## First-principle derivation of gain in high-index-contrast waveguides

Optics Express, Vol. 16, Issue 21, pp. 16659-16669 (2008)

http://dx.doi.org/10.1364/OE.16.016659

Acrobat PDF (344 KB)

### 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

## 1. Introduction

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]

2. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. **29**, 1209–1211 (2004). [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]

_{2}or Erdoped Si

_{3}N

_{4}can be inserted into one [4

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]

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]

*D*=

*εE*) must be continuous across the interface, the electric field in the slot waveguide is enhanced by the ratio of the dielectric constant of silicon to that of the slot material. In semiconductor materials this enhancement can be as large as one order of magnitude.

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

**H**can be written in terms of the electric field

**E**and the impedance of the material according to:

**e**̂

*is a unit vector along the direction of propagation (which we have chosen to be the*

_{z}*z*-direction) and

*n*is the index of refraction of the material. This is often written in the form relating the major components of the electric and magnetic fields (for a TM mode in this case)[8

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]

*β*is the propagation constant defined as

*β*≡2

*πn*̄/

*λ*. Based on these relationships the electric field energy, and waveguide power stored in a given region can be used interchangeably since they differ only by a constant. In this case the percentage of power overlapping the gain medium can be used to calculate the resulting modal gain, and it is often assumed that the same can be said for waveguide modes.

*E*

*and*

_{y}*E*

*must be*

_{y}*continuous*across the dielectric interfaces to the left and right of the waveguide, and

*discontinuous*across the top and bottom interfaces. The

*H*

*on the other hand must be continuous across all interfaces since the magnetic susceptibility is the same in all regions. This leads to noticeable differences between the electric and magnetic field magnitudes and profiles. This difference becomes dramatic when the peak of the electric field is placed at a dielectric discontinuity as is the case for slot waveguides (Figs. 1(e) and 1(f)).*

_{x}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]

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]

## 2. Confinement factor for high-index-contrast waveguides

*g*

*) to the modal gain (*

_{b}*g*

*) in a guiding structure:*

_{m}*g*

*and*

_{m}*g*

*have units of inverse length. The bulk material gain can be determined from the magnitude of the electric field for a plane wave propagating along the z-direction through the gain medium:*

_{b}**e**̂ is the polarization vector, Ψ is the cross sectional mode profile, and

*β*̃ is the complex propagation constant defined as

*k*

_{0}is the angular wavenumber in free space, and

*n*̄

*and*

_{r}*n*̄

*are the real and imaginary parts of the effective index respectively. By writing Eq. (5) in the same form as Eq. (4) we can see that the modal gain is determined by the complex propagation constant. From Eq. (6) we can then write the gain of the guided mode in terms of the imaginary part of the effective index:*

_{i}*n*

*) in a given active region (*

_{Ai}*A*) and solve for the complex propagation constant of the guided mode. This can be performed using variational methods [11]:

*c*is the speed of light in vacuum, the integral in the numerator is carried out only over the area of the active gain region (since this is where Δ

*ε*̃≠0), and the integral in the denominator is carried out over the entire cross section of the mode. We recognize the term in the brackets as the proportionality constant in Eq. (3) and therefore we can express the confinement factor as:

**E**|

^{2}. Since this term is normalized to unit power, the confinement factor can be thought of as the amount of intensity overlapping the gain medium per unit input power. Note that most previous derivations of this confinement factor err by incorrectly substituting the electric for magnetic field in attempts to simplify the expression. This is commonly written as [12–14

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]

*the expression in Eq. (12) is not valid*for high-index-contrast waveguides since it is based on the relationship for plane waves in homogeneous media that

*d*(

*ωε*)/

*dω*in Eq. (11) [16, 17]. If silicon is the most dispersive material, the error introduced by making this approximation is less than 7% at a wavelength of 1.55 microns. The group velocity of the mode (

*v*

*) describes the speed with which energy flows through a given cross section. Therefore we can write the power flux through a given cross section of the waveguide as:*

_{g}*n*

*≡*

_{g}*c*/

*v*

*) we substitute Eq. (14) into Eq. (11) and rewrite the confinement factor as:*

_{g}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]

## 3. Numerical verification of analytical results

_{2}. Figure 2(b) shows the fundamental TM mode calculated using a Matlab-based finite difference mode solver. We simulate material gain by introducing an imaginary part of the dielectric constant to the low-index slot region. For each value of material gain we use the finite difference mode solver to calculate the complex propagation constant and calculate the corresponding modal gain according to Eq. (7).

*g*

*) versus material gain (*

_{m}*g*

*) in Fig. 1(c) we show excellent agreement with the analytically calculated confinement factor. The slope of the line*

_{b}*g*

*versus*

_{m}*g*

*(circles in Fig. 1(c)) represents the confinement factor according to Eq. (3). We also calculate the confinement factor Γ based on Eq. (15) and the calculated TM mode in Fig. 1(b). We plot Γ*

_{b}*g*

*as the dashed line in Fig. 1(c). As expected, our calculated confinement factor agrees very well with the relationship between the modal and material gain from numerical simulations. The difference between these two factors (0.4328 from Eq. (15) and 0.4368 from the numerical simulations) is less than 1%. To highlight the difference between this confinement factor and the confinement of power to the gain medium we also plot on this graph*

_{b}*P*

_{A}*g*

*where we define the power in the active region as:*

_{b}**H**field (See Figs. 1(e) and 1(f)).

## 4. Minimizing the lasing threshold

*n*

*is the refractive index of the bulk material and*

_{b}*γ*

*is the energy density confinement in the material similar to Eq. (16). In general there will be several loss mechanisms which can be written as sum of terms of the form Eq. (18). To achieve lasing, the threshold condition requires the modal gain per unit length be greater than the modal loss per unit length:*

_{b}*n*

*increases the time it takes light to propagate through the waveguide which increases both the gain*

_{g}*and*loss per unit length equally.

*γ*

*). Although*

_{A}*γ*

*is generally larger than the power confinement in the slot, it is always less than or equal to 1. Therefore the lowest lasing threshold for these structures is limited by the bulk material gain. An identical result can be derived by analyzing resonant cavities. In that case*

_{A}*γ*

*relates the material to modal gain per unit*

_{A}*time*[19].

*α*

*is often an experimentally measured parameter with units of inverse length, we can substitute this measured quantity into Eq. (19) and rewrite the lasing threshold condition as:*

_{m}## 5. Scaling behavior of gain versus slot thickness

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]

*γ*

*) as a function of slot thickness which is plotted in Fig. 3(a). The maximum near a slot width of 60 nm illustrates the important point that the tradeoff between emission rate (which*

_{A}*increases*as the slot is narrowed) and material volume (which

*decreases*as the slot is narrowed) results in an optimal slot width for minimizing the lasing threshold. Initially, as the slot narrows from 120 nm, the increased emission rate more than compensates for the decrease in volume of gain material, and

*γ*

*increases. Near a thickness of about 50 nm the emission rate begins to saturate as it approaches its maximum value determined by the index contrast between the high and low index regions. After this point, further reduction of the slot thickness decreases the volume of material contributing to the gain without much enhancement of the emission rate, and the result is a sharp drop in*

_{A}*γ*

*.*

_{A}*both*the magnitude of the modal gain

*and*the optimal geometry are miscalculated using this method. The gain calculated numerically (in agreement with Γ) is nearly twice as large as would be expected based on the power confined to the slot mode. Additionally, the optimal slot width is miscalculated by more than 10% using power confinement.

## 6. Optimizing slot waveguide geometry

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]

_{2}(1.46) as the high and low index material respectively, and a wavelength of 1.55 µm. We have assumed that gain only occurs in the slot region. Figures 4(b)–4(d) show respectively the total confinement factor (Γ), the group index normalized to the slot index (

*n*

*/*

_{g}*n*

*), and the energy density confinement factor (*

_{A}*γ*

*) as a function of height and width of the waveguide.*

_{A}*γ*

*) shows no well-defined optimum. This is because the lasing threshold will scale only with the energy density confinement. The value of*

_{A}*γ*

*(Fig. 4(d)) increases monotonically with waveguide width and asymptotically approaches the value for an infinitely wide slab waveguide, which we calculate to have a maximum of*

_{A}*γ*

*=0.137 for a waveguide height of 340 nm.*

_{A}## 7. Discussion and conclusions

10. F. Dell’Olio and V. M. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express **15**, 4977–4993 (2007). [CrossRef] [PubMed]

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]

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]

## Acknowledgments

## References and links

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 |

2. | V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. |

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. |

4. | C. A. Barrios and M. Lipson, “Electrically driven silicon resonant light emitting device based on slot-waveguide,” Opt. Express |

5. | F. Ning-Ning, J. Michel, and L. C. Kimerling, “Optical field concentration in low-index waveguides,” IEEE J. Quantum Electron . |

6. | J. T. Robinson, C. Manolatou, C. Long, and M. Lipson, “Ultrasmall mode volumes in dielectric optical microcavities,” Phys. Rev. Lett. |

7. | E. Burstein and C. Weisbuch, eds. |

8. | T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron . |

9. | C. A. Barrios, K. B. Gylfason, B. Sánchez, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, “Slot-waveguide biochemical sensor,” Opt. Lett. |

10. | F. Dell’Olio and V. M. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express |

11. | H. Kogelnik, |

12. | L. A. Coldren and S. W. Corzine, |

13. | C. Pollock and M. Lipson, |

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. |

15. | J. D. Jackson, |

16. | H. A. Haus, |

17. | L. D. Landau and E. M. Lifshitz, |

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 . |

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, |

21. | J. T. Robinson, L. Chen, and M. Lipson, “On-chip gas detection in silicon optical microcavities,” Opt. Express |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(140.5680) Lasers and laser optics : Rare earth and transition metal solid-state lasers

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: July 15, 2008

Revised Manuscript: September 22, 2008

Manuscript Accepted: September 30, 2008

Published: October 3, 2008

**Citation**

Jacob T. Robinson, Kyle Preston, Oskar Painter, and Michal Lipson, "First-principle derivation of gain in high-index-contrast waveguides," Opt. Express **16**, 16659-16669 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16659

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### References

- 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]
- V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and confining light in void nanostructure," Opt. Lett. 29, 1209-1211 (2004). [CrossRef] [PubMed]
- 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]
- 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]
- 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. T. Robinson, C. Manolatou, C. Long, and M. Lipson, "Ultrasmall mode volumes in dielectric optical microcavities," Phys. Rev. Lett. 95, 143901 (2005). [CrossRef] [PubMed]
- E. Burstein and C. Weisbuch, eds., Confined electrons and photons, (Plenum Press: New York, NY,1995) [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]
- 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]
- F. Dell'Olio and V. M. Passaro, "Optical sensing by optimized silicon slot waveguides," Opt. Express 15, 4977-4993 (2007). [CrossRef] [PubMed]
- 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).
- 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]
- 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).
- 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]
- 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).
- J. T. Robinson, L. Chen, and M. Lipson, "On-chip gas detection in silicon optical microcavities," Opt. Express 16, 4296-4301 (2008). [CrossRef] [PubMed]

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