## Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators

Optics Express, Vol. 15, Issue 8, pp. 4452-4473 (2007)

http://dx.doi.org/10.1364/OE.15.004452

Acrobat PDF (326 KB)

### Abstract

We present a generalized formulation for the treatment of both bending (whispering gallery) loss and scattering loss due to edge roughness in microdisk resonators. The results are applicable to microrings and related geometries. For thin disks with radii greater than the bend-loss limit, we find that the finesse limited by the scattering losses induced by edge roughness is independent of radii. While a strong lateral refractive index contrast is necessary to prevent bending losses, unless the radii are of the order of a few microns, lateral air-cladding is detrimental and only enhances scattering losses. The generalized formulation provides a framework for selecting the refractive index contrast that optimizes the finesse at a given radius.

© 2007 Optical Society of America

## 1. Introduction

1. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. **10**, 23952419, (1971). [CrossRef]

2. R. J. Deri and E. Kapon, “Low-Loss III-V Semiconductor Optical Waveguides.” IEEE J. Quantum Electron. , **27**, 626–640, (1991). [CrossRef]

3. V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, “Propagation loss in single-mode GaAs-AlGaAs microring resonators: measurement and model,” J. Lightwave Technol. **19**, 1734–1739, (2001). [CrossRef]

4. M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE. J. Quantum Electron. **QE–19**, 1505–1514, (1983). [CrossRef]

5. B. E. Little and S. T. Chu, “Estimating surface-roughness loss and output coupling in microdisk resonators,” Opt. Lett. **21**, 1390–1392, (1996). [CrossRef] [PubMed]

6. B. E. Little, J. P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. **22**, 4–6, (1997). [CrossRef] [PubMed]

7. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl Phys Lett. **85**, 3693–3695, (2004). [CrossRef]

8. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**, 1515–1530, (2005). [CrossRef] [PubMed]

9. P. Rabiei, “Calculation of Losses in Micro-Ring Resonators With Arbitrary Refractive Index or Shape Profile and Its Applications,” J. Lightwave Technol. **23**, 1295–1301, (2005). [CrossRef]

10. H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. **13**, pp. 1857–1862, (1974). [CrossRef] [PubMed]

## 2. Review of the optical whispering gallery dispersion relation

_{z}(

*r*)

*e*. Here, Ψ

^{imφ}_{z}refers to the axial electric (TM) or magnetic (TE) modal field amplitude and

*m*is the azimuthal quantization number. The Bessel equation for the radial field dependence becomes:

*J*(

_{m}*k̃*

_{1}

*r*) inside the disk boundary, (

*r*<

*R*) and Hankel functions of the first kind,

*H*

_{m}^{(1)}(

*k̃*

_{2}

*r*) outside the disk boundary, (

*r*>

*R*). Here, a complex propagation constant and frequency

*k̃*=

_{j}*n*/

_{j}ω̃*c*are introduced for reasons that will become apparent later. Applying the boundary condition that the axial field components be continuous across the interface gives the complete radial dependence:

*Z*

_{0}is the impedance of free space. Satisfying boundary conditions for both the electric and magnetic fields across the interface results in a dispersion relation that can be written respectively for TM and TE modes as:

11. J. E. Heebner, R. W. Boyd, and Q. Park, “SCISSOR Solitons & other propagation effects in microresonator modified waveguides,” J. Opt. Soc. Am. B. **19**, 722–731, (2002). [CrossRef]

_{i}=

*Q*/

_{i}*m*) vs. normalized radius for a variety of azimuthal mode numbers and index ratios. These results will later be combined with the results of the normalized volume current formulation for edge scattering.

## 3. Review of the volume current method formulation

*exp*(-

*iωt*). A dielectric perturbation on the disk edge may be written as a spatially dependent permittivity distribution Δ

*ε*(

*r*′,

*z*′,

*φ*′). The dielectric perturbation introduces dipole currents which contribute to outwardly radiated (scattered) fields. The expressions for the perturbed current densities manifest themselves in the form of surface-parallel and surface-perpendicular contributions.

*n*

_{1}for the core, and

*n*

_{2}for the cladding),

## 4. Spectral density formulation for edge roughness

*z*. This justifies a decomposition of the radial variation into a Fourier series expansion of corrugation harmonics of azimuthal quantization number,

*M*

*C*(0) =

*σ*

^{2}. A Gaussian correlation function can be defined as:

*f*,

_{s}*f*=

_{s}*M*/2

*πR*and the spectral density is integrated around each integer

*M*.

*S*/2

_{c}*πR*< < 1 allowing the final (trapezoidal) approximation to the integral to hold.

## 5. Far field scattered power

*d*< <

*R*the small polar angle approximations about θ= 90° can be made,

**N**= 4

*πr*

**A**/

*μ*

_{0}is introduced for convenience [9

9. P. Rabiei, “Calculation of Losses in Micro-Ring Resonators With Arbitrary Refractive Index or Shape Profile and Its Applications,” J. Lightwave Technol. **23**, 1295–1301, (2005). [CrossRef]

*P*is the power in the guided mode.

_{g}## 6. TM scattering losses

*R*< <

_{M}*R*and localized to the disk edge surface all radial variables are replaced with the nominal disk radius and the integral is collapsed.

## 7. TE scattering losses

*E*and azimuthal,

_{r}*E*components. While the azimuthal component may be negligible for very low index contrast microresonators, in general it can be quite strong and cannot be neglected. Second, each of these components couples to both polar and azimuthal components of the radiated fields. Third, discontinuities in the planar electric field components exist unless the roughness is locally flat [12

_{φ}12. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Pertubation theory for Maxwell’s equations with shifting material boundaries,” Phys Rev E. **65**, 066611, (2002). [CrossRef]

*σ*< <

*S*). For a treatment of deep perturbations, see Johnson [13

_{c}13. S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl Phys B. **81**, 283–293, (2005). [CrossRef]

*θ*and azimuthal φ components each arising from these modal field components.

*R*< <

_{M}*R*and localized to the disk edge surface, all radial variables can be replaced with the disk radius to collapse the integral.

## 8. Normalized formulation for edge scattering losses

*d*when the height exceeds the wavelength as we will show later. The quality factor can be expressed in normalized units, from which useful limiting approximate forms can be derived. The quality factor is given by the radians per cycle divided by the fractional loss per cycle. Its association with the scattering loss is given by

*Q*

_{s}^{-1}=

*α*/

_{s}R*m*,

*X*, ξ,

*ℓ*respectively are normalized quantities representing the radius

_{c}*X*=

*n*

_{1}2

*πR*/

*λ*, roughness ξ =

*n*

_{1}

*σ*/

*λ*, and correlation length

*∓*=

_{c}*n*

_{1}

*S*/

_{c}*λ*. The superscript symbol

*p*refers to the polarization state (TM, TE

_{r}, or TE

_{φ}). The quantity Γ

*=*

^{p}_{z}*P*/

_{c}*P*is the ratio of power vertically confined to total power guided or simply the

_{g}*vertical*confinement factor. The usual confinement factor for a planar waveguide can be applied here when using the effective index method. This requires solution of the simple planar slab waveguide dispersion relation (

*k*vs.

_{z}*ω*) as in the Kogelnik formulation [10

10. H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. **13**, pp. 1857–1862, (1974). [CrossRef] [PubMed]

*is the*

^{p}_{r}*edge*confinement factor and is defined as the ratio of the intensity at the disk edge, (∣

*E*(

*R*)∣

^{2}/2

*Z*

_{0}) to the characteristic intensity of the mode, (2

*kP*/

_{c}*d*). This requires solution of the simple infinite cylinder whispering gallery mode dispersion relation (

*X,Q*vs.

_{i}*m*,

*n*). The radial mode profile is normalized to a power per unit disk height of

*P*/

_{c}*d*by integrating the azimuthal component of the Poynting vector along the radial dimension from the disk center out to the radiation boundary (

*R*=

_{r}*mλ*/2

*πn*

_{2}). Finally, a geometric factor resulting from the sum/integral of the far-field scattering pattern into azimuthal and polar angles is defined,

*d*>>

*λ*, but where

*d*<<

*R*such that the small polar angle approximation, Eqn. 37 still holds), the scattered radiation is directed outward at

*θ*= 90°. The following expressions result for the polar integrals where the derivation is assisted by making the change of variables

*d*cos

*θ*/

*λ*≡

*τ*′)

*d*<<

*λ*), the following expressions result for the polar integrals:

9. P. Rabiei, “Calculation of Losses in Micro-Ring Resonators With Arbitrary Refractive Index or Shape Profile and Its Applications,” J. Lightwave Technol. **23**, 1295–1301, (2005). [CrossRef]

*φ*through a modified grating equation,

*Xλ*/

*n*

_{1}

*S*) is much wider than that of the polar integral (2

_{c}*X*/

*n*), then the summation term is greatly simplified. For typical fabrication processes, the relation (

*S*<<

_{c}*λ*/2

*n*

_{2}) is generally the case.

*δ*=

*d*/

*λ*. Prior work [8

8. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**, 1515–1530, (2005). [CrossRef] [PubMed]

*N*and

_{θ,ϕ}*N*resulting in geometric factors of (4/3,2/3,2). The corrected set of geometric factors incorporates the cross-terms and a neglected factor of 1/2:

_{ϕ,r}*d*<

*λ*) microresonators, they are equivalent:

1. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. **10**, 23952419, (1971). [CrossRef]

2. R. J. Deri and E. Kapon, “Low-Loss III-V Semiconductor Optical Waveguides.” IEEE J. Quantum Electron. , **27**, 626–640, (1991). [CrossRef]

3. V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, “Propagation loss in single-mode GaAs-AlGaAs microring resonators: measurement and model,” J. Lightwave Technol. **19**, 1734–1739, (2001). [CrossRef]

5. B. E. Little and S. T. Chu, “Estimating surface-roughness loss and output coupling in microdisk resonators,” Opt. Lett. **21**, 1390–1392, (1996). [CrossRef] [PubMed]

8. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**, 1515–1530, (2005). [CrossRef] [PubMed]

*m*). With increasing refractive index, bending losses decrease while edge scattering losses increase. The tradeoff is thus clear: if index ratio is a design variable, one desires enough to negate the effects of bending loss - but only just enough - as any more leads to increased edge scattering loss in a practical device.

## 9. Conclusion

## References and links

1. | P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. |

2. | R. J. Deri and E. Kapon, “Low-Loss III-V Semiconductor Optical Waveguides.” IEEE J. Quantum Electron. , |

3. | V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, “Propagation loss in single-mode GaAs-AlGaAs microring resonators: measurement and model,” J. Lightwave Technol. |

4. | M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE. J. Quantum Electron. |

5. | B. E. Little and S. T. Chu, “Estimating surface-roughness loss and output coupling in microdisk resonators,” Opt. Lett. |

6. | B. E. Little, J. P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. |

7. | M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl Phys Lett. |

8. | M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express |

9. | P. Rabiei, “Calculation of Losses in Micro-Ring Resonators With Arbitrary Refractive Index or Shape Profile and Its Applications,” J. Lightwave Technol. |

10. | H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. |

11. | J. E. Heebner, R. W. Boyd, and Q. Park, “SCISSOR Solitons & other propagation effects in microresonator modified waveguides,” J. Opt. Soc. Am. B. |

12. | S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Pertubation theory for Maxwell’s equations with shifting material boundaries,” Phys Rev E. |

13. | S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl Phys B. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(230.5750) Optical devices : Resonators

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 29, 2007

Revised Manuscript: March 8, 2007

Manuscript Accepted: March 9, 2007

Published: April 3, 2007

**Citation**

John E. Heebner, Tiziana C. Bond, and Jeff S. Kallman, "Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators," Opt. Express **15**, 4452-4473 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4452

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### References

- P. K. Tien, "Light waves in thin films and integrated optics," Appl. Opt. 10, 23952419, (1971). [CrossRef]
- R. J. Deri and E. Kapon, "Low-Loss III-V Semiconductor Optical Waveguides." IEEE J. Quantum Electron., 27, 626-640, (1991). [CrossRef]
- V. Van, P. P. Absil, J. V. Hryniewicz, P. T. Ho, "Propagation loss in single-mode GaAs-AlGaAs microring resonators: measurement and model," J. Lightwave Technol. 19, 1734-1739, (2001). [CrossRef]
- M. Kuznetsov and H. A. Haus, "Radiation loss in dielectric waveguide structures by the volume current method," IEEE. J. Quantum Electron. QE-19, 1505-1514, (1983). [CrossRef]
- B. E. Little and S. T. Chu, "Estimating surface-roughness loss and output coupling in microdisk resonators," Opt. Lett. 21, 1390-1392, (1996). [CrossRef] [PubMed]
- B. E. Little, J. P. Laine, and S. T. Chu, "Surface-roughness-induced contradirectional coupling in ring and disk resonators," Opt. Lett. 22, 4-6, (1997). [CrossRef] [PubMed]
- M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, "Rayleigh scattering, mode coupling, and optical loss in silicon microdisks," Appl Phys Lett. 85, 3693-3695, (2004). [CrossRef]
- M. Borselli, T. J. Johnson, and O. Painter, "Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment," Opt. Express 13, 1515-1530, (2005). [CrossRef] [PubMed]
- P. Rabiei, "Calculation of Losses in Micro-Ring Resonators With Arbitrary Refractive Index or Shape Profile and Its Applications," J. Lightwave Technol. 23, 1295-1301, (2005). [CrossRef]
- H. Kogelnik and V. Ramaswamy, "Scaling rules for thin-film optical waveguides," Appl. Opt. 13, pp. 1857-1862, (1974). [CrossRef] [PubMed]
- J. E. Heebner, R. W. Boyd, and Q. Park, "SCISSOR Solitons & other propagation effects in microresonator modified waveguides," J. Opt. Soc. Am. B. 19, 722-731, (2002). [CrossRef]
- S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Pertubation theory for Maxwell’s equations with shifting material boundaries," Phys Rev E. 65, 066611, (2002). [CrossRef]
- S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl Phys B. 81, 283-293, (2005). [CrossRef]

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