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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 8 — Apr. 14, 2008
  • pp: 5874–5875
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Reciprocal transmissions and asymmetric modal distributions in waveguide-coupled spiral-shaped microdisk resonators: Comment

Jan Wiersig  »View Author Affiliations


Optics Express, Vol. 16, Issue 8, pp. 5874-5875 (2008)
http://dx.doi.org/10.1364/OE.16.005874


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Abstract

In a recent article [Opt. Express 15, 14650 (2007)] Lee et al. claimed that optical modes in spiral-shaped microcavities come in pairs of clockwise and counterclockwise traveling-wave modes having the same frequencies and Q-factors but different modal distributions. In this comment, we show that the opposite is true: the modes are in general nondegenerate in terms of frequencies and Q-factors and the modal distributions are similar.

© 2008 Optical Society of America

Fig. 1. (Color online) Calculated magnetic field intensity of mode 1 (a) and 2 (b). Distribution of angular momentum α (1) m (solid line) and α (2) m (dashed) normalized to 1 at maximum: (c) absolute value squared, (d) real and (e) imaginary part. (f) Superpositions α + m=(α (1) m+α (2) m)/2 (solid) and α - m=(α (1) m-α (2) m)/2 (dashed, scaled by a factor of 5).

A deeper understanding can be gained by expanding the z-component of the internal magnetic field in cylindrical harmonics Hz(r,ϕ)=∑ m=-∞ αmJm(nkr)exp(imϕ) with wave number k and the mth order Bessel function Jm of the first kind [1

1. G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing InGaN multiple-quantum-well spiral-shaped micropillars” Appl. Phys. Lett. 83, 1710–1712 (2003). [CrossRef]

]. Positive (negative) values of the angular momentum index m correspond to CCW (CW) traveling-wave components. Figure 1(c) shows that also the angular momentum distributions |αm|2 are almost indistinguishable. For both modes the CCW component dominates, i.e. none of the two modes can be classified as CW traveling-wave mode as opposed to the assumption in Ref. [2

2. J. Y. Lee, X. Luo, and A. W. Poon, “Reciprocal transmissions and asymmetric modal distributions in waveguidecoupled spiral-shaped microdisk resonators,” Opt. Express 15, 14650–14666 (2007). [CrossRef] [PubMed]

]. The difference between the modes is revealed by Figs. 1(d) and (e). If m<0 then both the real and the imaginary part of αm have a different sign for the two modes. Consequently, we can construct superpositions with α ± m=(α (1) m±α (2) m)/2 being CW and CCW traveling-waves, respectively; see Fig. 1(f). However, these superpositions are not eigenmodes of the cavity as they are composed of two modes with different frequencies and Q-factors. In Ref. [2

2. J. Y. Lee, X. Luo, and A. W. Poon, “Reciprocal transmissions and asymmetric modal distributions in waveguidecoupled spiral-shaped microdisk resonators,” Opt. Express 15, 14650–14666 (2007). [CrossRef] [PubMed]

] such kinds of superpositions are plotted due to the chosen excitation by incoming waves. The small but finite splitting λ 2-λ 1 leads to a beating phenomenon in the time domain which can be interpreted as scattering between CW and CCW components. The difference in intensity in Fig. 1(f) reflects the fact that the notch scatters CW traveling-waves more efficiently into CCW traveling-waves than the other way round. Note that the difference in intensity is not in conflict with the reciprocity relations as the wave equation is linear. In the case of the nonlinear laser dynamics, however, the difference in intensity ensures that CCW traveling-waves dominate, consistent with experiments [1

1. G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing InGaN multiple-quantum-well spiral-shaped micropillars” Appl. Phys. Lett. 83, 1710–1712 (2003). [CrossRef]

].

References and links

1.

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing InGaN multiple-quantum-well spiral-shaped micropillars” Appl. Phys. Lett. 83, 1710–1712 (2003). [CrossRef]

2.

J. Y. Lee, X. Luo, and A. W. Poon, “Reciprocal transmissions and asymmetric modal distributions in waveguidecoupled spiral-shaped microdisk resonators,” Opt. Express 15, 14650–14666 (2007). [CrossRef] [PubMed]

3.

J. Wiersig, “Boundary element method for resonances in dielectric microcavities” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003). [CrossRef]

OCIS Codes
(230.3990) Optical devices : Micro-optical devices
(230.5750) Optical devices : Resonators

ToC Category:
Optical Devices

History
Original Manuscript: January 25, 2008
Revised Manuscript: March 7, 2008
Manuscript Accepted: March 11, 2008
Published: April 11, 2008

Citation
Jan Wiersig, "Reciprocal transmissions and asymmetric modal distributions in waveguide-coupled spiral-shaped microdisk resonators: Comment," Opt. Express 16, 5874-5875 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5874


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References

  1. G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, "Unidirectional lasing InGaN multiple-quantum-well spiral-shaped micropillars" Appl. Phys. Lett. 83, 1710-1712 (2003). [CrossRef]
  2. J. Y. Lee, X. Luo, and A.W. Poon, "Reciprocal transmissions and asymmetric modal distributions in waveguidecoupled spiral-shaped microdisk resonators," Opt. Express 15, 14650-14666 (2007). [CrossRef] [PubMed]
  3. J. Wiersig, "Boundary element method for resonances in dielectric microcavities" J. Opt. A: Pure Appl. Opt. 5, 53-60 (2003). [CrossRef]

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