Abstract

From first principles we develop figures of merit to determine the gain experienced by the guided mode and the lasing threshold for devices based on high-index-contrast waveguides. We show that as opposed to low-index-contrast systems, this quantity is not equivalent to the power confinement since in high-index-contrast structures the electric and magnetic field distributions cannot be related by proportionality constant. We show that with a slot waveguide configuration it is possible to achieve more gain than one would expect based on the power confinement in the gain media. Using the figures of merit presented here we optimize a slot waveguide geometry to achieve low-threshold lasing and discuss the fabrication tolerances of such a design.

© 2008 Optical Society of America

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  1. A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, "Electrically pumped hybrid AlGaInAs-silicon evanescent laser," Opt. Express 14, 9203-9210 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, "Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material," Opt. Lett. 29, 1626-1628 (2004).
    [CrossRef] [PubMed]
  4. C. A. Barrios and M. Lipson, "Electrically driven silicon resonant light emitting device based on slot-waveguide," Opt. Express 13, 10092-10101 (2005).
    [CrossRef] [PubMed]
  5. F. Ning-Ning, J. Michel, and L. C. Kimerling, "Optical field concentration in low-index waveguides," IEEE J. Quantum Electron. 42, 885-890 (2006).
  6. J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (2005).
    [CrossRef] [PubMed]
  7. E. Burstein and C. Weisbuch, eds., Confined electrons and photons, (Plenum Press: New York, NY,1995)
    [CrossRef]
  8. T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE J. Quantum Electron. 33, 1763-1766 (1997).
    [CrossRef]
  9. C. A. Barrios, K. B. Gylfason, B. Sánchez, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, "Slot-waveguide biochemical sensor," Opt. Lett. 32, 3080-3082 (2007).
    [CrossRef] [PubMed]
  10. F. Dell'Olio and V. M. Passaro, "Optical sensing by optimized silicon slot waveguides," Opt. Express 15, 4977-4993 (2007).
    [CrossRef] [PubMed]
  11. H. Kogelnik, Theory of optical waveguides, in Guided-wave optoelectronics, T. Tamir, ed., (Springer Verlag: Berlin, 1990). p. 7.
  12. L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits (J. Wiley & Sons, New York, NY, 1995).
  13. C. Pollock and M. Lipson, Integrated photonics (Kluwer Academic, Norwell, MA, 2003).
  14. J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
    [CrossRef]
  15. J. D. Jackson, Classical electrodynamics. 3rd ed., (John Wiley & Sons, Inc., Hoboken, NJ, 1999).
  16. H. A. Haus, Waves and fileds in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  17. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Reading, MA, 1960).
  18. G. J. Veldhuis, O. Parriaux, H. J. W. M. Hoekstra, and P. V. Lambeck, "Sensitivity enhancement in evanescent optical waveguide sensors," J. Lightwave Technol. 18, 677-682 (2000).
    [CrossRef]
  19. R. Perahia, O. Painter, V. Moreau, and R. Colombelli, "Design of quantum cascade lasers for intra-cavity sensing in the mid infrared," (in preparation).
  20. A. E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).
  21. J. T. Robinson, L. Chen, and M. Lipson, "On-chip gas detection in silicon optical microcavities," Opt. Express 16, 4296-4301 (2008).
    [CrossRef] [PubMed]

2008 (1)

2007 (2)

2006 (2)

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, "Electrically pumped hybrid AlGaInAs-silicon evanescent laser," Opt. Express 14, 9203-9210 (2006).
[CrossRef] [PubMed]

F. Ning-Ning, J. Michel, and L. C. Kimerling, "Optical field concentration in low-index waveguides," IEEE J. Quantum Electron. 42, 885-890 (2006).

2005 (2)

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (2005).
[CrossRef] [PubMed]

C. A. Barrios and M. Lipson, "Electrically driven silicon resonant light emitting device based on slot-waveguide," Opt. Express 13, 10092-10101 (2005).
[CrossRef] [PubMed]

2004 (2)

2000 (1)

1997 (2)

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE J. Quantum Electron. 33, 1763-1766 (1997).
[CrossRef]

Almeida, V. R.

Baets, R.

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

Barrios, C. A.

Blok, H.

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE J. Quantum Electron. 33, 1763-1766 (1997).
[CrossRef]

Bowers, J. E.

Casquel, R.

Chen, L.

Cohen, O.

Colombelli, R.

R. Perahia, O. Painter, V. Moreau, and R. Colombelli, "Design of quantum cascade lasers for intra-cavity sensing in the mid infrared," (in preparation).

Dell'Olio, F.

Demeulenaere, B.

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE J. Quantum Electron. 33, 1763-1766 (1997).
[CrossRef]

Fang, A. W.

Griol, A.

Gylfason, K. B.

Haes, J.

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

Hoekstra, H. J. W. M.

Holgado, M.

Jones, R.

Kimerling, L. C.

F. Ning-Ning, J. Michel, and L. C. Kimerling, "Optical field concentration in low-index waveguides," IEEE J. Quantum Electron. 42, 885-890 (2006).

Lambeck, P. V.

Lenstra, D.

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE J. Quantum Electron. 33, 1763-1766 (1997).
[CrossRef]

Lipson, M.

Long, C.

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (2005).
[CrossRef] [PubMed]

Manolatou, C.

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (2005).
[CrossRef] [PubMed]

Michel, J.

F. Ning-Ning, J. Michel, and L. C. Kimerling, "Optical field concentration in low-index waveguides," IEEE J. Quantum Electron. 42, 885-890 (2006).

Moreau, V.

R. Perahia, O. Painter, V. Moreau, and R. Colombelli, "Design of quantum cascade lasers for intra-cavity sensing in the mid infrared," (in preparation).

Ning-Ning, F.

F. Ning-Ning, J. Michel, and L. C. Kimerling, "Optical field concentration in low-index waveguides," IEEE J. Quantum Electron. 42, 885-890 (2006).

Painter, O.

R. Perahia, O. Painter, V. Moreau, and R. Colombelli, "Design of quantum cascade lasers for intra-cavity sensing in the mid infrared," (in preparation).

Panepucci, R. R.

Paniccia, M. J.

Park, H.

Parriaux, O.

Passaro, V. M.

Perahia, R.

R. Perahia, O. Painter, V. Moreau, and R. Colombelli, "Design of quantum cascade lasers for intra-cavity sensing in the mid infrared," (in preparation).

Robinson, J. T.

J. T. Robinson, L. Chen, and M. Lipson, "On-chip gas detection in silicon optical microcavities," Opt. Express 16, 4296-4301 (2008).
[CrossRef] [PubMed]

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (2005).
[CrossRef] [PubMed]

Sánchez, B.

Sohlström, H.

Veldhuis, G. J.

Visser, T. D.

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE J. Quantum Electron. 33, 1763-1766 (1997).
[CrossRef]

Xu, Q.

IEEE J. Quantum Electron. (2)

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE J. Quantum Electron. 33, 1763-1766 (1997).
[CrossRef]

F. Ning-Ning, J. Michel, and L. C. Kimerling, "Optical field concentration in low-index waveguides," IEEE J. Quantum Electron. 42, 885-890 (2006).

J. Lightwave Technol. (1)

Opt. Express (4)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

J. Haes, B. Demeulenaere, R. Baets, D. Lenstra, T. D. Visser, and H. Blok, "Difference between te and tm modal gain in amplifying waveguides: Analysis and assessment of two perturbation approaches," Opt. Quantum Electron. 29, 263-273 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (2005).
[CrossRef] [PubMed]

Other (9)

E. Burstein and C. Weisbuch, eds., Confined electrons and photons, (Plenum Press: New York, NY,1995)
[CrossRef]

J. D. Jackson, Classical electrodynamics. 3rd ed., (John Wiley & Sons, Inc., Hoboken, NJ, 1999).

H. A. Haus, Waves and fileds in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon Press, Reading, MA, 1960).

R. Perahia, O. Painter, V. Moreau, and R. Colombelli, "Design of quantum cascade lasers for intra-cavity sensing in the mid infrared," (in preparation).

A. E. Siegman, Lasers (University Science Books, Sausalito, CA,1986).

H. Kogelnik, Theory of optical waveguides, in Guided-wave optoelectronics, T. Tamir, ed., (Springer Verlag: Berlin, 1990). p. 7.

L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits (J. Wiley & Sons, New York, NY, 1995).

C. Pollock and M. Lipson, Integrated photonics (Kluwer Academic, Norwell, MA, 2003).

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Figures (4)

Fig. 1.
Fig. 1.

Fundamental TM modes at a wavelength of 1.5 µm for waveguides 500 nm wide and 600 nm tall. All modes are normalized to unit power. The high-index material (n=3.5) is outlined in black. The waveguides are clad with n=3.25 for (a) and (b) and n=1.5 for (c)–(f). The first and second columns show E y and ω μ 0 β H x respectively, plotted on the same color scale. The two fields become increasingly dissimilar as more electric field is concentrated at high-index-contrast boundaries.

Fig. 2.
Fig. 2.

Numerical study of modal gain. (a) Schematic of slot waveguide with gain material defined by an imaginary component of the refractive index confined to the slot region (pink). (b) Major field component of the fundamental TM mode for the same structure as (a) calculated using a finite difference mode solver. (c) Circles show the modal gain (g m ) calculated from the complex effective index of the fundamental TM mode as determined using a finite difference mode solver. Material gain is added via the imaginary part of the refractive index in the slot. Dashed line shows the modal gain calculated according to Eq. (3) based on the confinement factor Γ determined from the zero-gain mode profile from Eq. (15). Dotted line shows the product of the power in the active gain region (P A ) and the material gain. We see that the confinement factor proposed in this paper correctly predicts the modal gain simulated numerically, while the power confinement greatly underestimates the simulated modal gain.

Fig. 3.
Fig. 3.

(a). The spatial confinement factor γ A plotted as a function of slot thickness t, where the gain region is defined as the slot (pink region in inset) between the high-index rails (green). Narrow slots result greater emission rates of gain material while thicker slots provide more material which contributes to the gain. The peak in γ A near a slot width of 60 nm indicates the condition where the combination of enhanced emission rate and volume of gain material result in the lowest lasing threshold. (b). The total confinement factor (Γ) (squares) and power in the slot region (P A ) (triangles) as a function of slot width. Dotted and dashed lines mark the slot widths which maximize Γ and P A respectively. The discrepancy between these two plots shows that the percentage of power in the gain media is not an accurate indication of either the magnitude or the optimal design for modal gain.

Fig. 4.
Fig. 4.

Optimization of width and height of Si/SiO2/Si slot waveguide with a 10 nm thick slot assumed to contain a gain medium. (a) Schematic of slot waveguide. (b) Total confinement factor Γ, proportional to the total modal gain. (c) Group index n g divided by the slot index (1.46), which is responsible for the difference between the lasing threshold and modal gain. (d) Electric field energy confinement γ A , inversely proportional to the lasing threshold. The maximum total modal gain is marked by the square in (a). The white contour shows the region which corresponds to a 5% change from the maximum values of Γ and γ A .

Equations (20)

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

H = c ε n ( e ̂ z × E ) ,
E y = ω μ 0 β H x ,
Γ g m g b ,
E ( z ) 2 = E 0 2 e g b z .
E ( x , y , z ) = e ̂ j E j o ψ j ( x , y ) e i ( ω t β ˜ z ) ,
β ˜ k 0 ( n ̅ r + i n ̅ i ) ,
g m = 2 Im { β ˜ } = 2 k 0 n ̅ i .
g b = 2 k 0 n A i .
Δ β ˜ = ω Δ ε ˜ E 2 d x d y 1 2 Re { E × H * } e ̂ z d x d y ,
g m = [ n A c ε 0 A E 2 d x d y Re { E × H * } e ̂ z d x d y ] g b .
Γ = n A c ε 0 A E 2 d x d y Re { E × H * } e ̂ z d x d y .
1 2 Re { E × H * } e ̂ z d x d y = 1 2 β ω μ 0 E 2 d x d y ,
U / l = 1 2 ε E 2 d x d y .
1 2 Re { E × H * } e ̂ z d x d y = v g 1 2 ε E 2 d x d y
Γ = n g A ε E 2 d x d y n A ε E 2 d x d y n g n A γ A .
γ A A ε E 2 d x d y ε E 2 d x d y .
P A A Re { E × H * } e z d x d y Re { E × H * } e z d x d y .
α m = n g n b γ b α b ,
n g γ A n A g b n g i γ i n i α i > 0 .
g b > α m Γ .

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