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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 14 — Jul. 5, 2010
  • pp: 15039–15053
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Theory of plasmonic Fabry-Perot nanolasers

Shu-Wei Chang, Tzy-Rong Lin, and Shun Lien Chuang  »View Author Affiliations


Optics Express, Vol. 18, Issue 14, pp. 15039-15053 (2010)
http://dx.doi.org/10.1364/OE.18.015039


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Abstract

Semiconductor plasmonic lasers at submicron and nanometer scales exhibit many characteristics distinct from those of their conventional counterparts at micron scales. The differences originate from their small sizes and the presence of metal plasma surrounding the cavity. To design a laser of this type, features such as metal dispersion, optical energy confinement, and group velocity have to be taken into account properly. In this paper, we provide a comprehensive approach to the design and performance evaluation of plasmonic Fabry-Perot nanolasers. In particular, we show the proper procedure to obtain the key parameters, especially the quality factor and threshold gain, which are usually neglected in conventional semiconductor Fabry-Perot lasers but become important for nanolasers.

© 2010 Optical Society of America

1. Introduction

Fig. 1. Two Ag-coated circular structures: (a) the nanocavity with a short vertical post and a subwavelength cross section, and (b) the micron-long cylinder (with a subwavelength cross section) which acts as the FP nanolaser cavity. In both cases, the In0.53Ga0.47As bulk semiconductor is used as the gain medium.

In this paper, we first present the rate equations for a multi-longitudinal-mode plasmonic FP nanolaser. The laser structure is a silver (Ag) coated circular FP waveguide cavity [8

8. A. V. Maslov and C. Z. Ning, “Size reduction of a semiconductor nanowire laser by using metal coating,” Proc. SPIE 6468, 64680I (2007). [CrossRef]

]. While the real laser device may be more complicated due to various technical concerns, this simple structure is a good example to address the issues mentioned earlier. We then explore various physical parameters including guided-mode properties, facet reflectivities, confinement factors, group velocities, quality factors, and threshold material gain of this plasmonic FP nanolaser as well as their differences from those of typical dielectric FP lasers. Finally, we utilize these parameters in the rate equations to investigate the operation characteristics of this plasmonic FP nanolaser.

2. Rate Equations

Figure 1(a) shows a silver-coated nanocavity with a short vertical post and subwavelength cross section. For devices with electrical current injection, a thin insulator layer is present between the cladding and core to avoid short circuit. This structure is the limiting case of the FP nanocavity in Fig. 1(b) when the dimension of multiple standing waves is shrunk to about half of an effective wavelength. Additional feedback structures are usually required to reduce the significant radiation loss in these structures [3

3. M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007). [CrossRef]

]. For such a small cavity, only a few modes in the interested frequency range exist, and often only one mode (or a few if allowed by cavity symmetry) can reach the threshold condition for lasing. In this case, the rate equations with a single lasing mode are sufficient to understand the laser behavior. On the other hand, when the post length increases to micron size, modes correspond to multiple standing waves are present. Although their resonance frequencies may be quite separated, often the broad gain spectrum in semiconductors under high pumping does not mean the mode with lowest threshold material gain is the ultimate lasing mode. Under such circumstances, the interplay between semiconductor gains and parameters characterizing the modes becomes an important factor for the lasing behavior. The rate equations with multiple modes, which take these subtleties into account, are required to understand the lasing action of such a plasmonic FP nanolaser.

We use the general multimode rate equations to model the performance of a nanolaser:

nt=ηiIqVaRnr(n)Rsp,cont(n)ΣbRsp,b(n)ΣbRst,b(n)sb,
(1a)
Sbt=Sbτp,b+ΓE,bRsp,b(n)+ΓE,bRst,b(n)Sb,
(1b)
Rnr(n)=An+Cn3,
(1c)
Rsp,cont(n)=1τrad1VaΣc,v,kfc,k(1fv,k),
(1d)
Rst,b(n)=vg,a(ωb)g(ωb,n),
(1e)

where n is the carrier density in the active region; ηi is the injection efficiency; I is the injection current from the top contact to the active region; q is the electron charge; Va is the volume of the active region; R nr(n) is the nonradiative recombination rate modeled with a defect recombination coefficient A and Auger coefficient C; R sp,cont(n) and R sp,b(n) are the spontaneous emission rates directly into the continuum modes and a discrete mode b, respectively; R st,b(n), Sb, τ p,b, and ΓE,b are the stimulated emission coefficient, photon density, photon lifetime, and energy confinement factor of mode b; v g,a(ωb) and g(h̅ωb,n) are the material group velocity and material gain in the active region for mode b, respectively; and v g,a(ωb) = c/n g,a(ωb), where n g,a(ωb) is the material group index in the active region. The coefficient A in Eq. (1c) is assumed to originate mainly from surface recombination, namely, A = vsΩa/Va, where vs is the surface recombination velocity, and Ωa is the total surface area of the active region. Also, in Eq. (1d), the spontaneous emission R sp,cont(n) into continuum modes is described by an effective photon lifetime τ rad, and f c,k and f v,k are the occupation numbers at wave vector k in conduction band c and valence band v, respectively [9

9. S. W. Chang, C. Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008). [CrossRef] [PubMed]

].

Various properties of the modes determine the corresponding parameters entering the above rate equations.We will explore the key physical quantities and their expressions in the following sections. The details of other quantities can be found in Ref. [6

6. S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE. J. Quantum. Electron. 45, 1004–1013 (2009). [CrossRef]

].

3. Guided Modes in the Plasmonic Circular Waveguide

As shown in Fig. 1(b), the propagation direction of the guided modes is denoted as the z direction. The cladding is assumed to be infinitely thick, which is a good approximation if the metal cladding is at least several skin depths thick. The material in the core is In0.53Ga0.47As bulk semiconductor, which functions as the gain medium. The diameter D of the core is 300 nm, and the cavity length L is 3 µm. At the two ends of the circular cylinder, thin Ag films can be coated to reduce the mirror loss.

There are two cases for the guided modes. If the propagation constant kz < k 0(Re[ε s(ω)])1/2, where k 0 is the vacuum wave vector, the guided modes are similar to those of a typical fiber waveguide. On the other hand, if kz > k 0(Re[ε s(ω)])1/2, the guided mode is surface-wave-like. In the case of kz < k 0(Re[ε s(ω)])1/2, the guided modes can be classified into transverse-electric modes TE0n and transverse-magnetic modes TM0n (azimuthal mode number m = 0), as well as HEmn and EHmn modes (m > 0). We follow the mode convention in Refs. [11

11. A. Kapoor and G. S. Singh, “Mode classification in cylindrical dielectric waveguide,” J. Lightwave. Technol. 18, 849 (2000). [CrossRef]

] and [12

12. C. S. Lee, S. W. Lee, and S. L. Chuang, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34, 773 (1986). [CrossRef]

]. The HEmn and EHmn modes approach TEmn and TMmn modes, respectively, as the silver-coated waveguide approaches a PEC waveguide, for example, when metal permittivity approaches negative infinity, or the thickness of an insulator layer between metal and semiconductor core approaches zero. With this convention, the HEmn modes are TE-like, and EHmn modes are TM-like in the regime kz < k 0(Re[ε s(ω)])1/2. For the other case kz > k 0(Re[ε s(ω)])1/2, there is exactly one mode for each azimuthal mode number m. We denote these modes as surface-plasma-polariton modes (SPPm), which are TM-like.

Figure 2(a) shows the effective index n effkz/k 0 of the guided modes in the lossless waveguide. In Fig. 2(b), we also show the counterparts of various guided modes with the same core but covered by the perfect electric conductor (PEC). For the PEC circular waveguide, the classifications of the TEmn and TMmn modes are exact. The comparison of the modes from two waveguides indicates the existence of the cutoff frequencies at which kz and n eff approach zero. These waveguide cutoffs are different from those of the guided modes in a dielectric waveguide, at which kz and n eff do not approach zero, nevertheless, with guided modes turned into radiation modes. From this viewpoint, plasmonic waveguides are more similar to PEC waveguides than to dielectric waveguides in nature. On the other hand, plasmonic and PEC waveguides are still different. The TM01 and HEm1 modes of the plasmonic waveguide gradually evolve into the SPPm modes, which are absent in the PEC waveguides. Except for TM01 mode, the turning point of each HEm1 mode into the corresponding SPPm mode marks the transition from TE-like into the TM-like behavior. This fact also explains why the TE01 mode cannot evolve into a surface-plasma-polariton mode because it lacks the necessary field components 𝓗z(ρ) and 𝓔ρ (ρ) for a TM-like surface wave at any frequency. Also, the frequencies of the SPPm modes can never exceed the plasmon resonance frequency ω res at which Re[ε p(ω res) + ε s(ω res)] = 0. Since the SPPm modes (m ≥ 1) evolve from corresponding HEm1 modes, it implies that each HEm1 mode has to emerge below ω res. Note that the plasmon resonance creates a very steep dispersion for SPPm modes nearby. In reality, this steepness will be smeared out by the imaginary parts Im[ε p(ω)] and Im[ε s(ω)] of permittivities, and is not easily observed experimentally.

The imaginary parts Im[ε p(ω)] and Im[ε s(ω)] of the permittivities make the propagation constant kz complex. The real parts of the propagation constant and effective index are not affected much except near the cutoff and plasmon resonance. The imaginary parts Im[ε p(ω)] and Im[ε s(ω)] make Re[n eff] slightly nonzero and thus convert the strict cutoff into a soft cutoff. Figure 3(a) shows the effective index from two-dimensional (2D) finite-element method (FEM) for a few guided modes with Im[ε p(ω)] included but Im[ε s(ω)] still set to zero, corresponding to the condition of ideal cold cavity. The counterparts in the lossless waveguide are also plotted for comparison. The agreement of the two calculations is in general good except near the cutoff and plasmon resonance.

αi=ωε02PzAdρIm[ε(ρ,ω)](ρ)2=12η0PzAdρ(ρ)2n(ρ,ω)α(ρ,ω),
(2)
Pz=Adρ12Re[(ρ)×*(ρ)]·ẑ,
(3)

Fig. 2. (a) The effective index of the plasmonic circular waveguide shown in Fig. 1(b). The TM0 mode and each of HEm1 (m ≥ 1) will evolve into the SPPm modes as the photon energy increases. (b) The counterpart of (a) but with silver replaced by the perfect electric conductor. The guided modes in (a) and (b) both show the cutoff behavior (n eff → 0).
Fig. 3. (a) The real parts of effective indices of a few guided modes obtained by 2D FEM (symbol) and by the calculation of lossless waveguide (line). (b) The modal losses of the same modes obtained by 2D FEM (symbol) and variational approach (line). The results from two methods agree very well except near the cutoff and plasmon resonance.

In the rest part of the paper, we will use the results from the lossless waveguide and adopt the variational approach.

4. Facet Reflectivity

Fig. 4. (a) The standing-wave pattern of the the x-polarized field in the xz plane for the HE11 mode at λ = 1443.65 nm. The facet at z = 0 is waveguide/air interface. (b) The curve fitting of normalized ∣𝓔x(z)∣ at ρ = 0. Due to the contamination of the near field near the facet, the fitted curve slightly deviates from the original data there. The obtained reflection coefficient has a magnitude of 0.881 and phase angle of −0.165 π.
SWREmaxEmin,
(4)

r=SWR1SWR+1=EmaxEminEmax+Emin.
(5)

(z)=inceikz(zzfacet)+rinceikz(zzfacet)
=ince2Im[kz](zzfacet)+r2e2Im[kz](zzfacet)+2rcos{2Re[kz](zzfacet)θr},
(6)
r=reiθr,
(7)

where 𝓔 inc is the incident amplitude of dominant transverse component at the transverse coordinate ρ, z facet is the position of the reflection facet; and θr is the phase angle of the complex reflection coefficient. In Eq. (6), the three quantities ∣𝓔 inc∣, ∣r∣, and θr are the fitting parameters to be extracted while Re[kz] and Im[kz] can be obtained from other calculations such as 2D FEM and are given as the input parameters. When using Eq. (7), the reflection components of other guided modes should be projected out. In our calculations, the standing-wave patterns show that the amplitudes of other excited and reflected guided modes are insignificant, and we will not consider this effect.

Fig. 5. (a) The material group index n g,a(ω) of In0.53Ga0.47As and waveguide group indices n g,z(ω) of the HE11 and SPP1 modes as a function of photon energy. The two waveguide group indices are always larger than that of In0.53Ga0.47As, and the deviation can be significant near the waveguide cutoff of HE11 mode or plasmon resonance of SPP1 mode. (b) The waveguide confinement factor Γwg and energy confinement factor ΓE for the HE11 and SPP1 modes. The waveguide confinement factors follow the trends of waveguide group indices while the energy confinement factor is always smaller than unity.

In Fig. 4, we present an example of reflectivity calculation. The incident wave consists of the HE11 mode. The junction between the waveguide and air plays the role of reflection facet, and we set the facet position z facet = 0. The wavelength is 1443.65 nm, which is one of the resonance wavelengths of this plasmonic circular cavity. In Fig. 4(a), we show the standing-wave pattern of the x-polarized field in the yz plane. This 3D calculation is performed using the commercial FEM software COMSOL Multiphysics [15

15. COMSOL Inc., http://www.comsol.com.

]. To minimize the unwanted reflection from the computation domain, perfect matched layers (PMLs) are constructed at the boundaries of the free space. We can see that a clear standing-wave pattern of the HE11 mode is formed in the waveguide region, and part of the incident power has been transferred to the free space. From the normalized field strength ∣𝓔x(z)∣ [max(∣𝓔x(z)∣) = 1] at ρ = 0 [Fig. 4(b)], we can obtain ∣r∣ = 0.881 (R = 0.777), and θr = −0.165 π. When performing this kind of curve fitting, it is necessary to exclude the first one or two standing wave periods because they are often contaminated by the near field close to the facet, as can be seen from the slight deviation between the fitted curve and original data near z=0. The effective index n eff of this mode is 2.927+0.01007i. If we use the Fresnel formula with unity refractive index of air and effective index of the mode, we would obtain ∣r∣ = 0.491 (R = 0.241) and θr = 8.47 × 10−4 π, which underestimates the reflectivity. Although the enhancement of the reflectivity for this small waveguide compared with that of the conventional waveguide seems promising, the much shorter waveguide length nevertheless enlarges the mirror loss significantly. The Ag coating, which can further decrease the mirror loss, will be considered in Section 6.

5. Quality Factor and Threshold Material Gain

1τp=vg,z(ω)[αi+12Lln(1R2)],
(8a)
1Q=1ωτp=vg,z(ω)ω[αi+1Lln(1R)]=1Qabs+1Qmir,
(8b)
1Qabs=vg,z(ω)αiω,1Qmir=vg,z(ω)ω1Lln(1R),
(8c)

1τpvg,z(ω)Γwggth=vg,a(ω)ΓEgth,
(9)

Γwg=na(ω)2η0PzAadρ(ρ)2,
(10a)
ΓE=Aadρε04{εg,a(ω)+Re[εa(ω)]}(ρ)2Adρε04{εg(ρ,ω)+Re[ε(ρ,ω)]}(ρ)2.
(10b)

In Eq. (10b), ε g(ρ) is the group permittivity, defined as Re[ωε(ρ)]/∂ω, and the quantities with subscript “a” are physical parameters specific to the active gain medium (In0.53Ga0.47As). The two confinement factors in Eq. (9) with their respective group velocities will result in the same threshold material gain g th. In addition, if the waveguide confinement factor Γwg is used, Eqs. (9) and (10a) lead to the well-known gain-loss balance condition of a FP cavity. In Fig. 5(b), we show both Γwg and ΓE for the HE11 and SPP1 modes. While Γwg is always larger than unity and can become huge at the waveguide cutoff of HE11 mode and plasmon resonance of SPP1 mode, the energy confinement factor remains smaller than unity, which is physical because the sum of ε g(ρ) and Re[ε (ρ)] is always larger than zero, even for metals [17

17. S. W. Chang and S. L. Chuang, “Normal modes for plasmonic nanolasers with dispersive and inhomogeneous media,” Opt. Lett. 34, 91–93 (2009). [CrossRef]

]. In addition, in Eq. (9), while the effect of huge Γwg on the product v g,z(ωwg = cΓwg/n g,z(ω) will be eventually compensated by the waveguide group index n g,z(ω) that shows the same trend as that of Γwg near the HE11 waveguide cutoff and SPP1 plasmon resonance, the energy confinement factor ΓE is more indicative of how well the mode overlaps with the gain medium. We will show that it is the energy confinement factor ΓE rather than Q factor that predicts the right trend of threshold material gain.

Fig. 6. The resonance spectrum of the store energy for the FP cavity with waveguide/air interfaces.

To verify the expression of the Q factor in Eq. (8b), we use an alternative approach to obtain this quantity. If we use a source with a variable frequency to excite the modes in the cavity, the spectral response of the storage energy U in the cavity, defined as the energy-density integral in the active volume V a (In0.53Ga0.47As)

U=Vadrε04{εg,a(ω)+Re[εa(ω)]}(r)2,
(11)

will show resonance peaks corresponding to different mode wavelengths [18

18. M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micro-pillar cavity quality factors calculated with finite element methods,” Opt. Express 17, 1144 (2009). [CrossRef] [PubMed]

]. In Eq. (11), the 3D electric-field profile 𝓔 (r) will automatically contain the resonance condition of standing wave. For a particular cavity mode whose resonance wavelength and full-width-at-half-maximum (FWHM) linewidth are λ r and Δλ, respectively, its Q factor is estimated as

Table 1. The reflectivities, quality factors, and threshold material gain of the resonance modes for the FP cavity with waveguide/air interfaces. Q (FP) is obtained using Eq. (8b). Q (3D) is obtained using FEM in Fig. 6.

table-icon
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Fig. 7. (a) The resonance spectrum for a silver coating of thickness t = 10 nm on the waveguide facet at z = 0. (b) The standing-wave pattern of the cavity mode at λ = 1414.63 nm in the yz plane. The 10 nm silver coating reduces the mirror loss and sharpens the standingwave pattern. The transmitted power is lower than that of the uncoated waveguide/air interface at z = 0 shown in Fig. 4(a).
QλrΔλ.
(12)

6. The Effect of High-Reflectivity Metal Coating

The metal coating at FP cavity facets can increase the reflectivity R and thus reduce the mirror loss. Since the material properties of metal cannot be altered much, a proper design of coating thickness should reduce the overall loss so that a reasonable and achievable material gain is able to support lasing action. For bulk semiconductors, the typical material gain under a reasonable carrier density is about a few hundreds at room temperature. It is thus necessary to lower the mirror loss of air/waveguide interfaces, which have a threshold of over 1000 cm−1 as shown in Table 1.

Table 2. The reflectivities, quality factors, and threshold material gains of the resonance modes for the FP cavity with the 10 nm Ag coating. Q (FP) is obtained using Eq. (8b). Q (3D) is obtained using FEM in Fig. 7(a).

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Fig. 8. (a) The resonance spectrum for a silver coating of a thickness t = 30 nm. (b) The standing-wave pattern of the cavity mode at λ = 1405.65 nm in the yz plane. Most of the power is reflected back into the waveguide due to the thin silver coating on the facet, which results in an even sharper standing-wave pattern than that in Fig. 7(b).

Figure 8 shows similar plots to Fig. 7 with a silver coating thickness t = 30 nm. From Fig. 8(b), the standing-wave pattern of the mode at λ = 1405.65 nm shows an even smaller portion of the output power than that of the case with a 10 nm silver coating. From Table 3, the Q factor of each mode exceeds 240 due to the high reflectivity (> 97%). The corresponding threshold material gains all drop below 670 cm−1. At this thickness, due to the more accurate reflectivity calculations, the Q factors from the FP formula and 3D FEM calculations agree very well.

Table 3. The reflectivities, quality factors, and threshold material gains of the resonance modes for the FP cavity with the 30 nm Ag coating. Q (FP) is obtained using Eq. (8b). Q (3D) is obtained using FEM in Fig. 8(a).

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A drawback of the high-reflectivity coating is the poor extraction efficiency. An accompanying issue is the amount of output power under a high-reflectivity condition. Since the energy from the stimulated emissions inside the cavity will finally vanish, it is either dissipated by the material loss or escapes the cavity. Although reducing the amount of energy leaving the cavity can decrease the threshold, it implies that a significant percentage of the generated photons become metal heat rather than output, which reduces the extraction efficiency. Thus, a proper thickness should be designed so that the threshold material gain is achievable while a reasonable extraction efficiency and output power can still be maintained.

7. Numerical Results of the Plasmonic Fabry-Perot NanoLaser

The final issues are whether In0.53Ga0.47As has sufficient gain so that this plasmonic nanolaser can lase at room temperature, and which FP mode lases. In the following calculation, we focus on the case of 30 nm Ag coating on the waveguide facet. The photon densities corresponding to the four FP modes in Table 3 (each has a degeneracy of two for m = 1) are included in the rate equations. There may be other cavity modes corresponding to different transverse or standing-wave profiles. However, these modes usually have much higher thresholds, so we do not take them into account. The surface recombination velocity is set to 200 cm/s, corresponding to a coefficient A = 2.8 × 109 s−1 [19

19. E. Yablonovitch, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In0.53Ga0.47As heterojunction regrowth on chemically prepared In0.53Ga0.47As surfaces,” Appl. Phys. Lett. 60, 371 (1992). [CrossRef]

]. The Auger coefficient is set to 4.67 × 10−29 cm6s−1 [20

20. Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 µm compressively strained semiconductor lasers,” IEEE. J. Quantum. Electron. 29, 1565–1575 (1993). [CrossRef]

]. The injection efficiency is assumed to be unity, and the radiative lifetime τ rad for the continuum modes is set to 5 µs. In Fig. 9, we show the room-temperature gain spectra of bulk In0.53Ga0.47As under different carrier densities. From Table 3, the threshold material gains of these FP modes range from 610–670 cm−1. The gain spectra indicate that the required threshold carrier density is between 3–4 × 1018 cm−3, which is achievable for In0.53Ga0.47As at room temperature.

Fig. 9. The gain spectra of bulk In0.53Ga0.47As under different carrier densities.
Fig. 10. (a) The stimulated emission rates of the four FP modes and nonradiative recombination rate by solving coupled Eqs. (1a)–(1e) for four modes. Around a current of 0.79 mA, the stimulated emission rate of the mode at λ = 1529.59 nm surpasses the nonradiative recombination rate and indicates that the lasing action begins to dominate. (b) The spontaneous emission rates of the four FP modes. Although only the mode at λ = 1529.59 nm lases, the four spontaneous emission rates are close in magnitude.
Fig. 11. The output powers of four FP modes by solving coupled Eqs. (1a)–(1e). They resemble the trends of stimulated emission rates in Fig. 10(a) once the population inversion is reached.

Figure 11 shows the output powers of the four modes, taking into account that about 30–37% of the power is absorbed when passing through the Ag-coated facets. The light output power Pb of mode b is

Pb=ωbVaSbΓE,b[Tb1Rbvg,z(ωb)Lln(1Rb)],
(13)

When the cavity length L is short, the field normalization and energy confinement factor with the 3D field profile should be used, as formulated in Ref. [6

6. S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE. J. Quantum. Electron. 45, 1004–1013 (2009). [CrossRef]

]. The FP formulation assumes a field of the single-mode forward and backward wave propagations exp(±ikzz) in a long cavity, and thus the mode mixing near the facets and field variation of standing wave are ignored. An ultrashort cavity can have significant mode mixing near waveguide boundaries and a profile with only a few standing waves. Under such circumstances, the direct 3D field solution should be used, and the energy confinement factor based on the 3D rather than 2D field in Eq. (10b) needs to be applied.

8. Conclusion

We have presented a rigorous rate-equation model including important physical parameters on the analysis of plasmonic Fabry-Perot nanolaser using an In0.53Ga0.47As circular waveguide with a silver cladding. Many approximations valid in conventional semiconductor Fabry-Perot lasers cannot be directly applied to the plasmonic Fabry-Perot laser in the submicron or nanometer scales. We show the importance of the reflectivity and waveguide group velocity when evaluating the quality factor of the cold cavity, where the approximations of the Fresnel formula and material group velocity may not be appropriate. These effects can influence the estimation of the required threshold material gain. Our theory accounts for metal plasma and waveguide dispersion (waveguide group index), and energy confinement factor. The theoretical formulation not only provides the design and operation condition of the plasmonic nanolasers but also can be used to predict the performance, such as the light output power as a function of the injection current. Future work on the high-speed modulation of plasmonic nanolasers is in progress.

Acknowledgments

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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]

14.

S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053106 (2005). [CrossRef]

15.

COMSOL Inc., http://www.comsol.com.

16.

C. Y. Lu, S. W. Chang, S. H. Yang, and S. L. Chuang, “Quantum-dot laser with a metal-coated waveguide under continuous-wave operation at room temperature,” Appl. Phys. Lett. 95, 233507 (2009). [CrossRef]

17.

S. W. Chang and S. L. Chuang, “Normal modes for plasmonic nanolasers with dispersive and inhomogeneous media,” Opt. Lett. 34, 91–93 (2009). [CrossRef]

18.

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micro-pillar cavity quality factors calculated with finite element methods,” Opt. Express 17, 1144 (2009). [CrossRef] [PubMed]

19.

E. Yablonovitch, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In0.53Ga0.47As heterojunction regrowth on chemically prepared In0.53Ga0.47As surfaces,” Appl. Phys. Lett. 60, 371 (1992). [CrossRef]

20.

Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 µm compressively strained semiconductor lasers,” IEEE. J. Quantum. Electron. 29, 1565–1575 (1993). [CrossRef]

OCIS Codes
(230.7370) Optical devices : Waveguides
(250.5403) Optoelectronics : Plasmonics
(250.5960) Optoelectronics : Semiconductor lasers

ToC Category:
Optoelectronics

History
Original Manuscript: May 17, 2010
Revised Manuscript: June 17, 2010
Manuscript Accepted: June 21, 2010
Published: June 29, 2010

Citation
Shu-Wei Chang, Tzy-Rong Lin, and Shun Lien Chuang, "Theory of Plasmonic Fabry-Perot Nanolasers," Opt. Express 18, 15039-15053 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-15039


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References

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  5. C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010). [CrossRef]
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  9. S. W. Chang, C. Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008). [CrossRef] [PubMed]
  10. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
  11. A. Kapoor and G. S. Singh, “Mode classification in cylindrical dielectric waveguide,” J. Lightwave Technol. 18, 849 (2000). [CrossRef]
  12. C. S. Lee, S. W. Lee, and S. L. Chuang, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34, 773 (1986). [CrossRef]
  13. 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]
  14. S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053106 (2005). [CrossRef]
  15. COMSOL Inc, http://www.comsol.com.
  16. C. Y. Lu, S. W. Chang, S. H. Yang, and S. L. Chuang, “Quantum-dot laser with a metal-coated waveguide under continuous-wave operation at room temperature,” Appl. Phys. Lett. 95, 233507 (2009). [CrossRef]
  17. S. W. Chang and S. L. Chuang, “Normal modes for plasmonic nanolasers with dispersive and inhomogeneous media,” Opt. Lett. 34, 91–93 (2009). [CrossRef]
  18. M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micro-pillar cavity quality factors calculated with finite element methods,” Opt. Express 17, 1144 (2009). [CrossRef] [PubMed]
  19. E. Yablonovitch, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In0.53Ga0.47As heterojunction regrowth on chemically prepared In0.53Ga0.47As surfaces,” Appl. Phys. Lett. 60, 371 (1992). [CrossRef]
  20. Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 μm compressively strained semiconductor lasers,” IEEE J. Quantum Electron. 29, 1565–1575 (1993). [CrossRef]

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