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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 12 — Jun. 9, 2008
  • pp: 9155–9164
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Surface-emitting circular DFB, disk-, and ring-Bragg resonator lasers with chirped gratings: a unified theory and comparative study

Xiankai Sun and Amnon Yariv  »View Author Affiliations


Optics Express, Vol. 16, Issue 12, pp. 9155-9164 (2008)
http://dx.doi.org/10.1364/OE.16.009155


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Abstract

We have developed a theory that unifies the analysis of the modal properties of surface-emitting chirped circular grating lasers. This theory is based on solving the resonance conditions which involve two types of reflectivities of chirped circular gratings. This approach is shown to be in agreement with previous derivations which use the characteristic equations. Utilizing this unified analysis, we obtain the modal properties of circular DFB, disk-, and ring- Bragg resonator lasers. We also compare the threshold gain, single mode range, quality factor, emission efficiency, and modal area of these types of circular grating lasers. It is demonstrated that, under similar conditions, disk Bragg resonator lasers have the highest quality factor, the highest emission efficiency, and the smallest modal area, indicating their suitability in low-threshold, high-efficiency, ultracompact laser design, while ring Bragg resonator lasers have a large single mode range, high emission efficiency, and large modal area, indicating their suitability for high-efficiency, large-area, high-power applications.

© 2008 Optical Society of America

1. Introduction

2. Reflectivities of chirped circular Bragg gratings

Fig. 1. Illustration of chirped circular grating lasers: (a) Circular DFB laser; (b) Disk Bragg resonator laser; (c) Ring Bragg resonator laser. Laser radiation is coupled out of the resonators in vertical direction via the gratings.

In [31

31. X. K. Sun, J. Scheuer, and A. Yariv, “Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers,” IEEE J. Sel. Top. Quantum Electron. 13, 359–366 (2007). [CrossRef]

] we derived a comprehensive coupled-mode theory, by including the effect of resonant vertical radiation, to analyze such chirped circular grating structures in active media. Using the Green’s function method, the effect of vertical radiation is incorporated into the coupled in-plane wave equations, and a set of evolution equations for the amplitudes of the inplane waves is obtained:

{dAdx=uAvBe2iδ·xdBdx=uB+vAe2iδ·x
(1)

where A and B are the amplitudes of the in-plane outward- and inward- propagating cylindrical waves, respectively. x is the normalized radius x=βρ. δ=(β design-β)/β denotes the normalized frequency detuning factor. The coefficients u and v are defined as u=gA-h 1, v=h 1+ih 2, where gA is the normalized gain coefficient. The minimum value of gA required to achieve laser emission will be determined by the resonance condition. h 1 and h 2 are the radiation- and feedback- coupling coefficients, respectively. It should be noted that, although Eqs. (1) seem to be a set of coupled equations for in-plane (vertically confined) waves only, they implicitly include the effect of vertical radiation due to h 1. In terms of obtaining the threshold for the lasing modes, vertical radiation is treated as a loss term.

To solve for A and B, Ã=Ae -iδx and =Beiδx are introduced, and (1) becomes

{dA~dx=(uiδ)A~vB~dB~dx=(uiδ)B~+vA~
(2)

the generic solutions of which are

{A˜(x)=A˜(0)sinh[S(xL)]+cosh[S(xL)]sinh[SL]+cosh[SL]B˜(x)=A˜(0)v[(uiδ)S]sinh[S(xL)]+[(uiδ)S]cosh[S(xL)]sinh[SL]+cosh[SL]
(3)

where S is defined as S=(uiδ)2v2 , ℂ is a constant to be determined by specific boundary conditions, and L is a normalized length parameter.

Fig. 2. Illustration of the two types of boundary conditions for calculating reflectivities. Left: A(0)=B(0), r 1(L)=A(L)/B(L); Right: B(L)=0, r 2(L,x 0)=B(x 0)/A(x 0).

1. The grating extends from the center x=0 to x=L. An inward propagating wave with amplitude B(L) impinges from outside on the grating.

We have to ensure at the center A(0)=B(0),Ã(0)=(0), leading to =(uviδ)sinh[SL]+Scosh[SL]Ssinh[SL]+(uviδ)cosh[SL] and to the reflectivity

r1(L)=A(L)B(L)=A˜(L)eiδLB˜(L)eiδL=e2iδL(uviδ)sinh[SL]+Scosh[SL](uviδ)sinh[SL]+Scosh[SL].
(4)

The grating extends from x=x 0 to x=L. An outward propagating wave with amplitude A(x 0) impinges from inside on the grating.

Since there is no inward propagating wave coming from outside, we invoke B(L)=0,(L)=0, leading to =suiδ and to the reflectivity

r2(x0,L)=B(x0)A(x0)=B˜(x0)eiδx0A˜(x0)eiδx0=e2iδx0vsinh[S(Lx0)](uiδ)sinh[S(Lx0)]Scosh[S(Lx0)].
(5)

It is worth noting that, as seen from their definitions, the above reflectivities include the propagation phase.

Fig. 3. Reflectivities |r 1(100)| and |r 2(x 0,x 0+100)| for different gain levels gA=0, 5×10-4, 1×10-3, and 4×10-3, in the presence of vertical radiation.

3. Resonance conditions for circular DFB, disk-, and ring- Bragg resonator lasers

Utilizing the reflectivities for different boundary conditions, it is possible to derive the resonance conditions for each circular grating laser configuration.

In circular DFB lasers, the limiting cases r 1(xb)→∞ or r 2(0,xb)=1 lead to

tanh[Sxb]=Suviδ.
(6)

Disk Bragg resonator lasers have reflectivity equal to unity at the center so that the resonance condition is 1·e2gAx0·r2(x0,xb)=1 , which reads

e2(gAiδ)x0vsinh[S(xbx0)](uiδ)sinh[S(xbx0)]Scosh[S(xbx0)]=1.
(7)

The resonance condition of ring Bragg resonator lasers is r1(xL)·e2gA(xRxL)·r2(xR,xb)=1 , which reads

e2(gAiδ)(xRxL)(uviδ)sinh[SxL]+Scosh[SxL](uviδ)sinh[SxL]+Scosh[SxL]·vsinh[S(xbxR)](uiδ)sinh[S(xbxR)]Scosh[S(xbxR)]=1.
(8)

Though in much simpler forms, the above resonance conditions are essentially the same as those characteristic equations derived in Refs. [31

31. X. K. Sun, J. Scheuer, and A. Yariv, “Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers,” IEEE J. Sel. Top. Quantum Electron. 13, 359–366 (2007). [CrossRef]

] and [32

32. X. K. Sun and A. Yariv, “Modal properties and modal control in vertically emitting annular Bragg lasers,” Opt. Express 15, 17323–17333 (2007). [CrossRef] [PubMed]

]. The foundation for validity of the “resonance condition theory” is the continuity of the amplitudes of the cylindrical waves at the interfaces. The in-plane electric field is expressed as E(x)=A(x)H (1) m(x)+B(x)H (2) m(x). The continuity conditions for the outward and inward propagating waves require that A(x - 0)=A(x + 0) and B(x - 0)=B(x + 0) hold at any interface x 0. They adequately guarantee the continuity of the in-plane electric field and its first derivative, which are the only requirements for matching the fields at the interfaces in deriving the characteristic equations. The restrictions at the center and exterior boundary are automatically satisfied by choosing proper reflectivities. Therefore, using the resonance condition theory, we simplify and unify the derivation of threshold conditions for the family of circular grating coupled surface emitting lasers.

Table 1. Modal threshold gains (gA), frequency detuning factors (δ), and modal fields of the circular DFB, disk-, and ring- Bragg resonator lasers (xb=200).

table-icon
View This Table

4. Single mode range, quality factor, emission efficiency, and modal area for circular DFB, disk-, and ring- Bragg resonator lasers

4.1 Single mode range

Fig. 4. Evolution of threshold gains of the 5 lowest-order modes of circular DFB, disk-, and ring- Bragg resonator lasers.

4.2 Quality factor

The quality factor Q for optical resonators is usually defined as ωεP where ω denotes the radian resonance frequency, ε the total energy stored in the resonator, and P the power loss. The quality factor is a measure of the speed with which a resonator dissipates its energy. In the circular grating lasers investigated here, the power loss P has two contributions: vertical radiation coupled out of the resonator due to the first-order Bragg diffraction, and peripheral leakage due to the finite radial length of the Bragg reflector.

Jebali et al. recently developed an analytical formalism to calculate the Q factor for firstorder circular grating resonators in a 2D configuration [34

34. A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bachtold, “Analytical calculation of the Q factor for circular-grating microcavities,” J. Opt. Soc. Am. B 24, 906–915 (2007). [CrossRef]

]. Due to the 2D aspect of their model, only in-plane peripheral leakage was considered as the source of power loss. Here we study a 3D case and include vertical radiation in the power loss. Nevertheless, we do not plan to derive an exact expression for Q. Rather, by considering that the energy stored in a volume is proportional to ∫|E|2 dV and that the outflow power through a surface is proportional to ∫|E|2 dS, we define an unnormalized quality factor

Q=0Ddz02πdφ0ρbE(ρ,z)2ρdρgratingΔE(ρ,z=0)2ρdρdφ+0Ddz02πE(ρ=ρb,z)2ρbdφ
=0DZ2(z)dz·0xbE(x)2xdxgratingΔE(x,y=0)2xdx+0DZ2(z)dz·E(x=xb)2βxb
(9)

where Z(z) denotes the vertical mode profile for the layer structure and D the thickness of the resonator. Due to the angular symmetry (for m=0 under consideration), the angular integration factors are canceled out. Detailed expressions for the in-plane field E and radiated field ΔE can be found in [31

31. X. K. Sun, J. Scheuer, and A. Yariv, “Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers,” IEEE J. Sel. Top. Quantum Electron. 13, 359–366 (2007). [CrossRef]

]. To within a scaling factor, the unnormalized quality factor Q′ is essentially the same as the usually defined Q, however it is more intuitive and convenient for calculational purposes. The results of our calculations for the unnormalized quality factor for each laser configuration are displayed in Fig. 5. As expected, increases in the device size (xb) resulted in an enhanced Q′ value. Additionally, the disk Bragg resonator lasers exhibited a much higher Q′ than the other laser structures of identical dimensions. As an example, for xb=100, the Q′ value of the disk Bragg resonator device is approximately 3 times greater than that of the circular DFB or ring Bragg resonator structures. This is in agreement with their lower-threshold behavior shown in Table 1.

Fig. 5. Unnormalized quality factors of circular DFB, disk-, and ring- Bragg resonator lasers.

4.3 Emission efficiency

It is natural to define the emission efficiency η as the fraction of the total power loss which is represented by the useful vertical radiation. Figure 6 depicts the η of the first lasing modes, within each single mode range, for each device structure. As expected, all the lasers possess a larger η with a larger device size. Comparing devices of identical dimensions, only the diskand ring- Bragg resonator lasers achieve high emission efficiencies. This is a result of the first lasing modes of these structures being bandgap modes while the first lasing mode of the circular DFB laser is a bandgap-edge (in-band) mode. Bandgap modes experience much stronger reflection from the Bragg gratings, yielding less peripheral power leakage than inband modes.

Fig. 6. Emission efficiencies of circular DFB, disk-, and ring- Bragg resonator lasers.

4.4 Modal area

Fig. 7. Modal areas of circular DFB, disk-, and ring- Bragg resonator lasers. Their top surface area (πx 2 b) is marked with a black dashed line as a reference.
Fig. 8. Comparison of radial electric field profiles of ring Bragg resonator lasers (xb=90, 113, and 150). For xb=113, the electric field has equal amplitude at the center (x=0) and the defect (x=xb/2).

5. Summary

Acknowledgment

This work was supported in part by the Defense Advanced Research Projects Agency (DARPA) and in part by the National Science Foundation. X. Sun is grateful to J. Sendowski for his kind help in paper revision.

References and links

1.

S. A. Shakir, T. C. Salvi, and G. C. Dente, “Analysis of Grating-Coupled Surface-Emitting Lasers,” Opt. Lett. 14, 937–939 (1989). [CrossRef] [PubMed]

2.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, “Low-Threshold Grating-Coupled Surface-Emitting Lasers,” Appl. Phys. Lett. 55, 813–815 (1989). [CrossRef]

3.

R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, “High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers,” Electron. Lett. 26, 125–127 (1990). [CrossRef]

4.

T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, “Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning,” IEEE Photon. Technol. Lett. 5, 1149–1152 (1993). [CrossRef]

5.

F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, “Very Low Threshold 1.55 µm Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect,” Appl. Phys. Lett. 67, 2777–2779 (1995). [CrossRef]

6.

R. G. Waarts, “Optical Characterization of Grating Surface Emitting Semiconductor Lasers,” Appl. Opt. 29, 2718–2721 (1990). [CrossRef] [PubMed]

7.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor lasers: An analysis,” J. Appl. Phys. 68, 1435–1444 (1990). [CrossRef]

8.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, “Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating,” Electron. Lett. 27, 1819–1821 (1991). [CrossRef]

9.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992). [CrossRef]

10.

C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, “Room temperature operation of electrically pumped Surface-Emitting Circular Grating DBR Laser,” Electron. Lett. 28, 1037–1039 (1992). [CrossRef]

11.

C. Wu, T. Makino, M. Fallahi, R. G. A. Craig, G. Knight, I. Templeton, and C. Blaauw, “Novel Circular Grating Surface-Emitting Lasers with Emission from Center,” Jpn. J. Appl. Phys. 33-Pt. 2, L427–L429 (1994). [CrossRef]

12.

R. H. Jordan, D. G. Hall, O. King, G. W. Wicks, and S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997). [CrossRef]

13.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998). [CrossRef]

14.

K. J. Kasunic and E. M. Wright, “Nonlinear dynamics of circular-grating distributed-feedback semiconductor devices,” J. Opt. Soc. Am. B 16, 96–102 (1999). [CrossRef]

15.

A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, “Above-Threshold Analysis of Second-Order Circular-Grating DFB Lasers,” IEEE J. Quantum Electron. 36, 259–267 (2000). [CrossRef]

16.

C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, “A Surface-Emitting Circular Grating Polymer Laser,” Adv. Mater. 13, 1161–1164 (2001). [CrossRef]

17.

P. L. Greene and D. G. Hall, “Effects of Radiation on Circular-Grating DFB Lasers—Part I: Coupled-Mode Equations,” IEEE J. Quantum Electron. 37, 353–364 (2001). [CrossRef]

18.

P. L. Greene and D. G. Hall, “Effects of Radiation on Circular-Grating DFB Lasers—Part II: Device and Pump-Beam Parameters,” IEEE J. Quantum Electron. 37, 364–371 (2001). [CrossRef]

19.

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, “Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings,” IEEE J. Quantum Electron. 39, 1086–1098 (2003). [CrossRef]

20.

G. F. Barlow, A. Shore, G. A. Turnbull, and I. D. W. Samuel, “Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser,” J. Opt. Soc. Am. B 21, 2142–2150 (2004). [CrossRef]

21.

W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, “Vertically emitting annular Bragg lasers using polymer epitaxial transfer,” Appl. Phys. Lett. 85, 3669–3671 (2004). [CrossRef]

22.

A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G.-L. Bona, and W. Bachtold, “Lasing in organic circular grating structures,” J. Appl. Phys. 96, 3043–3049 (2004). [CrossRef]

23.

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, “Low-threshold two-dimensional annular Bragg lasers,” Opt. Lett. 29, 2641–2643 (2004). [CrossRef] [PubMed]

24.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, “Lasing from a circular Bragg nanocavity with an ultrasmall modal volume,” Appl. Phys. Lett. 86, 251101 (2005). [CrossRef]

25.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraouhi, T. F. Krauss, K. A. Shore, and I. D. W. Samuel, “Influence of grating characteristics on the operation of circular-grating distributed-feedback polymer lasers,” J. Appl. Phys. 98, 023105 (2005). [CrossRef]

26.

G. A. Turnbull, A. Carleton, A. Tahraouhi, T. F. Krauss, I. D. W. Samuel, G. F. Barlow, and K. A. Shore, “Effect of gain localization in circular-grating distributed feedback lasers,” Appl. Phys. Lett. 87, 201101 (2005). [CrossRef]

27.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, “Smallest possible electromagnetic mode volume in a dielectric cavity,” IEE Proc.-Optoelectron. 145, 391–397 (1998). [CrossRef]

28.

E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).

29.

J. Scheuer and A. Yariv, “Coupled-Waves Approach to the Design and Analysis of Bragg and Photonic Crystal Annual Resonators,” IEEE J. Quantum Electron. 39, 1555–1562 (2003). [CrossRef]

30.

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

31.

X. K. Sun, J. Scheuer, and A. Yariv, “Optimal design and reduced threshold in vertically emitting circular Bragg disk resonator lasers,” IEEE J. Sel. Top. Quantum Electron. 13, 359–366 (2007). [CrossRef]

32.

X. K. Sun and A. Yariv, “Modal properties and modal control in vertically emitting annular Bragg lasers,” Opt. Express 15, 17323–17333 (2007). [CrossRef] [PubMed]

33.

R. F. Kazarinov and C. H. Henry, “Second-Order Distributed Feedback Lasers with Mode Selection Provided by First-Order Radiation Losses,” IEEE J. Quantum Electron. QE-21, 144–150 (1985). [CrossRef]

34.

A. Jebali, D. Erni, S. Gulde, R. F. Mahrt, and W. Bachtold, “Analytical calculation of the Q factor for circular-grating microcavities,” J. Opt. Soc. Am. B 24, 906–915 (2007). [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(130.0130) Integrated optics : Integrated optics
(130.2790) Integrated optics : Guided waves
(140.5960) Lasers and laser optics : Semiconductor lasers
(230.1480) Optical devices : Bragg reflectors
(250.7270) Optoelectronics : Vertical emitting lasers

ToC Category:
Optoelectronics

History
Original Manuscript: March 10, 2008
Revised Manuscript: June 3, 2008
Manuscript Accepted: June 3, 2008
Published: June 5, 2008

Citation
Xiankai Sun and Amnon Yariv, "Surface-emitting circular DFB, disk- and ring- Bragg resonator lasers with chirped gratings: a unified theory and comparative study," Opt. Express 16, 9155-9164 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-9155


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References

  1. S. A. Shakir, T. C. Salvi, and G. C. Dente, "Analysis of Grating-Coupled Surface-Emitting Lasers," Opt. Lett. 14, 937-939 (1989). [CrossRef] [PubMed]
  2. D. F. Welch, R. Parke, A. Hardy, W. Streifer, and D. R. Scifres, "Low-Threshold Grating-Coupled Surface-Emitting Lasers," Appl. Phys. Lett. 55, 813-815 (1989). [CrossRef]
  3. R. Parke, R. Waarts, D. F. Welch, A. Hardy, and W. Streifer, "High-Efficiency, High Uniformity, Grating Coupled Surface Emitting Lasers," Electron. Lett. 26, 125-127 (1990). [CrossRef]
  4. T. Kjellberg, M. Hagberg, N. Eriksson, and A. G. Larsson, "Low-Threshold Grating-Coupled Surface-Emitting Lasers with Etch-Stop Layer for Precise Grating Positioning," IEEE Photon. Technol. Lett. 5, 1149-1152 (1993). [CrossRef]
  5. F. S. Choa, M. H. Shih, J. Y. Fan, G. J. Simonis, P. L. Liu, T. Tanbunek, R. A. Logan, W. T. Tsang, and A. M. Sergent, "Very Low Threshold 1.55 ?m Grating-Coupled Surface-Emitting Lasers for Optical Signal Processing and Interconnect," Appl. Phys. Lett. 67, 2777-2779 (1995). [CrossRef]
  6. R. G. Waarts, "Optical Characterization of Grating Surface Emitting Semiconductor Lasers," Appl. Opt. 29, 2718-2721 (1990). [CrossRef] [PubMed]
  7. T. Erdogan and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990). [CrossRef]
  8. C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optical Pumped Surface-Emitting DFB GaInAsP/InP Lasers with Circular Grating," Electron. Lett. 27, 1819-1821 (1991). [CrossRef]
  9. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992). [CrossRef]
  10. C. Wu, M. Svilans, M. Fallahi, I. Templeton, T. Makino, J. Glinski, R. Maciejko, S. I. Najafi, C. Maritan, C. Blaauw, and G. Knight, "Room temperature operation of electrically pumped Surface-Emitting Circular Grating DBR Laser," Electron. Lett. 28, 1037-1039 (1992). [CrossRef]
  11. C. Wu, T. Makino, M. Fallahi, R. G. A. Craig, G. Knight, I. Templeton, and C. Blaauw, "Novel Circular Grating Surface-Emitting Lasers with Emission from Center," Jpn. J. Appl. Phys. 33-Pt. 2, L427-L429 (1994). [CrossRef]
  12. R. H. Jordan, D. G. Hall, O. King, G. W. Wicks, and S. Rishton, "Lasing behavior of circular grating surface-emitting semiconductor lasers," J. Opt. Soc. Am. B 14, 449-453 (1997). [CrossRef]
  13. C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998). [CrossRef]
  14. K. J. Kasunic and E. M. Wright, "Nonlinear dynamics of circular-grating distributed-feedback semiconductor devices," J. Opt. Soc. Am. B 16, 96-102 (1999). [CrossRef]
  15. A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, "Above-Threshold Analysis of Second-Order Circular-Grating DFB Lasers," IEEE J. Quantum Electron. 36, 259-267 (2000). [CrossRef]
  16. C. Bauer, H. Giessen, B. Schnabel, E.-B. Kley, C. Schmitt, and U. Scherf, "A Surface-Emitting Circular Grating Polymer Laser," Adv. Mater. 13, 1161-1164 (2001). [CrossRef]
  17. P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part I: Coupled-Mode Equations," IEEE J. Quantum Electron. 37, 353-364 (2001). [CrossRef]
  18. P. L. Greene and D. G. Hall, "Effects of Radiation on Circular-Grating DFB Lasers??????Part II: Device and Pump-Beam Parameters," IEEE J. Quantum Electron. 37, 364-371 (2001). [CrossRef]
  19. A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron 39, 1086-1098 (2003). [CrossRef]
  20. G. F. Barlow, A. Shore, G. A. Turnbull, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser," J. Opt. Soc. Am. B 21, 2142-2150 (2004). [CrossRef]
  21. W. M. J. Green, J. Scheuer, G. DeRose, and A. Yariv, "Vertically emitting annular Bragg lasers using polymer epitaxial transfer," Appl. Phys. Lett. 85, 3669-3671 (2004). [CrossRef]
  22. A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, G.-L. Bona, and W. Bachtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96, 3043-3049 (2004). [CrossRef]
  23. J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, "Low-threshold two-dimensional annular Bragg lasers," Opt. Lett. 29, 2641-2643 (2004). [CrossRef] [PubMed]
  24. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005). [CrossRef]
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