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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 17323–17333
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Modal properties and modal control in vertically emitting annular Bragg lasers

Xiankai Sun and Amnon Yariv  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 17323-17333 (2007)
http://dx.doi.org/10.1364/OE.15.017323


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Abstract

The modal properties, including the resonant vertical radiation, of a type of laser structures based on the annular Bragg resonance (ABR) are studied in detail. The modal threshold gains and the resonance frequencies of such lasers are obtained from the derived governing characteristic equation. Two kinds of ABR lasers, one with a π/2 phase shift in the outer grating and the other without, are analyzed. It is numerically demonstrated that, it’s possible to get a large-area, high-efficiency, single defect mode lasing in ABR lasers if we choose the kind without a π/2 phase shift in the outer grating and also a device size smaller than a critical value.

© 2007 Optical Society of America

1. Introduction

2. Comprehensive coupled mode theory

Fig. 1. Illustration of an annular Bragg laser.

As illustrated in Fig. 1, an annular Bragg laser consists of a circumferentially guiding defect and the surrounding annular Bragg gratings in a gain medium. The inner grating spans from the center to ρL while the outer grating spans from ρR to ρb. In the case that the polarization effects due to the waveguide structure are not concerned, we can introduce the “weak guidance approximation,” under which all the field components can be obtained from the z component of the electric field which satisfies the scalar wave equation in cylindrical coordinates

[1ρρ(ρρ)+1ρ22φ2+k02n2(ρ,z)+2z2]Ez(ρ,φ,z)=0,
(1)

where k0=ωc=2πλ0 is the wave number in vacuum. For an azimuthally propagating eigenmode, the Ez in a passive uniform medium in which the dielectric constant n 2(ρ,z)=εr(z) can be expressed as

Ez(ρ,φ,z)=Ez(m)(ρ,z)exp(imφ)
=[AHm(1)(βρ)+BHm(2)(βρ)]Z(z)exp(imφ),
(2)

where m is the azimuthal mode number, β=k 0 neff is the in-plane propagation constant, and Z(z) is the fundamental mode profile of the planar slab waveguide satisfying

(k02εr(z)+2z2)Z(z)=β2Z(z).
(3)

In a radially perturbed gain medium, the dielectric constant can be expressed as n 2(ρ,z)=εr(z)+i(z)+Δε(ρ,z) where |εi(z)|<<εr(z) represents the gain/loss and Δε(ρ,z) reflects the contribution of perturbation. For optimal field confinement the perturbation Δε(ρ,z) has to be expanded in Hankel-phased plane wave series [8

8. 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]

]

Δε(ρ,z)=Δε0l=±1,±2al(z)exp(ilΦ[Hm(1)(βdesignρ)])
=Δε0l=±1,±2al(z)exp(ilΦ[Hm(1)(x)])exp(ilδ·x)
=Δε0(a2(z)Hm(2)Hm(1)e2iδ·x+a2(z)Hm(1)Hm(2)e2i·x+a1(z)Hm(2)Hm(1)eiδ·x+a1(z)Hm(1)Hm(1)eiδ·x),
(4)

In the above expression, al(z) is the expansion coefficient of Δε(ρ,z) at a given z. x is the normalized radius defined as x=βρ. δ=(β design-β)/β (|δ|≪1), the normalized frequency detuning factor, represents the relative frequency shift from the optimal coupling design.

To account for the vertically radiating fields, we include an additional term ΔE(x,z) so that

Ez(m)(x,z)=[A(x)Hm(1)(x)+B(x)Hm(2)(x)]Z(z)+ΔE(x,z).
(5)

Assuming that the radiating field ΔE(x,z) has an exp(±ik 0 z) dependence on z in free space, i.e.

[1ρρ(ρρ)m2ρ2]ΔE=0,
(6)

substituting (4), (5), (6) into (1), introducing the large-radius approximations [8

8. 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]

]

Hm(1,2)(x)x<<dHm(1,2)(x)dx,dnHm(1,2)(x)dxn(±i)nHm(1,2)(x),
(7)

2iZ(dAdxHm(1)dBdxHm(2))+ik02εiβ2(AHm(1)Z+BHm(2)Z)+1β2(k02εr+ik02εi+2z2)ΔE
=k02Δε0β2(a2Hm(2)Hm(1)e2iδ·x+a2Hm(1)Hm(2)e2iδ·x
+a1Hm(2)Hm(1)eiδ·x+a1Hm(1)Hm(1)eiδ·x)(AHm(1)Z+BHm(2)Z+ΔE).
(8)

The phase-matching condition requires that the source and wave have close phase dependence. Grouping the terms with the same kind of Hankel functions leads to the following set of coupled equations

{2idAdxHm(1)Z+ik02εiβ2AHm(1)Z=k02Δε0β2(a2BHm(1)e2iδ·xZ+a1ΔEHm(1)Hm(1)eiδ·x)(a)2idBdxHm(2)Z+ik02εiβ2BHm(2)Z=k02Δε0β2(a2AHm(2)e2iδ·xZ+a1ΔEHm(1)Hm(2)eiδ·x)(b)(k02εr+2z2)ΔE=k02Δε0(a1AHm(1)eiδ·xZ+a1BHm(1)eiδ·xZ)(c)
(9)

From (9c), ΔE can be expressed as

ΔE=(s1Aeiδ·x+s1Beiδ·x)Hm(1),
(10)

where

sl(z)=k02Δε0+al(z')Z(z')G(z,z')dz',
(11)

and G(z, z′) is the Green’s function satisfying (k02εr(z)+2z2)G(z,z')=δ(zz')..

Substituting (10) into (9a) and (9b), multiplying both sides by Z(z) and integrating over z, we arrive at

{dAdx=(gAh1,1)A(h1,1+ih2)Be2iδ·xdBdx=(gAh1,1)B+(h1,1+ih2)Ae2iδ·x,
(12)

where the gain coefficient gAk022Pβ2+εi(z)Z2(z)dz, the radiation coupling coefficients h±1,±1=ik02Δε02Pβ2+a±1(z)s±1(z)Z(z)dz, the feedback coupling coefficient h2=h±2=k02Δε02Pβ2+a±2(z)Z2(z)dz, and the normalization constant P≡∫+∞ -∞ Z 2(z)dz.

In the case of index grating, we can choose the phase of the grating such that a -1=a 1, then all the radiation coupling coefficients are the same and can be denoted as h 1. Let u=gA-h 1 and ν=h 1+ih 2, then the generic solution to (12) is

{A(x)=[C1exp(Sx)+C2exp(Sx)]exp(iδ·x)B(x)=1ν[C1(Su+iδ)exp(Sx)C2(S+uiδ)exp(Sx)]exp(iδ·x),
(13)

where S(uiδ)2ν2. In analogy to the case of a linear grating [11

11. A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, New York, 1997).

], the modes with a real S manifest themselves as band-gap modes since they are located within the band gap in the band diagram and their fields are reflected in the grating region. They are mostly confined in the guiding defect so that they are also termed as “defect modes.” In the unperturbed region where Δε=0, we have h 1=h 2=0, and the solution to (12) is simply

{A(x)=A(0)exp(gAx)B(x)=B(0)exp(gAx).
(14)

3. Modal fields and characteristic equation of annular Bragg lasers

For an ABR laser as shown in Fig. 1, the electric field E (m) z(x,z) in different regions takes different forms

Ez(m)(x,z)={A1(x)Hm(1)(x)Z(z)+B1(x)Hm(2)(x)Z(z)+ΔE1(x,z),regionI:x<xLA2egAxHm(1)(x)Z(z)+B2egAxHm(2)(x)Z(z),regionII:xL<x<xRA3(x)Hm(1)(x)Z(z)+B3(x)Hm(2)(x)Z(z)+ΔE3(x,z)regionIII:xR<x<xb.
(15)

where xL, xR, and xb are normalized ρL, ρR, and ρb, respectively.

Designed in a passive model, the demonstrated ABR lasers in [9

9. 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]

] introduced a π/2 phase shift in their outer gratings. This, however, as will be discussed later, is unfavorable for single defect mode operation. Therefore, we will study two cases: (1) the outer grating (region III) has an additional π/2 phase shift compared to the inner grating (region I); (2) both the inner grating and the outer grating have the same phase dependence Φ[H (1) m(x)]. So in case (1), we need to change a 1 to ia 1, a -1 to -ia -1, and a 2 to -a 2 in region III. From their definitions, h 1,1, h -1,-1 and h 2 have a sign flip while h 1,-1 and h -1,1 keep the same, which means that the additional phase shift doesn’t have an effect on the vertical radiation mechanism. Thus in region III, A 3 and B 3 can still be expressible as (13) provided that we replace v by v′=-v. For the same reason, the radiation field ΔE 3=(s1 Ae -iδ·x+s′ -1 Be iδ·x)|H (1) m| where s′ 1=is 1 and s′ 1=is -1.

We invoke the following boundary conditions for TE modes:

(1) At the center x=0, the total amplitude must remain finite and it should be satisfied at any z. Since in region I, E(x,z)=A 1(x)H (1) m(x)Z(z)+B 1(x)H (2) m(x)Z(z)+ΔE 1(x,z) and |ΔE 1(x,z)|≪|A 1(x)H (1) m(x)Z(z)+B 1(x)H (2) m(x)Z(z)|, we can set A 1(0)=B 1(0).

(2) At the exterior boundary xb, no incoming wave comes from outside (x>xb), thus B 3(xb)=0.

(3) At the interfaces xL and xR, the electric field Ez is continuous, i.e., EI(xL)=EII(xL) and EII(xR)=EIII(xR).

(4) At the interfaces xL and xR, the first order derivative of the electric field E′z is continuous, i.e., E′I(xL)=E′II(xL) and E′II(xR)=E′III(xR).

By matching the boundary conditions (1) and (2), then multiplying by Z(z) and integrating over z, we get the integrated E (m) z(x) in the 3 different regions:

{EI(x)=PC11[eS·x+iδ·x+Su+ν+iδS+uνiδeS·x+iδ·x]Hm(1)(x)PC11ν[(Su+iδ)eS·xiδ·xSu+ν+iδS+uνiδ(S+uiδ)eS·xiδ·x]Hm(2)(x)EII(x)=P[A2egAxHm(1)(x)+B2egAxHm(2)(x)]EIII(x)=PC31eS·x+iδ·x[1+Su+iδS+uiδe2S(xbx)]Hm(1)(x)PC31(Su+iδ)ν'eS·xiδ·x[1e2S(xbx)]Hm(2)(x),
(16)

where P is the normalization constant defined before. By satisfying the boundary conditions (3) and (4), we finally arrive at the characteristic equation for the annular Bragg lasers:

(gA+i)(LHS)I1(gA+i)(LHS)I+1.(gA+i)(RHS)III+1(gA+i)(RHS)III1=e2gA(xRxL)Hm(1)(xR)Hm(2)(xR).Hm(2)(xL)Hm(1)(xL),
(17)

where

(LHS)I={[eS·xL+iδ·xL+Su+ν+iδS+uνiδ]Hm(1)(xL)1ν[(Su+iδ)eS·xLiδ·xLSu+ν+iδS+uνiδ(S+uiδ)eS·xLiδ·xL]Hm(2)(xL)}{[(S+i(δ+1))eS·xL+iδ·xL+Su+ν+iδS+uνiδ(S+i(δ+1))eS·xLiδ·xL]Hm(1)(xL)1ν[(Su+iδ)(Si(δ+1))eS·xLiδ·xL+Su+ν+iδS+uνiδ(S+i(δ+1))eS·xLiδ·xL]Hm(2)(xL)}

and

(RHS)III=eS·xR+iδ·xR[1+Su+iδS+uiδe2S(xbxR)]Hm1(xR)(Su+iδ)v'eS·xRiδ·xR[1e2S(xbXR)]Hm(2)(xR){(S+i(δ+1))eS·xR+iδ·xR[1+Su+iδS+uiδe2S(xbxR)]Hm(1)(xR)2SSu+iδS+uiδeS·xR+iδ·xRe2S(xbxR)Hm(1)(xR)(Su+iδ)v'(Si(δ+1)eS·xR+iδ·xR[1e2S(xbxR)]Hm(2)(xR)2S(Su+iδ)v'eS·xRiδ·xRe2S(xbxR)Hm(2)(xR)}.

4. Numerical results and modal control in annular Bragg lasers

Without loss of generality, we assume an annular Bragg laser fabricated in a layer structure as described in [12

12. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, “InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties,” IEEE J. Sel. Top. Quantum Electron. 11, 476–484 (2005). [CrossRef]

] which was designed for 1.55-µm laser emission. We approximate the complicated layer structure by an effective index profile comprising five layers: lower cladding, n=1.54; first layer, n=3.281 and thickness of 60.5 nm; second layer (the active region), n=3.4057 and thickness of 129 nm; third layer, n=3.281 and thickness of 60.5 nm; upper cladding, n=1.54. Numerical calculations of the mode profile and the effective index of the approximated layer structure indicate negligible deviations from those of the exact one. Here we focus our analysis on the case of a shallow grating with an etch depth of ~185 nm. The vertical mode profile Z(z), the effective index neff, and the Green’s function are numerically calculated. For the in-plane grating, we assume a rectangular profile with a Hankel-phased modulation [8

8. 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]

]Θ(Φ[Hm(1)(x)],α)={1,cos(Θ[Hm(1)(x)])α,which can be0,cos(Θ[Hm(1)(x)])<α, which can be expanded in Fourier series as

Θ(Φ[Hm(1)(x)],α)=arccosαπ+2πl=1sin(larccosα)lcos(lΦ[Hm(1)(x)]).
(18)

This yields the expansion coefficients a2=a2=sin(2πdc)2π and a1=a1=sin(2πdc)2π where dcarccosαπ(1<α<1,0<dc<1) is the duty cycle of the Hankel-phase-modulated rectangular grating. We have pointed out in [10

10. 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]

] that, to get both strong radiation coupling out of the resonator and in-plane feedback from the grating, dc=0.25 is a good choice since h 2 is maximal while Re(h 1) is not small. For m=0, we get h 1=0.0072+0.0108i and h 2=0.0601.

Table 1. Modal threshold gains, detuning factors, and modal field patterns of the ABR lasers (xb=200) which have a π/2 phase shift in the outer grating.

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Table 2. Modal threshold gains, detuning factors, and modal field patterns of the ABR lasers (xb=200) which have the same phase dependence in the inner and outer gratings.

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In such grating-coupled surface emitting lasers, the total power loss is composed of two contributions: the coherently scattered, vertically emitted light comprises our useful signal, while the in-plane transverse loss from the resonator is the power leakage [10

10. 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 define the emission efficiency η as the ratio between the useful vertical radiation power and the total power loss. We vary the exterior boundary radius xb while fixing the defect size and locating the defect always at the middle (xb/2), and calculate η for both the defect mode and the first in-band mode as a function of xb. The results are plotted in Fig. 2. As can be seen, the emission efficiency, for both modes, improves as the device size (xb) increases, and more impressively, the defect mode has much higher emission efficiency than the first in-band mode for the same device size.

Fig. 2. Emission efficiency η of the defect mode and the first in-band mode, as a function of the normalized exterior boundary radius xb.

Fig. 3. Threshold gain gA and detuning factor δ, of the defect mode and the first in-band mode, as a function of the normalized exterior boundary radius xb.

We also notice the periodic oscillation in gA and δ. This can be understood by the phase factor in the mode resonance condition. Derived from the solutions to (12), the reflectivity of a eigenwave incident from outward to inward on the interface xL subject to the boundary condition A(-xb/2)=B(-xb/2) is

r1=ei(δ+iu)xb(vei(δ+iu)xb+iδu)sinh(Sxb2)+Scosh(Sxb2)(vei(δ+iu)xb+iδu)sinh(Sxb2)+Scosh(Sxb2),
(19)

while from inward to outward on the interface xR subject to the boundary condition B(xb/2)=0 is

r2=vsinh(Sxb2)(iδu)sinh(Sxb2)+Scosh(Sxb2).
(20)

The phase difference caused by the interface xL is

exp[iΦ(Hm(2)(xb2)Hm(1)(xb2))]e2i(xb2mπ2π4)=ieixb,
(21)

where m=0 has been assumed. The mode resonance condition requires that r1r2·(ieixb)=1, thus the phase factor eixbeiδxb=ei(1δ)xb is responsible for the oscillation in gA and δ.

5. Conclusion

Acknowledgment

This work was supported in part by the Defense Advanced Research Projects Agency (DARPA) and in part by the National Science Foundation. The authors thank the anonymous reviewers for their helpful comments.

References and links

1.

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

2.

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]

3.

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]

4.

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]

5.

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, “Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993). [CrossRef]

6.

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]

7.

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]

8.

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]

9.

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]

10.

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]

11.

A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, New York, 1997).

12.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, “InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties,” IEEE J. Sel. Top. Quantum Electron. 11, 476–484 (2005). [CrossRef]

13.

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]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(140.5960) Lasers and laser optics : Semiconductor lasers
(230.1480) Optical devices : Bragg reflectors
(250.7270) Optoelectronics : Vertical emitting lasers

ToC Category:
Rings, Disks, and Other Cavities

History
Original Manuscript: September 5, 2007
Revised Manuscript: November 1, 2007
Manuscript Accepted: November 28, 2007
Published: December 10, 2007

Virtual Issues
Physics and Applications of Microresonators (2007) Optics Express

Citation
Xiankai Sun and Amnon Yariv, "Modal properties and modal control in vertically emitting annular Bragg lasers," Opt. Express 15, 17323-17333 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-17323


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References

  1. T. Erdogan, and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990). [CrossRef]
  2. 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]
  3. 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]
  4. 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]
  5. C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold Gain and Threshold Current Analysis of Circular Grating DFB and DBR Lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993). [CrossRef]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, New York, 1997).
  12. J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP Annular Bragg Lasers: Theory, Applications, and Modal Properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484 (2005). [CrossRef]
  13. 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]

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