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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 1 — Jan. 5, 2009
  • pp: 1–6
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Surface-emitting circular DFB, disk-, and ring-Bragg resonator lasers with chirped gratings. II: nonuniform pumping and far-field patterns

Xiankai Sun and Amnon Yariv  »View Author Affiliations


Optics Express, Vol. 17, Issue 1, pp. 1-6 (2009)
http://dx.doi.org/10.1364/OE.17.000001


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Abstract

This is a continuation of our previous work [Opt. Express 16, 9155 (2008)]. In this paper we investigate the effect of nonuniform pumping on the modal properties of surface-emitting chirped circular grating lasers. By numerically solving the coupled-mode equations and matching the boundaries we compare and discuss the threshold pump levels and frequency detuning factors for three pumping profiles: uniform, Gaussian, and annular. Depending on the overlap of the pumping and modal profiles, Gaussian pumping results in the lowest threshold pump levels except for the fundamental mode of ring Bragg resonator laser, and annular pumping provides larger threshold discrimination between the fundamental and first-order modes of circular DFB and ring Bragg resonator lasers, which is favorable for single-mode operation in these lasers. We also study the far-field patterns of the fundamental modes of circular DFB, disk-, and ring-Bragg resonator lasers. Circular DFB and ring Bragg resonator lasers have the first-order dominating peak, while disk Bragg resonator laser exhibits the zeroth-order dominating peak.

© 2009 Optical Society of America

1. Introduction

Circular grating coupled surface emitting lasers are able to produce circularly-symmetric, large-emission-aperture, narrow-divergence laser beams, which makes them not only excellent light emitters with efficient coupling to optical fibers but also superior elements for on-chip 2D laser array integration for coherent beam combination. As shown in Fig. 1, three configurations of such circular grating lasers have been extensively investigated: (a) Circular DFB lasers, in which the grating extends from the center to the exterior boundary xb. (b) Disk Bragg resonator lasers, in which a center disk is surrounded by a radial Bragg grating extending from x 0 to xb. (c) Ring Bragg resonator lasers, in which an annular defect is surrounded by inner and outer gratings on both sides. The inner grating extends from the center to xL while the outer from xR to xb. In these lasers, the gratings serve two purposes -providing feedback for the in-plane fields to form a radial resonator, and coupling the vertical laser radiation out of the plane as an output coupler. Detailed in [1

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

] and [2

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

], the gratings have to be designed radially chirped in order to optimally interact with the optical fields since the eigenmodes of the wave equation in cylindrical coordinates, the Bessel functions, have nonperiodic zeros.

Fig. 1. Surface-emitting 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.

2. Coupled-mode equations, boundary conditions, and numerical methods

The coupled-mode equations for the Hankel-phased circular gratings in active media were derived in [12

12. 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 [13

13. 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]

] including the effect of resonant vertical radiation. By using the Green’s function method, the contribution from vertical radiation is incorporated as a coefficient into the coupled in-plane wave equations, yielding a set of evolution equations for the amplitudes of the in-plane waves

{dA(x)dx=u(x)A(x)vB(x)e2iδxdB(x)dx=u(x)B(x)+vA(x)e2iδx.
(1)

All the circular grating lasers including the three configurations shown in Fig. 1 have to satisfy the common boundary conditions: (i) A(0)=B(0) at the center; (ii) B(xb)=0 at the exterior boundary xb; (iii) A(x) and B(x) are continuous for all 0<x<xb. As already discussed in [3

3. X. K. Sun and A. 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). [CrossRef] [PubMed]

], boundary condition (iii) is equivalent to the continuities of both the in-plane electric field and its first derivative.

Fig. 2. Illustration of different gain distribution profiles: (a) uniform; (b) Gaussian; (c) annular.

  1. Uniform: g(x) = gA, 0 ≤ xxb, pumplevel=0xbgAxdx=12gAXb2,,
  2. Gaussian: g(x)=gAexp(x2wp2),x0,pumplevel=0gAexp(x2wp2)xdx=12gAwp2,,
  3. Annular: g(x)=gA[exp((xxp)2wp2)+exp((x+xp)2wp2)],x0,pumplevel=0gA[exp((xxp)2wp2)+exp((x+xp)2wp2)]xdx=gA[wp2exp(xp2wp2)+πwpxperf(xpwp)],, where the error function erf(x)2π0xet2dt.

3. Numerical results and discussions

3.1. Nonuniform pumping

Fig. 3. Threshold pump level Pth and detuning factor δ of circular DFB, disk-, and ring- Bragg resonator lasers under uniform, Gaussian and annular pumping profiles.

3.2. Far-field patterns

U(r)apertureΔEρψexp(ikrr)4πrrdreikr4πrapertureΔEρψexp(ik(rr̂))dr
=eikr4πrψ=02πρ=0ρbΔE(ρ)exp[ikρsinθcos(ψϕ)]ρdρdψ=eikr2r0ρbΔE(ρ)J0(kρsinθ)ρdρ
(2)

where r′ = ρ cosψ x̂ + ρ sinψ ŷ is the source point and r = r sinθ cosϕ x̂ + r sinθsinϕ ŷ + r cosθ ẑ is the field point. The far-field intensity pattern is then given by I(r) = U *(r)U(r) = |U(r)|2 and plotted in Fig. 4 for the fundamental modes of circular DFB, disk-, and ring- Bragg resonator lasers.

Fig. 4. Far-field intensity patterns of the fundamental modes of circular DFB, disk-, and ring-Bragg resonator lasers.

As expected, the different lobes correspond to different diffraction orders of the light from the circular emission aperture. For circular DFB and ring Bragg resonator lasers, they have most of the energy located in the first-order Fourier component thus their first-order diffraction peaks dominate, while for disk Bragg resonator laser, the zeroth-order peak dominates. These theoretical results are similar to previously published experimental data for circular DFB and DBR lasers [16

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

, 17

17. M. Fallahi, M. Dion, F. Chatenoud, I. M. Templeton, R. Barber, and J. Sedivy, “Low Threshold CW Operation of Circular-Grating Surface-Emitting DBR Lasers Using MQW and a Self-Aligned Process,” IEEE Photon. Technol. Lett. 6, 1280–1282 (1994). [CrossRef]

].

4. Conclusions

Acknowledgments

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 Dr. P. Chak for his kind help in numerical calculations.

References and links

1.

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

2.

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

3.

X. K. Sun and A. 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). [CrossRef] [PubMed]

4.

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]

5.

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]

6.

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]

7.

K. J. Kasunic, E. M. Wright, and N. Peyghambarian, “Numerical modeling of inhomogeneously-pumped circular-grating DFB lasers,” Proc. SPIE 2398, 125–134 (1995). [CrossRef]

8.

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]

9.

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]

10.

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]

11.

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]

12.

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]

13.

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]

14.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

15.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

16.

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

17.

M. Fallahi, M. Dion, F. Chatenoud, I. M. Templeton, R. Barber, and J. Sedivy, “Low Threshold CW Operation of Circular-Grating Surface-Emitting DBR Lasers Using MQW and a Self-Aligned Process,” IEEE Photon. Technol. Lett. 6, 1280–1282 (1994). [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(130.2790) Integrated optics : Guided waves
(140.5560) Lasers and laser optics : Pumping
(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: November 7, 2008
Revised Manuscript: December 18, 2008
Manuscript Accepted: December 18, 2008
Published: December 22, 2008

Citation
Xiankai Sun and Amnon Yariv, "Surface-emitting circular DFB, disk-, and ring-Bragg resonator lasers with chirped gratings. II: nonuniform pumping and far-field patterns," Opt. Express 17, 1-6 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-1-1


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References

  1. J. Scheuer and A. Yariv, "Coupled-Waves Approach to the Design and Analysis of Bragg and Photonic Crystal Annular Resonators," IEEE J. Quantum Electron. 39, 1555-1562 (2003). [CrossRef]
  2. J. Scheuer and A. Yariv, "Annular Bragg defect mode resonators," J. Opt. Soc. Am. B 20, 2285-2291 (2003). [CrossRef]
  3. X. K. Sun and A. 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). [CrossRef] [PubMed]
  4. 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]
  5. 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]
  6. 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]
  7. K. J. Kasunic, E. M. Wright, and N. Peyghambarian, "Numerical modeling of inhomogeneously-pumped circular-grating DFB lasers," Proc. SPIE 2398, 125-134 (1995). [CrossRef]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).
  15. E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).
  16. R. H. Jordan, D. G. Hall, O. King, G. Wicks, and S. Rishton, "Lasing behavior of circular grating surface-emitting semiconductor lasers," J. Opt. Soc. Am. B 14, 449-453 (1997). [CrossRef]
  17. 17. M. Fallahi, M. Dion, F. Chatenoud, I. M. Templeton, R. Barber, and J. Sedivy, "Low Threshold CW Operation of Circular-Grating Surface-Emitting DBR Lasers Using MQW and a Self-Aligned Process," IEEE Photon. Technol. Lett. 6, 1280-1282 (1994). [CrossRef]

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