## Modal properties and modal control in vertically emitting annular Bragg lasers

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

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

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]

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]

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]

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

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]

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]

*π*/2 phase shift in the outer grating and the other without, then find the conditions for a single defect mode lasing. In Section 5 we present a conclusion.

## 2. Comprehensive coupled mode theory

*ρ*while the outer grating spans from

_{L}*ρ*to

_{R}*ρ*. 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

_{b}*z*component of the electric field which satisfies the scalar wave equation in cylindrical coordinates

*E*in a passive uniform medium in which the dielectric constant

_{z}*n*

^{2}(

*ρ,z*)=

*ε*(

_{r}*z*) can be expressed as

*m*is the azimuthal mode number,

*β*=

*k*

_{0}

*n*is the in-plane propagation constant, and

_{eff}*Z*(

*z*) is the fundamental mode profile of the planar slab waveguide satisfying

*n*

^{2}(

*ρ,z*)=

*ε*(

_{r}*z*)+

*iε*(

_{i}*z*)+Δ

*ε*(

*ρ,z*) where |

*ε*(

_{i}*z*)|<<

*ε*(

_{r}*z*) represents the gain/loss and Δ

*ε*(

*ρ,z*) reflects the contribution of perturbation. For optimal field confinement the perturbation Δ

*ε*(

*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]

*a*(

_{l}*z*) is the expansion coefficient of Δ

*ε*(

*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.

*E*(

*x,z*) so that

*E*(

*x,z*) has an exp(±

*ik*

_{0}

*z*) dependence on

*z*in free space, i.e.

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]

*A*(

*x*) and

*B*(

*x*), and applying the modal solution in the passive unperturbed case, we find

*E*can be expressed as

*G*(

*z*,

*z′*) is the Green’s function satisfying

*Z*(

*z*) and integrating over

*z*, we arrive at

*P*≡∫

^{+∞}

_{-∞}

*Z*

^{2}(

*z*)

*dz*.

*a*

_{-1}=

*a*

_{1}, then all the radiation coupling coefficients are the same and can be denoted as

*h*

_{1}. Let

*u*=

*g*-

_{A}*h*

_{1}and

*ν*=

*h*

_{1}+

*ih*

_{2}, then the generic solution to (12) is

*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 Δ

*ε*=0, we have

*h*

_{1}=

*h*

_{2}=0, and the solution to (12) is simply

## 3. Modal fields and characteristic equation of annular Bragg lasers

*E*

^{(m)}

*(*

_{z}*x,z*) in different regions takes different forms

*x*,

_{L}*x*, and

_{R}*x*are normalized

_{b}*ρ*,

_{L}*ρ*, and

_{R}*ρ*, respectively.

_{b}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]

*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 Δ

*E*

_{3}=(

*s*′

_{1}

*Ae*

^{-iδ·x}+

*s′*

_{-1}

*Be*

*)|*

^{iδ·x}*H*

^{(1)}

*m*| where

*s′*

_{1}=

*is*

_{1}and

*s′*

_{1}=

*is*

_{-1}.

*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*)+Δ

*E*

_{1}(

*x*,

*z*) and |Δ

*E*

_{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).

*x*, no incoming wave comes from outside (

_{b}*x*>

*x*), thus

_{b}*B*

_{3}(

*x*)=0.

_{b}*x*and

_{L}*x*, the electric field

_{R}*E*is continuous, i.e.,

_{z}*E*(

_{I}*x*)=

_{L}*E*(

_{II}*x*) and

_{L}*E*(

_{II}*x*)=

_{R}*E*(

_{III}*x*).

_{R}*x*and

_{L}*x*, the first order derivative of the electric field

_{R}*E′*is continuous, i.e.,

_{z}*E′*(

_{I}*x*)=

_{L}*E′*(

_{II}*x*) and

_{L}*E′*(

_{II}*x*)=

_{R}*E′*(

_{III}*x*).

_{R}*Z*(

*z*) and integrating over

*z*, we get the integrated

*E*

^{(m)}

*(*

_{z}*x*) in the 3 different regions:

*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:

## 4. Numerical results and modal control in annular Bragg lasers

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]

*µ*m 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

*n*, and the Green’s function are numerically calculated. For the in-plane grating, we assume a rectangular profile with a Hankel-phased modulation [8

_{eff}**39**, 1555–1562 (2003). [CrossRef]

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]

*d*=0.25 is a good choice since

_{c}*h*

_{2}is maximal while Re(

*h*

_{1}) is not small. For

*m*=0, we get

*h*

_{1}=0.0072+0.0108

*i*and

*h*

_{2}=0.0601.

*ρ*=17.5

_{b}*µ*m (

*x*=

_{b}*βρ*≈200) used in [9

_{b}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]

*x*/2, with its width (

_{b}*x*-

_{R}*x*) being a wavelength of the cylindrical waves therein. So (

_{L}*x*+

_{L}*x*)/2=

_{R}*x*/2, and

_{b}*x*-

_{R}*x*=2π

_{L}*≈*6.3 since the approximation of Hankel functions

*x*,

_{L}*x*,

_{R}*x*into (17), solve for all the allowed pairs of

_{b}*g*and

_{A}*δ*, and pick up those within the range 0<

*g*<0.01, -0.1<

_{A}*δ*<0.1. Table 1 shows the threshold gains

*g*, the detuning factors

_{A}*δ*, and the in-plane modal field patterns of the first five resonant modes of the ABR lasers whose outer grating has an additional

*π*/2 phase shift.

*δ*=0). This is because we are using a mixed-order Bragg grating, and the interference of the radiation due to first-order diffraction breaks the mode degeneracy of in-plane (guided) waves, which was first proposed for longitudinal mode selection in linear DFB lasers [13

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]

*π*/2 phase shift in the outer grating. On the other hand, the additional

*π*/2 phase shift separates the whole resonator into two coupled resonators. This is like a Febry-Perot resonator in which a

*λ*/4 plate is inserted at the middle point. The difference in the amount of feedback from its two end facets breaks the degeneracy of the eigenmodes of the new structure, as can be seen from a comparison between Mode 1 and 2, and also between Mode 3 and 4. Due to the coupling loss between the two separated resonators, the defect mode whose maximal field is at the middle point has a relatively high

*g*, as evidenced by Mode 5. To reduce the threshold gain of the defect mode, we consider the ABR lasers whose outer grating has the same phase dependence Φ[

_{A}*H*/

^{(1)}

*(*

_{m}*x*)] as the inner grating. The calculated results are listed in Table 2. As expected, the defect mode now possesses the lowest threshold gain, which is almost an order of magnitude lower than that in the previous case. The higher-order (in-band) modes resemble their counterparts in a non-periodic circular grating DFB laser (in which no defect is introduced in the middle and the Hankel-phased grating spreads from the center to the exterior boundary).

**13**, 359–366 (2007). [CrossRef]

*η*as the ratio between the useful vertical radiation power and the total power loss. We vary the exterior boundary radius

*x*while fixing the defect size and locating the defect always at the middle (

_{b}*x*/2), and calculate

_{b}*η*for both the defect mode and the first in-band mode as a function of

*x*. The results are plotted in Fig. 2. As can be seen, the emission efficiency, for both modes, improves as the device size (

_{b}*x*) increases, and more impressively, the defect mode has much higher emission efficiency than the first in-band mode for the same device size.

_{b}*x*for a single defect mode operation. The calculated threshold gain

_{b}*g*and detuning factor

_{A}*δ*as a function of the exterior boundary radius

*x*are displayed in Fig. 3. We see that, for

_{b}*x*>250 (

_{b}*ρ*>21.8

_{b}*µ*m), the first in-band mode has a lower threshold gain than the defect mode, so

*x*has to be less than 250 to guarantee a single defect mode lasing.

_{b}*g*and

_{A}*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

*x*subject to the boundary condition

_{L}*A*(-

*x*/2)=

_{b}*B*(-

*x*/2) is

_{b}*x*subject to the boundary condition

_{R}*B*(

*x*/2)=0 is

_{b}*x*is

_{L}*m*=0 has been assumed. The mode resonance condition requires that

*g*and

_{A}*δ*.

## 5. Conclusion

*π*/2 phase shift in the outer grating and the other without, were analyzed. It was pointed out that the additional

*π*/2 phase shift in the outer grating actually separates the whole resonator into two, thus raising the threshold gain of the defect mode. We also numerically demonstrated that, in order to get a single high-efficiency defect mode lasing in the ABR lasers, we can choose the kind without a

*π*/2 phase shift in the outer grating, and also an exterior boundary radius smaller than a critical value.

## Acknowledgment

## References and links

1. | T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor lasers: An analysis,” J. Appl. Phys. |

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. |

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. |

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. |

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. |

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. |

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 |

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. |

9. | J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, “Low-threshold two-dimensional annular Bragg lasers,” Opt. Lett. |

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. |

11. | A. Yariv, |

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. |

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. |

**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

- T. Erdogan, and D. G. Hall, "Circularly symmetric distributed feedback semiconductor lasers: An analysis," J. Appl. Phys. 68, 1435-1444 (1990). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. Yariv, Optical Electronics in Modern Communications (Oxford Univ. Press, New York, 1997).
- 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]
- 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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