## Finite-difference time-domain algorithm for modeling Sagnac effect in rotating optical elements

Optics Express, Vol. 16, Issue 8, pp. 5227-5240 (2008)

http://dx.doi.org/10.1364/OE.16.005227

Acrobat PDF (323 KB)

### Abstract

Electrodynamics in rotating optical elements has attracted much interest due to its potential application to ultra-sensitive rotating sensing. And it is important to investigate the Sagnac effect in some novel photonic structures for it may lead to a variety of unusual manifestations. We propose a Finite-Difference Time-Domain (FDTD) method to model the Sagnac effect, which is based on the modified constitutive relation in rotating frame. The time-stepping expressions for the FDTD routine are derived and discussed, and the classical Sagnac phase shift along a waveguide is calculated. Further discussions about numerical dispersion, dielectric boundary condition and perfect matched layer (PML) absorbing boundary conditions in the rotating FDTD model are also presented respectively. The theoretical analysis and simulation results prove that the numerical algorithm can analyze the Sagnac effect effectively, and can be applied to general cases with various material properties and complex geometric structures. The proposed algorithm provides a promising systematic tool to study the properties of rotating optical elements, and to accurately analyze, design and optimize rotation sensitive optical devices.

© 2008 Optical Society of America

## 1. Introduction

1. U. Leonhardt and P. Piwnitski, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A **62**, 055801 (2000). [CrossRef]

3. J. Scheuer and A. Yariv, “Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures,” Phys. Rev. Lett.. **96**, 053901 (2006). [CrossRef] [PubMed]

4. C. Peng, Z. Li, and A. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express **15**, 3864–3875 (2007). [CrossRef] [PubMed]

5. B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E. **71**, 056621 (2005). [CrossRef]

6. B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. **24**, 142–151 (2006). [CrossRef]

7. S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A **74**, 021801 (2006). [CrossRef]

9. C. Peng, Z. Li, and A. Xu, “Rotation sensing based on a slow-light resonating structure with high group dispersion,” Appl. Opt. **46**, 4125–4131 (2007). [CrossRef] [PubMed]

10. B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B **24**, 1216–1224 (2007). [CrossRef]

9. C. Peng, Z. Li, and A. Xu, “Rotation sensing based on a slow-light resonating structure with high group dispersion,” Appl. Opt. **46**, 4125–4131 (2007). [CrossRef] [PubMed]

3. J. Scheuer and A. Yariv, “Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures,” Phys. Rev. Lett.. **96**, 053901 (2006). [CrossRef] [PubMed]

4. C. Peng, Z. Li, and A. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express **15**, 3864–3875 (2007). [CrossRef] [PubMed]

9. C. Peng, Z. Li, and A. Xu, “Rotation sensing based on a slow-light resonating structure with high group dispersion,” Appl. Opt. **46**, 4125–4131 (2007). [CrossRef] [PubMed]

5. B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E. **71**, 056621 (2005). [CrossRef]

12. B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E **74**, 016608 (2006). [CrossRef]

## 2. Theory of the FDTD algorithm in rotating frame

*R*≪

*c*|,

*R*is the rotating radius and

*c*is the speed of light in vacuum. As some early works postulated [13

13. T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE **61**, 1694–1702 (1973). [CrossRef]

14. J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. **181**, 1765–1775 (1969). [CrossRef]

6. B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. **24**, 142–151 (2006). [CrossRef]

13. T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE **61**, 1694–1702 (1973). [CrossRef]

14. J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. **181**, 1765–1775 (1969). [CrossRef]

*r*=

*xx̂*+

*zẑ*+

*zẑ*, Ω⃗=

*ẑ*Ω. The derivation can be started with Eq. 2.1 and Eq. 2.2. As Ω is assumed varying slowly, Ω=

*const*,

*∂*Ω/

*∂t*=0. And the axis is fixed in rotation frame, which leads to

*∂r⃗*/

*∂t*=0. Thus, we get:

*E*,

_{x}*E*,

_{y}*H*for TE modes and

_{z}*H*,

_{x}*H*,

_{y}*E*for TM modes. Compared with the ordinary form of Maxwell’s Equations, some extra terms which represent the Sagnac effect are induced, whose magnitudes are in the order of

_{z}*c*

^{-2}and proportional to rotation velocity. Without loss of generality, we focus the discussion on TE mode, and

*∂E*/

_{x}*∂t*,

*∂E*/

_{y}*∂t*,

*∂H*/

_{z}*∂t*can be resolved with discarding the

*c*

^{-4}terms, then it leads to:

*H*can be split to two components

_{z}*H*and

_{zx}*H*as Berenger’s solution for the perfect matched layer (PML) [16], therefore, Eq. 2.11 can be written as:

_{zy}*E*,

*H*components are interleaved at intervals Δ

*x*,Δ

*y*in space. The

*H*components are located at point

_{z}*i*,

*i*+1, …, and surrounded by

*E*,

_{x}*E*components which are located at point

_{y}*i*+1/2,

*i*+3/2, …. On the other hand, the leapfrog scheme is also commonly used [17]. The temporal locations of the

*E*and

*H*components are interleaved at intervals of half time-step (Δ

*t*/2).

*H*components are calculated at time-step

*n*from the previous field components, and

*E*components are calculated at time-step

*n*+1/2.

18. D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE , **4**268–270, (1994). [CrossRef]

*σ*,

*σ*)=0, the exponential term should be treated carefully, as in

_{m}*H*|

_{z}

^{n}_{i,j+1/2}, which is required by Eq. 2.14, but it is not stored at point (

*i*,

*j*+1/2). A way to resolve the problem is to approximately express it as (

*H*|

_{z}^{n}

_{i,j}+

*H*|

_{z}^{n}

_{i,j+1})/2, which are all readily available. This strategy can also be applied to

*E*|

_{x}^{n+1/2}

_{i,j}and

*E*|

_{y}^{n+1/2}

_{i,j}.

*H*and

_{zx}*H*are stored at time-step

_{zy}*n*,

*E*and

_{x}*E*are stored at time-step

_{y}*n*+1/2. This scheme also leads to another problem that some field components are not available at certain temporal locations. For instance, to calculate

*E*|

_{x}^{n+1/2}

_{i,j+1/2}by Eq. 2.14, we need

*∂E*/

_{x}*∂y*,

*∂E*/

_{y}*∂x*at time-step

*n*, but they are not available in the computer’s memory. However, an extrapolation method can be applied to estimate their values, which is based on the values at previous time steps. For a simple linear extrapolation, only the values at time step

*n*-1/2 and

*n*-3/2 are required. We can extend this extrapolation to higher orders, if higher accuracy is required.

*∂H*/

_{z}*∂y*|

^{n}

_{i,j}, which is required by Eq. 2.17, can be expressed as (

*H*|

_{z}^{n}

_{i,j+1}-

*H*|

_{z}^{n}

_{i,j-1})/2Δ

*y*. But it doesn’t work for the cells at the boundary because they can not be “centered”. We also need an extrapolation method to estimate their values, which are based on the spatial derivatives in internal cells. At least two adjacent internal cells are required for the simplest extrapolation, and more cells may be needed for some sophisticated extrapolation methods.

*c*

^{-4}terms, then the temporal differentials

*∂H*/

_{x}*∂t*and

*∂H*/

_{y}*∂t*can be expressed by spatial differentials. We can repeat this procedure and obtain six equations with temporal differentials on one side and spatial differentials on the other side. Then by discretizing these equations, the time-stepping equations for FDTD routine can be obtained. The discussion about the extrapolation technique can be also extended to the 3D case.

## 3. Modeling 2D rotating waveguide by the FDTD algorithm

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

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

19. R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. **93**, 143901 (2004). [CrossRef] [PubMed]

*i*,

*j*), the value of the field component at every time step can be recorded as

*f*(

*n*Δ

*t*), and the spectrum

*F*(

*ω*) for this time domain signal can be calculated by Fourier transformation, as

*F*(

*ω*)=∫

^{NΔt}

_{0}

*f*(

*t*)

*e*

^{-jωt}

*dt*⋍Σ

^{N-1}

_{n=0}

*f*(

*n*Δ

*t*)

*e*

^{-jωnΔt}Δ

*t*. The phase angle of the complex frequency represents the phase information. Thus the phase difference between two positions A and B can be obtained from the respective spectrum immediately, as Δ

*ϕ*=arg(

*F*(

_{A}*ω*))-arg(

*F*(

_{B}*ω*)). Indeed, how to extract the unknown frequencies and phases from discrete time signals, sometimes called “harmonic inversion”, is a well investigated problem and various algorithms are proposed to resolve it [20

20. V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. **107**, 6756–6769 (1997). [CrossRef]

*V*=-

_{x}*y*Ω and

*V*=

_{y}*x*Ω. Because the Sagnac phase shift is proportional to the scalar product of the wave propagating direction and the velocity, thus

*V*≠0 leads to an additional phase shift, and it is a constant when

_{x}*V*varies. It is also well known that the waveguide dispersion can in no way influence the magnitude of the Sagnac effect. Different width and refractive index of the waveguide may cause different propagation modes, but the Sagnac phase shift should be unchanging.

_{y}*ϕ*=Δ

_{sagnac}*ϕ*-Δ

_{rotating}*ϕ*. We calculate it for a series of positions along the center of the waveguide, and the results are plotted in Fig. 3(a)–3(c). The simulation results agree well with Eq. 3.1, which proves that the Sagnac phase shift is a constant regardless of the velocity in the perpendicular direction to the waveguide, the width of the waveguide and the refractive index of the material. These exactly confirm the theory’s assumptions. Therefore the proposed FDTD algorithm can effectively model the classical Sagnac effect along the waveguide. Thus it is believed that: this algorithm can be applied to many other photonic structures with various material properties and complex geometric structures, to study their behavior in rotating frame.

_{stationary}## 4. Analysis and discussion about the FDTD model in rotating frame

### 4.1. Numerical dispersion and stability

*ε*,

*µ*with position in the grid, and there are no magnetic or electric loss, then the system yields

*V*=-

_{x}*y*Ω and

*V*=

_{y}*x*Ω,

*ω*is the wave angular frequency,

*k̃*and

_{x}*k̃*are the x- and y-components of the numerical wavevector. It should be noticed that the numerical wavevector here contains the contribution from the Sagnac effect, therefore it is slightly different with the wavevector in the stationary frame.

_{y}*k*,

_{x}*k*are the wavevectors in the stationary frame without taking accout the Sagnac effect; Δ

_{y}*ϕ*presents the Sagnac effect shift, as Eq. 3.1 shows, which depends on the scalar product of the wave propagating direction and the velocity vector of the medium, then it can be rewritten as:

*nω*/

*c*)

^{2}=(

*k*)

_{x}^{2}+(

*k*)

_{y}^{2}, and Eq. 4.12, it is

*x*,Δ

*y*,Δ

*t*approach zero. Qualitatively, this suggests that if fine enough gridding is chosen, the impact of numerical dispersion can be reduced to any desired degree. The numerical wavevector in rotating medium is proportional to the numerical wavevector in stationary frame with the factor

*G*, which represents the Sagnac effect numerically. So that the Sagnac phase shift will not be submerged in the numerical phase error due to the algorithm’s nature.

*G*, it is apparent that the numerical wave experiences an anisotropic Sagnac phase shift. These errors represent a fundamental limitation of all grid-based Maxwell’s equations’ algorithms, and can be trouble-some when modeling electronically large structures. Further work to improve the computation accuracy is required.

### 4.2. Dielectric boundary conditions in rotating frame

*ẑ*, the boundary conditions can be further simplified. Form Eq. 4.15, it is notice that the continuity of the tangential

*E*and

*H*fields still preserved across the dielectric interface. If Ω⃗ is in the

*z*-direction, for a small region at the media interface,

### 4.3. Absorbing boundary conditions (ABC) in rotating FDTD model

*x*<0), onto the material half-space having the eletric conductivity σ and magnetic loss

*σ** (Region 2,

*x*>0). Substituting the expressions for

*∂Ẽ*/

_{y}*∂x*and

*∂Ẽ*/

_{x}*∂y*form Eq. 4.20,4.21 into Eq. 4.22,4.23, we obtain the wave-equation in rotating medium:

*σ*

_{x}=

*σ*

_{y}=0 and

*σ**

_{x}=

*σ**

_{y}=0, it reduces to a lossless homogenous medium. In this case, the above dispersion relation Eq. 4.27 consists with Eq. 4.13. Therefore, in Region 1, the total field are given by

*τ*is the

*H*-field reflection and transmission coefficients. As we discussed in Section. 4.2, the dielectric boundary conditions still hold for 2D rotating medium, thus the tangential

*E*and

*H*fields must be preserved across the

*x*=0 interface. For non-rotating cases, the PML-ABC requires

*s*=

_{y}*s**

_{y}=1,

*s*=

_{x}*s**

_{x},

*ε*

_{1}=

*ε*

_{2},

*µ*

_{1}=

*µ*

_{2}, or equivalently

*σ*

_{y}=0=

*σ**

_{y},

*σ*

_{x}/

*ε*

_{1}=

*σ**

_{x}/

*µ*

_{1}. Apply them to Eq. 4.28–4.33, it is noticed that the phase-matching condition

*k*

_{x1}=

*k*

_{x2},

*k*

_{y1}=

*k*still hold for the rotating case. Further, we derive the

_{y2}*H*-field reflection and transmission coefficients:

*ν*/

*c*. For a slowly-rotating case, the reflection should be considerably small in quantity. Therefore, the PML-ABC which originally developed for non-rotating case still has good accuracy for the rotating medium.

## 5. Conclusion

## Acknowledgments

## References and links

1. | U. Leonhardt and P. Piwnitski, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A |

2. | A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. |

3. | J. Scheuer and A. Yariv, “Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures,” Phys. Rev. Lett.. |

4. | C. Peng, Z. Li, and A. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express |

5. | B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E. |

6. | B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. |

7. | S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A |

8. | S. Sunada and T. Harayama, “Design of resonant microcavities: application to optical gyroscopes,” Opt. Express |

9. | C. Peng, Z. Li, and A. Xu, “Rotation sensing based on a slow-light resonating structure with high group dispersion,” Appl. Opt. |

10. | B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B |

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

12. | B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E |

13. | T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE |

14. | J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. |

15. | J. Van Bladel, “Relativity and Engineering”, Springer, Berlin, (1984) |

16. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic,” J. Computational Physics |

17. | A. Taflove, “Computational Electrodynamics: The Finite-Difference Time-Domain Method,” Artech House, Boston, (1995). |

18. | D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE , |

19. | R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. |

20. | V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.2800) Fiber optics and optical communications : Gyroscopes

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

(140.3370) Lasers and laser optics : Laser gyroscopes

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 25, 2008

Revised Manuscript: March 21, 2008

Manuscript Accepted: March 22, 2008

Published: April 1, 2008

**Citation**

Chao Peng, Rui Hui, Xuefeng Luo, Zhengbin Li, and Anshi Xu, "Finite-difference time-domain algorithm
for modeling Sagnac effect in rotating
optical elements," Opt. Express **16**, 5227-5240 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5227

Sort: Year | Journal | Reset

### References

- U. Leonhardt and P. Piwnitski, "Ultrahigh sensitivity of slow-light gyroscope," Phys. Rev. A 62, 055801 (2000). [CrossRef]
- A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko and L. Maleki, "Optical gyroscope with whispering gallery mode optical cavities," Opt. Commun. 233, 107-112 (2004). [CrossRef]
- J. Scheuer and A. Yariv, "Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures," Phys. Rev. Lett. 96, 053901 (2006). [CrossRef] [PubMed]
- C. Peng, Z. Li, and A. Xu, "Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency," Opt. Express 15, 3864-3875 (2007). [CrossRef] [PubMed]
- B. Z. Steinberg, "Rotating photonic crystals: A medium for compact optical gyroscopes," Phys. Rev. E. 71, 056621 (2005). [CrossRef]
- B. Z. Steinberg and A. Boag "Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes," J. Opt. Soc. Am. B. 24, 142-151 (2006). [CrossRef]
- S. Sunada and T. Harayama, "Sagnac effect in resonant microcavities," Phys. Rev. A 74, 021801 (2006). [CrossRef]
- S. Sunada and T. Harayama, "Design of resonant microcavities: application to optical gyroscopes," Opt. Express 15, 16245-16254 (2007). [CrossRef] [PubMed]
- C. Peng, Z. Li, and A. Xu, "Rotation sensing based on a slow-light resonating structure with high group dispersion," Appl. Opt. 46, 4125-4131 (2007). [CrossRef] [PubMed]
- B. Z. Steinberg, J. Scheuer, and A. Boag, "Rotation-induced superstructure in slow-light waveguides with modedegeneracy: optical gyroscopes with exponential sensitivity," J. Opt. Soc. Am. B 24, 1216-1224 (2007). [CrossRef]
- E. J. Post, "Sagnac Effect," Rev. Mod. Phys. 39, 475-493 (1967). [CrossRef]
- B. Z. Steinberg, A. Shamir, and A. Boag, "Two-dimensional Green’s function theory for the electrodynamics of a rotating medium," Phys. Rev. E 74, 016608 (2006). [CrossRef]
- T. Shiozawa, "Phenomenological and electron-theoretical study of the electrodynamics of rotating systems," Proc. IEEE 61, 1694-1702 (1973). [CrossRef]
- J. L. Anderson and J. W. Ryon, "Electromagnetic radiation in accelerated systems," Phys. Rev. 181, 1765-1775 (1969) [CrossRef]
- J. Van Bladel, Relativity and Engineering (Springer, Berlin, 1984)
- J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic," J. Computational Physics 114, 185-200 (1994).
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).
- D. S. Katz, E. T. Thiele, and A. Taflove, "Validation and extension to three dimensions of the Berenger PMLabsorbing boundary condition for FD-TD meshes," Microwave and Guided Wave Letters, IEEE, 4, 268-270 (1994). [CrossRef]
- R. Wang, Y. Zheng, and A. Yao "Generalized Sagnac Effect," Phys. Rev. Lett. 93, 143901 (2004). [CrossRef] [PubMed]
- V. A. Mandelshtam and H. S. Taylor, "Harmonic inversion of time signals," J. Chem. Phys. 107, 6756-6769 (1997). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.