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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 21 — Oct. 13, 2008
  • pp: 16659–16669
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First-principle derivation of gain in high-index-contrast waveguides

Jacob T. Robinson, Kyle Preston, Oskar Painter, and Michal Lipson  »View Author Affiliations


Optics Express, Vol. 16, Issue 21, pp. 16659-16669 (2008)
http://dx.doi.org/10.1364/OE.16.016659


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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

A critical photonic component yet to be demonstrated on a silicon-based platform is an electrically pumped device with optical gain for amplification or lasing. Achieving this optical gain in a silicon-based device is extremely challenging since silicon is an indirect band-gap semiconductor and therefore an inefficient photon source. Hybrid silicon devices based on direct band-gap III–V materials bonded to silicon photonic elements present an interim solution [1

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]

], however, the reliance on a wafer bonding step prohibits a high throughput fabrication process possible with a silicon-based process.

One possible configuration for silicon-based gain is the recently proposed slot waveguide design [2

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

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]

]. In this configuration a low-index gain material such as Er-doped SiO2 or Erdoped Si3N4 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]

] or multiple [5

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).

] thin slots between two silicon rails. Electrical excitation of the gain material could be achieved by passing a tunneling current through the slot-region.

One key advantage of this configuration is the large optical field enhancement in the slot-region due to the boundary conditions imposed on the electric field normal the slot interface [6

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]

]. Since the normal electric displacement (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.

In standard low-index-contrast waveguides, optical gain and power confinement are considered proportional based on the following arguments. For electromagnetic plane waves in homogenous media the magnetic field H can be written in terms of the electric field E and the impedance of the material according to:

H=cεn(êz×E),
(1)

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

]:

Ey=ωμ0βHx,
(2)

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 ωμ0βHx respectively, plotted on the same color scale. The two fields become increasingly dissimilar as more electric field is concentrated at high-index-contrast boundaries.

For high-index-contrast waveguides, however, the linear relationships between the electric and magnetic fields (Eqs. (1) and (2)) do not hold since they must satisfy different boundary conditions. This is shown in Fig. 1. For low-index-contrast waveguides, Eqs. (1) and (2) are close approximations. Figures 1(a) and 1(b) shows the fundamental TM mode with an index difference of 0.25 between the core and cladding. We see close agreement in the magnitude and spatial profiles of these two fields. However, as the index-contrast is increased to 2.5 (Fig. 1(c) and (d)) there is a noticeable difference between E y and ωμ0βHx. Notice that E y must be continuous across the dielectric interfaces to the left and right of the waveguide, and discontinuous across the top and bottom interfaces. The H x 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)).

2. Confinement factor for high-index-contrast waveguides

Γgmgb,
(3)

where g m and g b 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:

E(z)2=E02egbz.
(4)

To calculate the effective confinement factor for a given waveguide mode profile, we begin with an expression for the electric field of a guided mode propagating along the z-direction. This can be written as the sum of all three vector components of the electric field using Einstein summation notation:

E(x,y,z)=êjEjoψj(x,y)ei(ωtβ˜z),
(5)

where ê is the polarization vector, Ψ is the cross sectional mode profile, and β̃ is the complex propagation constant defined as

β˜k0(n̅r+in̅i),
(6)

gm=2Im{β˜}=2k0n̅i.
(7)

Similarly we can write the bulk material gain in terms of the imaginary part of refractive index of the active gain material:

gb=2k0nAi.
(8)

Δβ˜=ωΔε˜E2dxdy12Re{E×H*}êzdxdy,
(9)

where Δε̃=(n A+iΔn Ai)2. According to Eqs. (6)(8) this can be written in the form:

gm=[nAcε0AE2dxdyRe{E×H*}êzdxdy]gb.
(10)

Here 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 Δε̃≠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:

Γ=nAcε0AE2dxdyRe{E×H*}êzdxdy.
(11)

12Re{E×H*}êzdxdy=12βωμ0E2dxdy,
(12)

and thus Eq. (11) could be written as the percentage of power or intensity confined to the active region. However, as shown in Section 1, 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 H=εcn̅(êz×E) [15

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

].

U/l=12εE2dxdy.
(13)

Note that here we have neglected material dispersion when writing the stored energy per unit length. To account for material dispersion one should replace epsilon with d(ωε)/ in Eq. (11) [16

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

, 17

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

]. 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 g) 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:

12Re{E×H*}êzdxdy=vg12εE2dxdy
(14)

Using the definition of group index (n gc/v g) we substitute Eq. (14) into Eq. (11) and rewrite the confinement factor as:

Γ=ngAεE2dxdynAεE2dxdyngnAγA.
(15)

γAAεE2dxdyεE2dxdy.
(16)

3. Numerical verification of analytical results

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.

PAARe{E×H*}ezdxdyRe{E×H*}ezdxdy.
(17)

4. Minimizing the lasing threshold

αm=ngnbγbαb,
(18)

ngγAnAgbngiγiniαi>0.
(19)

We see immediately that the group index can be divided out of Eq. (19). This is because increasing n g increases the time it takes light to propagate through the waveguide which increases both the gain and loss per unit length equally.

gb>αmΓ.
(20)

This quantity αmΓ is a useful figure of merit since it is equal to the minimum bulk material gain needed to reach the lasing threshold in the waveguide. Because the group index cancels out in Eq. (18), this figure of merit can be directly compared for waveguides of different geometries to determine which can achieve lasing with the lowest material gain coefficient.

5. Scaling behavior of gain versus slot thickness

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.

To minimize the lasing threshold we look for a maximum in the spatial confinement factor (γ A) 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 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 A 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 γ A.

6. Optimizing slot waveguide geometry

To optimize the dimensions of a slot waveguide for an electrically pumped silicon laser, we apply the principles in the proceeding sections to achieve a waveguide design with a minimal lasing threshold. We estimate that the slot thickness should be no larger than 10 nm in order to achieve electrical injection via tunneling into an oxide-based gain media using bias voltages on the order of volts [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]

]. Therefore we keep the slot thickness fixed at 10 nm and compute the confinement factors as we vary the height and width of the waveguide. Here we have used the refractive indices of Si (3.48) and SiO2 (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 A), and the energy density confinement factor (γ A) as a function of height and width of the waveguide.

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.

We see from Fig. 4 that the optimal device geometry is relatively insensitive to variation in waveguide dimensions. Around the optimal design, variations of approximately ±50 nm in the total height and width of the waveguide result in changes in the confinement factors of less than 5%. This allows the device performance to be relatively unaffected by size variations which can occur during fabrication.

7. Discussion and conclusions

We have shown that some commonly applied metrics are not appropriate for determining gain in high-index-contrast waveguides, and from first principles developed several figures of merit to characterize waveguide structures for gain. In particular we have shown that the concept of power confinement to the gain region significantly miscalculates the gain experienced by the waveguide mode. Instead we have shown that the true confinement factor which determines gain per unit length results from the combination of group index and confinement of the electric field energy to the gain region. These terms can combine and in some cases exceed unity meaning that one can achieve greater gain per unit length than would be possible in the bulk material. The lasing threshold on the other hand only depends on the percentage of electric field energy in the gain region. To account for this we have introduced a new figure of merit to describe the suitability of a waveguide to achieve low-threshold lasing. This figure of merit is the experimentally measured propagation loss divided by the confinement factor introduced in this paper. The evaluation of this ratio determines the minimal material gain required to achieve lasing in the waveguide structure.

Additionally we have applied our analysis to the design of slot waveguide structures. We have shown that the lasing threshold has a minimum for a particular slot width and increases dramatically as the slot is made thinner. Also we have shown that gain characteristics of the waveguides are fairly insensitive to variations in overall waveguide dimensions.

Since the confinement factors presented here were derived from perturbation theory, they can be applied to other phenomena in high-index-contrast waveguides including refractive index sensing. In deriving the confinement factors presented here we have studied gain as a perturbation of the imaginary part of the refractive index over a given region of the guided mode. The same formalism holds true for perturbations to the real part of the refractive index and therefore the confinement factors for gain presented in the paper can also be used as confinement factors for refractive index sensing [10

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

, 18

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]

] and have shown good agreement with experiment [21

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]

].

In summary, this work provides the qualitative and quantitative analysis necessary in developing high-index-contrast waveguides for applications such as amplification and lasing.

Acknowledgments

The authors gratefully acknowledge Thomas L. Koch for his helpful discussions and Christina Manolatou for the use of her finite difference mode solver. Research support is gratefully acknowledged from the National Science Foundation Center on Materials and Devices for Information Technology Research (CMDITR), DMR-0120967, the National Science Foundation’s CAREER Grant No. 0446571, and the U.S. Air Force MURI program on “Electrically-Pumped Silicon-Based Lasers for Chip-Scale Nanophotonic Systems” supervised by Dr. Gernot Pomrenke.

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 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]

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]

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

  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]
  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]

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