## Optical annular resonators based on radial Bragg and photonic crystal reflectors

Optics Express, Vol. 11, Issue 21, pp. 2736-2746 (2003)

http://dx.doi.org/10.1364/OE.11.002736

Acrobat PDF (827 KB)

### Abstract

A ring resonator based on Bragg reflection is studied in detail. Closed form expressions for the field and dispersion curves for radial Bragg gratings and photonic crystals based resonators are derived and compared to FDTD simulations. For strong confinement, the required gratings exhibit a chirped period and a varying index profile. Small bending radii and low radiation losses are shown to be possible due to the Bragg confinement. The sensitivity of the resonator characteristics to fabrication errors is analyzed quantitatively. A mixed confinement configuration utilizing both Bragg reflection and total internal reflection is also suggested and analyzed.

© 2003 Optical Society of America

## 1. Introduction

1. A. Yariv, “Critical Coupling and Its Control in Optical Waveguide-Ring Resonator Systems,” IEEE Photonics Technol. Lett. **14**, 483–485 (2002). [CrossRef]

4. R. W. Boyd and J. E. Heebner, “Sensitive Disk Resonator Photonic Biosensor,” Appl. Opt. **40**, 5742–5747 (2001). [CrossRef]

5. J. Scheuer and A. Yariv, “Two-dimensional optical ring resonators based on radial Bragg resonance,” Opt. Lett. **28**, 1528–1530 (2003). [CrossRef] [PubMed]

6. P. Yeh and A. Yariv, “Bragg Reflection Waveguides,” Opt. Commun. **19**, 427–430 (1976). [CrossRef]

8. J. Scheuer and A. Yariv, “Annular Bragg-defect-mode Resonators,” J. Opt. Soc. Am. B. **20**2285–2291 (2003). [CrossRef]

9. S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. **81**, 2499–2501 (2002). [CrossRef]

9. S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. **81**, 2499–2501 (2002). [CrossRef]

10. A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. **27**, 936–938 (2002). [CrossRef]

## 2. Resonator design

*ρ*and

*θ*are the radial and azimuthal coordinates respectively and

*k*

_{0}is the wavenumber in vacuum. In order to transform the annular waveguide to a straight one, we utilize the following conformal transformation [11

11. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. **QE-11**, 75–83 (1975). [CrossRef]

*R*is an arbitrary parameter. The transformation (2) maps a circle in the real plane with radius

*R*

_{0}to a straight line at

*U*

_{0}=

*R*·ln(

*R*

_{0}/

*R*). Thus, the structure in Fig. 1A is transformed into a series of straight lines. The wave equation in the (

*U*,

*V*) plane is obtained by transforming (1):

*n*

_{eq}(

*U*)=

*n*(

*U*)exp(

*U*/

*R*) is the profile of the refractive index in the (

*U*,

*V*) plane. It is important to notice that the transformed wave equation (3) is identical to the wave equation in Cartesian coordinates. The corresponding index profile in the (

*U*,

*V*) plane is, however, distorted – increasing exponentially as a function of

*U*. The inverse transformation of (2) is given by:

*U*,

*V*) plane. Since the requirements for a confined straight Bragg waveguide and specifically,

*n*

_{eq}(

*U*) in the (

*U*,

*V*) plane are known [6

6. P. Yeh and A. Yariv, “Bragg Reflection Waveguides,” Opt. Commun. **19**, 427–430 (1976). [CrossRef]

10. A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. **27**, 936–938 (2002). [CrossRef]

*n*(

*ρ*) in the real plane can be simply found by the inverse transformation (4).

*U*,

*V*) plane is transformed to the (

*ρ*,

*θ*) plane. It should be noted that the gratings in the (

*ρ*,

*θ*) plane are spatially “chirped” i.e., their period changes as a function of the radius

*ρ*. In addition, the index of the gratings and their index contrast become smaller for larger

*ρ*. This effect is caused by the 1/

*ρ*factor multiplying the inverse-transformed index as shown in Eq. (4). Similarly, a line defect PC waveguide [7] can be transformed to an annular defect PC resonator.

*R*. This later parameter merely determines to which radius

*U*=0 transforms to. Therefore, using different

*R*for transforming the same line defect waveguide in the (

*U*,

*V*) plane result in different resonator designs in the (

*ρ*,

*θ*) plane. It is important to understand that the equivalent

*n*

_{eq}does not have to be larger than 1 but the real index profile, in the (

*ρ*,

*θ*) plane, must satisfy this requirement (see e.g., Fig 2).

## 3. Modal field properties

*U*,

*V*) plane, the field solution can be readily transformed to the (

*ρ*,

*θ*) plane. The guided modal solution for the transformed problem in the (

*U*,

*V*) plane is given by:

*β*can be either negative or positive, indicating a clockwise or counterclockwise propagating wave. The electric field is given by a superposition of sine and cosine functions in the defect and by Bloch waves in the gratings [6

6. P. Yeh and A. Yariv, “Bragg Reflection Waveguides,” Opt. Commun. **19**, 427–430 (1976). [CrossRef]

*U*

_{L}and

*U*

_{R}are the left and right boundaries of the defect,

*E*

_{K}

*(U)*is a periodic function of

*U*with the same period as the gratings.

5. J. Scheuer and A. Yariv, “Two-dimensional optical ring resonators based on radial Bragg resonance,” Opt. Lett. **28**, 1528–1530 (2003). [CrossRef] [PubMed]

*U*

_{cent}is the coordinate of the center of the defect,

*ε*

_{eq,0}is the equivalent dielectric constant in the core,

*W*is the defect width,

*b*is the perturbation period and

*l*indicates the Bragg order.

*U*,

*V*) plane (see e.g., Fig. 2). Unlike the equivalent waveguide, the modal field in the circular resonator

*E*(

*ρ*,

*θ*)=

*E̅*(

*ρ*)·exp(

*iβRθ*) must satisfy the cyclic boundary condition

*E*(

*ρ*,

*θ*)=

*E*(

*ρ*,

*θ*+2

*π*) and, therefore, the azimuthal propagation coefficient (the phase shift per revolution) must be an integer:

*r*

_{defect}and

*n*

_{defect}are respectively the radius and refractive index of the resonator structure at the defect. For a given

*m*, the field

*E*

_{m}(

*ρ*,

*θ*) corresponds to a mode of the resonator formed by the annular defect. The FSR of this resonator is given by:

*c*is the speed of light in vacuum and

*ν*is the optical frequency. The FSR increases if the second term in the numerator is made as small as possible, i.e., if

*l*=1 and

*b*is as large as possible. If

*b*is large enough that (2

*λ*)≫

*b*

^{-1}then the FSR is given by:

*ρ*(external) side of the defect the loss can be made arbitrarily small. In addition, because of the strong Bragg confinement, the defect can be located at any arbitrary radius.

*a*=0.372µm, rods diameter

*D*=0.6

*a*, and

*λ*=1.56µm. The rods and surroundings refractive indices are

*n*

_{c}=3.5 and

*n*

_{s}=1.5 respectively and the transformation radius is

*R*=3.55µm. The effective index of modal field propagated in the line defect waveguide at

*λ*=1.56µm is approximately

*n*

_{eff}=0.84 (see Fig. 3I). In order to construct a corresponding annular resonator whit that resonance wavelength, the transformation radius must satisfy the followings:

*m*is the azimuthal wavenumber of the modal field. For different

*m*, a different

*R*is needed and accordingly – a different resonator structure. The modal field shown in Fig. 3II has an azimuthal wavenumber of

*m*=12 which corresponds to a transformation radius of

*R*=3.55µm.

*ρ, θ*) plane and compared it with the profile of the annular resonator (Fig. 3II). Figure 4 depicts the comparison between the transformed solution (green) and the resonator modal field calculated by the FDTD simulation (blue). The match is practically perfect – indicating that the analysis method is indeed accurate.

*m*. The circles indicate the resonance wavelengths and the solid line represent a quadratic interpolation with the polynomial coefficients shown in the figure. The resonator FSR around 1.55µm is approximately 28nm and it increases for shorter wavelengths. The dispersion curve was calculated by an FDTD simulation of the structure shown in Fig. 3II with a short pulse (10fs) excitation.

## 4. Sensitivity to fabrication errors

## 5. Mixed confinement methods

*ρ*,

*θ*) plane might approach large values at small radii. This could be problematic because practically, refractive indices cannot be arbitrarily large. This problem, however, could be quite easily solved without sacrificing the advantages of the structure, by confining the mode on the smaller

*ρ*(internal) side of the defect utilizing total internal reflection. A Similar approach was suggested and studied in regular (straight) Bragg waveguides [6

6. P. Yeh and A. Yariv, “Bragg Reflection Waveguides,” Opt. Commun. **19**, 427–430 (1976). [CrossRef]

12. L. Djaloshinski and M. Orenstein, “Disk and Ring Microcavity Lasers and Their Concentric Coupling,” IEEE J. Quantum Electron. **35**, 737–744, 1999. [CrossRef]

*n*

_{L}is the

*real*refractive index on the internal side of the defect,

*U*

_{0}is the coordinate of the right hand side of the waveguide (see Fig. 8I),

*W*is the defect width and

*m*is the Bessel order given by

*m*=

*βR*.

*U*,

*V*) plane is:

*W*is found by requiring the continuity of the field and its derivative at the defect boundaries:

*λ*is the wavelength and the prime indicates a derivative.

*m*impinge on the interface with smaller angles relatively to the perpendicular. Since TIR occurs only for incident angles which are larger than the critical angle, these fields would not be totally reflected from the inner interface and, therefore, would not be confined in the defect. Nevertheless, these waves would not radiate because they would still be confined by the external Bragg reflector.

*m*=16.

## 6. Summary and conclusions

*ρ*. The corresponding modal field profile exhibits similar chirped oscillations in the Bragg reflectors regime. We studied the sensitivity of the device performances to fabrication errors using the FDTD method. While small errors in the holes positions do not introduce substantial performances deviation from the optimal case, errors in the holes radii generate additional low-Q resonances because of the formation of local defects.

## Acknowledgments

## References and links

1. | A. Yariv, “Critical Coupling and Its Control in Optical Waveguide-Ring Resonator Systems,” IEEE Photonics Technol. Lett. |

2. | B. E. Little, “Ultracompact Si-SiO2 microring resonator optical dropping filter,” Opt. Lett. |

3. | C. K. Madsen and J. H. Zhao, |

4. | R. W. Boyd and J. E. Heebner, “Sensitive Disk Resonator Photonic Biosensor,” Appl. Opt. |

5. | J. Scheuer and A. Yariv, “Two-dimensional optical ring resonators based on radial Bragg resonance,” Opt. Lett. |

6. | P. Yeh and A. Yariv, “Bragg Reflection Waveguides,” Opt. Commun. |

7. | K. Sakoda, Optical Properties of Photonic Crystals (Springer, New-York1999). |

8. | J. Scheuer and A. Yariv, “Annular Bragg-defect-mode Resonators,” J. Opt. Soc. Am. B. |

9. | S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee, and J. Kim, “Two-dimensional photonic crystal hexagonal waveguide ring laser,” Appl. Phys. Lett. |

10. | A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. |

11. | M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. |

12. | L. Djaloshinski and M. Orenstein, “Disk and Ring Microcavity Lasers and Their Concentric Coupling,” IEEE J. Quantum Electron. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.5750) Optical devices : Resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 8, 2003

Revised Manuscript: October 10, 2003

Published: October 20, 2003

**Citation**

Jacob Scheuer and Amnon Yariv, "Optical annular resonators based on radial Bragg and photonic crystal reflectors," Opt. Express **11**, 2736-2746 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-21-2736

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### References

- A. Yariv, �??Critical Coupling and Its Control in Optical Waveguide-Ring Resonator Systems,�?? IEEE Photonics Technol. Lett. 14, 483-485 (2002). [CrossRef]
- B. E. Little, �??Ultracompact Si-SiO2 microring resonator optical dropping filter,�?? Opt. Lett. 23, 1570-1572 (1998).
- C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley- Interscience Publications, New-York 1999).
- R. W. Boyd and J. E. Heebner, "Sensitive Disk Resonator Photonic Biosensor," Appl. Opt. 40, 5742-5747 (2001). [CrossRef]
- J. Scheuer and A. Yariv, �??Two-dimensional optical ring resonators based on radial Bragg resonance,�?? Opt. Lett. 28, 1528-1530 (2003). [CrossRef] [PubMed]
- P. Yeh and A. Yariv, �??Bragg Reflection Waveguides,�?? Opt. Commun. 19, 427-430 (1976). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals (Springer, New-York 1999).
- J. Scheuer and A. Yariv, �??Annular Bragg-defect-mode Resonators,�?? J. Opt. Soc. Am. B. 20 2285-2291 (2003). [CrossRef]
- S. Kim, H. Ryu, H. Park, G. Kim, Y. Choi, Y. Lee and J. Kim, �??Two-dimensional photonic crystal hexagonal waveguide ring laser,�?? Appl. Phys. Lett. 81, 2499-2501 (2002). [CrossRef]
- A. Yariv, �??Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,�?? Opt. Lett. 27, 936-938 (2002). [CrossRef]
- M. Heiblum and J. H. Harris, �??Analysis of curved optical waveguides by conformal transformation,�?? IEEE J. Quantum Electron. QE-11, 75-83 (1975). [CrossRef]
- L. Djaloshinski and M. Orenstein, �??Disk and Ring Microcavity Lasers and Their Concentric Coupling,�?? IEEE J. Quantum Electron. 35, 737-744, 1999. [CrossRef]

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