Abstract

Intra-cavity diffraction in VCSELs is a loss mechanism that potentially can cause a significant decrease in efficiency and a rise in the threshold current, particularly in cavities with small lateral features with a high index contrast. One such VCSEL type is the 2.3 µm GaSb-based buried tunnel junction (BTJ) VCSEL studied in this work, where the BTJ induced topology of the top layers gives rise to excess loss through diffraction. Diffraction loss is difficult to measure, and also the numerical estimation must be done with care because of the non-axial propagation of the diffracted fields. We present a simulation method with spatially varying dimensionality, such that the field is three-dimensional (3D) in the entire cavity, whereas the material structure of the cavity is modelled in 3D near the BTJ and the layers with a varying topology, but elsewhere is assumed to be 1D like in a regular DBR structure. We find that the diffraction loss displays a non-monotonic behaviour as a function of the BTJ diameter, but as expected it rapidly increases below a certain diameter of the BTJ and may even become the dominant cause of loss in some device designs. We also show that the diffraction loss can be much reduced if the layers above the BTJ can be deposited such that the surface profile becomes smoother with increasing distance from the BTJ.

© 2008 Optical Society of America

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References

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  1. M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
    [CrossRef]
  2. A. Krier, Mid-infrared semiconductor optoelectronics, (Springer-Verlag 2006).
    [CrossRef]
  3. A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
    [CrossRef]
  4. P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999).
    [CrossRef]
  5. E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997).
    [CrossRef]
  6. G. R. Hadley and K. L. Lear, "Diffraction loss of confined modes in microcavities," 1997 Digest of the IEEE/LEOS Summer Topical Meetings (Cat. No. 97TH8276), 65-66 (1997).
  7. R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995).
    [CrossRef]
  8. G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
    [CrossRef]
  9. M. J. Noble, J. P. Loehr, and J. A. Lott, "Semi-analytic calculation of diffraction losses and threshold currents in microcavity VCSELs," Proc. LEOS’98. IEEE Lasers and Electro-Optics Society 1998 Annual Meeting (Cat. No. 98CH36243) 1, 212-213 (1998).
  10. P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003).
    [CrossRef]
  11. S. Riyopoulos and D. Dialetis, "Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure," J. Opt. Soc. Am. B 18, 1497-1511 (2001).
    [CrossRef]
  12. P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
    [CrossRef]
  13. A. G. Fox and T. Li, "Resonant modes in maser interferometer," Bell System Tech. J. 40, 453-488 (1961).
  14. D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993).
    [CrossRef]
  15. P. Yeh, Optical waves in layered media, (Wiley 2005).
  16. J. W. Goodman, Introduction to Fourier optics, (McGraw-Hill 1996).

2008 (1)

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

2006 (1)

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

2003 (1)

P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003).
[CrossRef]

2001 (2)

S. Riyopoulos and D. Dialetis, "Radiation scattering by apertures in vertical-cavity surface-emitting laser cavities and its effects on mode structure," J. Opt. Soc. Am. B 18, 1497-1511 (2001).
[CrossRef]

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

1999 (1)

P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999).
[CrossRef]

1997 (1)

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997).
[CrossRef]

1996 (1)

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

1995 (1)

R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995).
[CrossRef]

1993 (1)

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

1961 (1)

A. G. Fox and T. Li, "Resonant modes in maser interferometer," Bell System Tech. J. 40, 453-488 (1961).

Amann, M.-C.

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Babic, D. I.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Bachmann, A.

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

Baets, R.

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

Bienstman, P.

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

Böhm, G.

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Bowers, J. E.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Burton, R. R.

R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995).
[CrossRef]

Carlsten, J. L.

P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999).
[CrossRef]

Choquette, K. D.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

Chung, Y.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Coldren, L. A.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997).
[CrossRef]

Corzine, S. W.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

Dagli, N.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Dialetis, D.

Dier, O.

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Fox, A. G.

A. G. Fox and T. Li, "Resonant modes in maser interferometer," Bell System Tech. J. 40, 453-488 (1961).

Grau, M.

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Hadley, G. R.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

Hegblom, E. R.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997).
[CrossRef]

Kashani-Shirazi, K.

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

Kendall, P. C.

R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995).
[CrossRef]

Kilper, D. C.

P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999).
[CrossRef]

Lauer, C.

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

Lear, K. L.

P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999).
[CrossRef]

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

Li, T.

A. G. Fox and T. Li, "Resonant modes in maser interferometer," Bell System Tech. J. 40, 453-488 (1961).

Lim, T.

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

Mackowiak, P.

P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003).
[CrossRef]

Nakwaski, W.

P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003).
[CrossRef]

Ortsiefer, M.

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

R¨onneberg, E.

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Riyopoulos, S.

Robson, P. N.

R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995).
[CrossRef]

Roos, P. A.

P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999).
[CrossRef]

Rosskopf, J.

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Sarzala, R. P.

P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003).
[CrossRef]

Scott, J. W.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

Shau, R.

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Stern, M. S.

R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995).
[CrossRef]

Thibeault, B. J.

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997).
[CrossRef]

Warren, M. E.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

Wasiak, M.

P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003).
[CrossRef]

Windhorn, K.

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

P. A. Roos, J. L. Carlsten, D. C. Kilper, and K. L. Lear,"Diffraction from oxide confinement apertures in verticalcavity lasers," Appl. Phys. Lett. 75, 754-756 (1999).
[CrossRef]

Bell System Tech. J. (1)

A. G. Fox and T. Li, "Resonant modes in maser interferometer," Bell System Tech. J. 40, 453-488 (1961).

Electron. Lett. (2)

M. Ortsiefer, G. Böhm, M. Grau, K. Windhorn, E. R¨onneberg, J. Rosskopf, R. Shau, O. Dier, and M.-C. Amann, "Electrically pumped room temperature CW VCSELs with 2.3 μm emission wavelength," Electron. Lett. 42,640-641 (2006).
[CrossRef]

A. Bachmann, T. Lim, K. Kashani-Shirazi, O. Dier, C. Lauer, and M.-C. Amann, "Continuous-wave operation of electrically pumped GaSb-based vertical cavity surface emitting laser at 2.3 μm," Electron. Lett. 44,202-203 (2008).
[CrossRef]

IEE Proc. Optoelectron. (1)

R. R. Burton, M. S. Stern, P. C. Kendall, and P. N. Robson, "VCSEL diffraction-loss theory," IEE Proc. Optoelectron. 142, 77-81 (1995).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. D (1)

P. Mackowiak, R. P. Sarzala, M. Wasiak and W. Nakwaski, "Radial optical confinement in nitride VCSELs," J. Phys. D 36, 2041-2045 (2003).
[CrossRef]

J. Quantum Electron. (2)

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, "Comprehensive numerical modeling of vertical-cavity surface-emitting lasers," J. Quantum Electron. 32, 607-616 (1996).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

J. Sel. Top. Quantum Electron. (1)

E. R. Hegblom, D. I. Babic, B. J. Thibeault, and L. A. Coldren, "Scattering losses from dielectric apertures in vertical-cavity lasers," J. Sel. Top. Quantum Electron. 3, 379-389 (1997).
[CrossRef]

Opt. Quantum Electron. (1)

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

Other (5)

P. Yeh, Optical waves in layered media, (Wiley 2005).

J. W. Goodman, Introduction to Fourier optics, (McGraw-Hill 1996).

G. R. Hadley and K. L. Lear, "Diffraction loss of confined modes in microcavities," 1997 Digest of the IEEE/LEOS Summer Topical Meetings (Cat. No. 97TH8276), 65-66 (1997).

A. Krier, Mid-infrared semiconductor optoelectronics, (Springer-Verlag 2006).
[CrossRef]

M. J. Noble, J. P. Loehr, and J. A. Lott, "Semi-analytic calculation of diffraction losses and threshold currents in microcavity VCSELs," Proc. LEOS’98. IEEE Lasers and Electro-Optics Society 1998 Annual Meeting (Cat. No. 98CH36243) 1, 212-213 (1998).

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Figures (9)

Fig. 1.
Fig. 1.

Illustration of the relation between the actual VCSEL structure and the numerical model. The numerical model for the central VCSEL region is built up by coupled layers, or coupled cavities, where each may have a refractive index with an arbitrary lateral variation. Also indicated are the four optical fields in each layer, and their relations via transmission, reflection and propagation.

Fig. 2.
Fig. 2.

Amplitude reflectance (top) and phase shift upon reflection (bottom), here calculated for the starting wavelength of 2340 nm, of a plane wave as a function of the incidence angle from the x- and y-axis, respectively, for the top DBR structure used in the simulation.

Fig. 3.
Fig. 3.

Flowchart of the numerical algorithm to simulate the VCSEL cavity field and calculate the diffraction loss. The variables N p and Δλ p are decision parameters set by the user.

Fig. 4.
Fig. 4.

Schematic of the BTJ VCSEL cavity structure modeled in this study.

Fig. 5.
Fig. 5.

The on-axis refractive index variation in the VCSEL structure (top), and the numerical representation of the coupled-cavity section (bottom). Case A is for a perfect reproduction of the steep walls of the BTJ throughout the top layers, i.e., D trans,m =0, while Case B is for a transition region increasing linearly with the distance from the BTJ (to the right) having a maximum value D trans,m =1=1 µm.

Fig. 6.
Fig. 6.

Evolution towards convergence of the iterative simulation for a VCSEL with D BTJ =6 µm and D trans , m =0. The inset shows the laterally spatially averaged phase value for the field in a fixed longitudinal position as a function of the number of iterations (cavity round-trips); the sudden phase jumps result from the field averaging in the simulation method. The main figure shows the calculation wavelength, which is dynamically adjusted during the iterations to enforce true repetition of the field, so that the field phase does not change between the round-trips; the inset shows that this goal is accomplished after ~1000 iterations.

Fig. 7.
Fig. 7.

Cross section in the yz-plane of the absolute value of the standing-wave field in the coupled-cavity section for the VCSEL with D BTJ =6 µm and D trans,m =0, calculated from the fields E(x,y) m,d,p obtained in the final iteration. The inset shows the amplitude profiles of the left-propagating field at the left boundary in all the 15 layers.

Fig. 8.
Fig. 8.

Diffraction loss per round-trip and resonance wavelength as functions of the diameter of the BTJ, for D trans,m =0.

Fig. 9.
Fig. 9.

Diffraction loss normalized to the total round-trip loss as a function of the BTJ diameter, for D trans,m =0. The inset shows the normalized diffraction loss as a function of the transition distance between the high and the low levels of the topmost layer in the dielectric top mirror, for D BTJ =6 µm. A non-zero value for D trans,m =1 implies that a gradual transition is assumed to occur also in all other layers in the physical top DBR. Also shown is the resonance wavelength for these cases.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

W d = W tot W mirrors, normal ; η d = W d W tot
E ( x , y ) m , r , l = t ( x , y ) m 1 , r , r E ( x , y ) m 1 , r , r + r ( x , y ) m , l , l E ( x , y ) m , l , l
E ( x , y ) m , l , r = t ( x , y ) m + 1 , l , l E ( x , y ) m + 1 , l , l + r ( x , y ) m , r , r E ( x , y ) m , r , r

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