation on Ω

*r*/

*c*:

*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.

*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:

*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]:

**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 :

*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}.

**g**with components

*g*

_{α}= -

*g*

_{0α}/

*g*

_{00}. In the first approximation on Ω

*r*/

*c*, the vector

**g**is rewritten as:

_{αβ}= δ

_{αβ}, γ = 1, where δ

_{αβ}stands for the usual Kronecker symbol.

## 4. The macroscopic equations of a field

*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:

**D**= 〈

**ẽ**〉 + 4π

**P**and

**H**= 〈

*h̃*〉-4π

**M**. Averaging Eqs. (8), we obtain the constitutive equations:

2. E.J. Post, “Sagnac Effect,” Rev. Mod. Phys. **39**, 475–493 (1967) [CrossRef]

**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.

**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

*u*

^{i}on an interval

*ds*;

*e*is the charge of electron. Expressing

*g*

_{ik}(see ref. [9]), we find the following nonzero components of

*r*/

*c*:

**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*|.

_{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.

**P**and

**E**in a rotating medium could be written in the following form:

**g**

_{C}is determined by an angular velocity of the system:

^{(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

**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:

**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:

*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:

*L*is:

**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:

**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

*Ln*cosθ/

*c*. This angle is of the same module and the opposite sign to the cavity rotation angle during a round trip.

*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*:

*m*(-

*m*) has the left (right) circular polarization, the lower sign corresponds to the opposite case.

*m*= 0. This splitting is purely caused by circular anisotropy of the cavity.

## 8. Conclusion

*∂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

## References

1. | C.V. Heer, “Resonant Frequencies of an Electromagnetic Cavity in an Accelerated System of Reference,” Phys. Rev. |

2. | E.J. Post, “Sagnac Effect,” Rev. Mod. Phys. |

3. | A.M. Khromykh, “Ring Generator in a Rotating Reference System,” Zh. Eksp. Teor. Fiz. |

4. | A.M. Volkov and V.A. Kiselyev, “Proper Frequencies of a Rotating Ring Resonator,” Zh. Eksp. Teor. Fiz. |

5. | A.M. Volkov and V.A. Kiselyev, “Rotating Ring Cavity with a Nonreciprocal Element,” Opt. Spektrosk. |

6. | A.M. Volkov and V.A. Kiselyev, “Rotating Ring Cavity with a Moving Medium as the Nonreciprocal Element,” Opt. Spektrosk. |

7. | V.E. Privalov and Yu.V. Filatov, “A Study of the Output Characteristics of the Rotating Gas Ring Laser,” Kvant. Electron. (Moscow) |

8. | A.M. Belonogov, “Electromagnetic Oscillations in a Three-Dimentional Resonator in a Rotating Reference Frame,” Zh. Tekh. Fiz. |

9. | L.D. Landau and E.M. Lifshitz, |

10. | L.D. Landau and E.M. Lifshitz, |

11. | Yu.A. Anan’ev, |

**OCIS Codes**

(120.2230) Instrumentation, measurement, and metrology : Fabry-Perot

(120.5790) Instrumentation, measurement, and metrology : Sagnac effect

(140.3370) Lasers and laser optics : Laser gyroscopes

(350.5720) Other areas of optics : Relativity

**ToC Category:**

Focus Issue: Control of loss and decoherence in quantum systems

**History**

Original Manuscript: December 11, 1997

Published: April 27, 1998

**Citation**

Dmitri Boiko, "Circular anisotropy of a rotating Fabry-Perot cavity," Opt. Express **2**, 397-403 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-9-397

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### References

- C.V. Heer, "Resonant Frequencies of an Electromagnetic Cavity in an Accelerated System of Reference," Phys. Rev. 134, A799-804 (1964) [CrossRef]
- E.J. Post, "Sagnac Effect," Rev. Mod. Phys. 39, 475-493 (1967) [CrossRef]
- A.M. Khromykh, "Ring Generator in a Rotating Reference System," Zh. Eksp. Teor. Fiz. 50, 281-282 (1966)
- A. M. Volkov, V .A. Kiselyev, "Proper Frequencies of a Rotating Ring Resonator," Zh. Eksp. Teor. Fiz. 57, 1353-1360 (1969)
- A. M. Volkov, V. A. Kiselyev, "Rotating Ring Cavity with a Nonreciprocal Element," Opt. Spektrosk. 29, 365-370 (1970)
- A. M. Volkov, V. A. Kiselyev, "Rotating Ring Cavity with a Moving Medium as the Nonreciprocal Element," Opt. Spektrosk. 30, 332-339 (1971)
- V. E. Privalov, Yu. V. Filatov, "A Study of the Output Characteristics of the Rotating Gas Ring Laser," Kvant. Electron. (Moscow) 4, 1418-1425 (1977)
- A. M. Belonogov, "Electromagnetic Oscillations in a Three-Dimentional Resonator in a Rotating Reference Frame," Zh. Tekh. Fiz. 39, 1175-1179 (1969)
- L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Nauka, Moscow, 1988)
- L .D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Nauka, Moscow, 1992)
- Yu. A. Ananev, Optical Resonators and Laser Beams (Nauka, Moscow, 1990)

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