14 March 2016 Optical gain in surface plasmon nanocavity oscillators
Author Affiliations +
J. of Nanophotonics, 10(1), 016015 (2016). doi:10.1117/1.JNP.10.016015
Abstract
An analytical model of optical gain is developed for three types of surface plasmon nano-oscillators: (i) a metal film with a gain medium nanostrip, (ii) a metal film, with nanohole, deposited on a layer of gain medium, and (iii) a nanoparticle coated with a gain medium. The operating frequency of the plasmon laser is close to surface plasmon resonance, hence the cavity size is strongly reduced. The evanescent field of the oscillator stimulates the electron–hole recombination in the gain medium, amplifying the cavity field. With electron–hole occupation probabilities in the relevant energy states in the gain medium just exceeding 50% each, the growth rate exceeds 1013  s1. The excited cavity mode acts as an oscillatory dipole to emit optical radiation.
Liu and Tripathi: Optical gain in surface plasmon nanocavity oscillators

1.

Introduction

The surface plasma wave (SPW), supported by a conductor–dielectric interface, has an important property: as the wave frequency approaches the surface plasmon resonance, the wavelength and transverse extent of the mode resonantly shrink, confining optical frequency waves to nanodimensions, far below the diffraction limit prescribed for body waves.1,2 This has led to the development of nanoelectronics and nano-devices into a major field12.3.4.5 of research. Of significant importance is the development of nanolaser or spaser.67.8.9.10 Noginov et al.11 have recently operated a 531 nm spaser-based nanolaser employing 44 nm diameter nanoparticles with gold core and dye-doped silica shell. Lu et al.12,13 have developed a multilayer plasminic nanolaser using smooth silver film with SiO2 or Al2O3 nanolayer and gallium nitride nanorod deposited on it. The nanorod comprises a layer of indium gallium nitride that acts as gain medium. The theoretical formalisms of spaser are usually based on quantum density matrix approach.6 However, some employ classical model due to its transparent simplicity. Kumar et al.14 developed a classical analytical formalism of two-layer surface plasmon laser amplifier pumped by a forward biased p–n junction.

In this paper, we present theoretical analysis of optical gain in three surface plasmonic nano-oscillators, (i) a metal film with a nanostrip of gain medium, (ii) a nanohole in a metal film deposited on a gain layer,15 and (iii) a nanoparticle coated with gain medium. These are the configurations experimentally studied in recent years and have promise for building arrays of nanoradiators with desired directivity and power. Here we pursue analytical treatment to reveal physics with some clarity which at times gets masked in numerical simulations.

The oscillators operate near the surface plasmon resonance frequency and the quality factor of the cavity is determined by the free electron collisional damping. In the first oscillator, nanostrip provides for lateral localization of the surface plasmons16 by reducing the phase velocity of the SPW (as compared to the metal–vacuum interface). In the second oscillator, mode is localized around the nanohole. In the third case, space charge oscillations of electron sphere with respect to ion sphere provide a natural oscillator. The field of the oscillator mode stimulates electron–hole recombination in the gain medium that in turn amplifies the mode.

In Sec. 2, we study the optical gain in a metal-nanostrip oscillator. In Secs. 3 and 4, we consider the nanohole and nanoparticle oscillators. In Sec. 5, we discuss the results.

2.

Metal Film—Nanostrip Spaser

Consider a metal film of free electron density n00, electron effective mass m, and lattice dielectric constant ϵL. On top of it (x>0) lies a nanostrip with 0<y<W, 0<z<L and dielectric constant ϵd (cf. Fig. 1). The thickness of the metal film is larger than the skin depth of the SPW, hence it can be treated like a semi-infinite medium (x<0). Grooves are created in the metal at z=0, L for longitudinal localization of plasmons.

Fig. 1

Schematic of nanostrip-loaded metal film. The nanostrip of length L, width W, and relative permittivity ϵd localizes the SPW. It contains a layer of gain medium that under optical pumping excites localized surface plasmons.

JNP_10_1_016015_f001.png

Had it been an infinite metal–dielectric interface, it would support a SPW with electric field17

(1)

E=F1(x)ei(ωtk||.r||),
F1=A(k^||ik||αIx^)eαIxfor  x<0
=A(k^||+ik||αIIx^)eαIIxfor  x>0
k||=ωc(ϵmϵdϵm+ϵd)1/2,

(2)

αI=ωc(ϵm2ϵm+ϵd)1/2,αII=ωc(ϵd2ϵm+ϵd)1/2,
where k||=kyy^+kzz^, r||=yy^+zz^, ϵm=ϵL(ωp2/ω2)(1iν/ω) is the effective relative permittivity of the metal, ωp=(n00e2/mϵ0)1/2 is the plasma frequency, e is the electron charge, ν is the electron collision frequency, ϵ0 is the free space permittivity. SPW exists for ω<ωR where

(3)

ωR=ωP/(ϵm+ϵd)1/2.

For a given ω, k|| increases with increasing ϵd.

When one limits the lateral size W of the dielectric layer, the parallel wave number of the SPW on the metal–vacuum interface is

(4)

k0||=ωc(ϵmϵm+1)1/2.

One may take the SPW to be propagating along z^ but with finite ky. Outside the strip, i.e., on the metal–vacuum interface, the SPW would have the same kz as inside the strip. However, since k0||<k||, the SPW field outside the strip would be evanescent in y. One may take SPW to be localized in y and z with

(5)

kyn1π/W,kzn2π/L,
where n1 and n2 are integers.

For the plasmon nano-oscillator, we may deduce the mode structure as follows. Take Hx=0, Ex0. Assuming x, t variations of fields as exp(αxiωt) and using ×E=iωμ0H and ×H=iωϵ0ϵE express Ey, Ez, Hy, Hz in terms of Ex. In different regions one may take the y, z variations of fields to be the same [with ky, kz given by Eq. (5)], in compliance with the boundary conditions at x=0 and allow α to have different values in different media. Thus we write in the region 0<y<W, 0<z<L

(6)

x<0Ex=A1FxeαIxeiωt,Ey,Hz=(αI,iωϵ0ϵm)kyA1k||2FyeαIxeiωt,Ez,Hy=(αI,iωϵ0ϵm)kzA1k||2FzeαIxeiωt,

(7)

x>0Ex=A2FxeαIIxeiωt,Ey,Hz=(αII,iωϵ0ϵd)kyA2k||2FyeαIIxeiωt,Ez,Hz=(αII,iωϵ0ϵd)kzA2k||2FzeαIIxeiωt,
Fx=sin(kyy)sin(kzz),Fy=cos(kyy)sin(kzz),Fz=sin(kyy)cos(kzz),
αI2=k||2ω2c2ϵm,αII2=k||2ω2c2ϵd.

Applying the continuity of ϵEx, Ez at x=0, we obtain A2=A1ϵd/  ϵm and the dispersion relation

(8)

F21+αIϵd/αIIϵm=0,
giving k|| as in Eq. (2). Since k|| is quantized [cf. Eq. (5)], we get discrete eigenfrequencies ω. In the absence of collisions (ν=0), F2 is real and so are the frequencies. For ν0, F2 has a finite imaginary part, F2=F2r(ω)+iF2i. Writing ω=ωriΓ, we obtain

(9)

ωr2=ωp22ϵL[1+(1+ϵLϵd)β2{[1+(1+ϵLϵd)β2]24ϵLϵdβ2}1/2],

(10)

Γ=F2i(F2r/ω)=ν2ωp2ωr2ϵdϵm2+ϵLϵd,
where β2=(π2c2/ωp2W2)(n12+n22W2/L2). We have plotted in Fig. 2(a) the real frequency of a surface plasmon eigenmode as a function of normalized width of the strip for typical parameters. The frequency steadily falls off with the width. Figure 2(b) displays the variation of normalized damping rate with normalized frequency. The damping increases rather rapidly as one approaches the surface plasmon resonance frequency.

Fig. 2

(a) Normalized eigenfrequency of the plasmon oscillator as a function of normalized width of the strip for ϵL=4, ϵd=2.6, n1=1, n2=5, L/W=3. (b) Normalized damping rate as a function of normalized frequency for ν/ω=3×104, ϵL=4, ϵd=2.6.

JNP_10_1_016015_f002.png

The energy density of the electromagnetic fields of the plasmon resonator is

(11)

WEM=ϵ04ω(ωϵ)E.E*+μ04H.H*=ϵ04A12αI2e2αIx[k||2(L+ωp2ω2)sin2(kyy)sin2(kzz)+(ϵm+2ωp2αI2k2ω2){kz2sin2(kyy)cos2(kzz)+ky2cos2(kyy)sin2(kzz)}]for  x<0,=ϵ0ϵdA124αII2e2αIIx[k||2sin2(kyy)sin2(kzz)+kz2sin2(kyy)cos2(kzz)+ky2cos2(kyy)sin2(kzz)]for  x>0,
where ϵ=ϵm for x<0, ϵ=ϵd for x>0. The total energy stored in the SPW is
EEM=0L0WWEMdxdydz=δ1A12,

(12)

δ1=ϵ0WLk216[ϵdαII3+1αI3(ϵLωp2ω2ϵmϵd)].

2.1.

Optical Gain

Now we allow the dielectric strip to comprise a gain medium (e.g., an optically pumped semiconductor layer) of x-width Δ, y-width W, and z-extent L, located at x=d. In the conduction and valence bands, the density of states and occupation probability for electrons and holes (of effective masses me, mh), respectively, are

(13)

ρe(Ee)=14π2(2  me2)3/2Ee1/2,ρh(Eh)=14π2(2mh2)3/2Eh1/2,

(14)

fe(Ee)=[1+exp((EeEFe)/T)]1,fh(Eh)=[1+exp((EhEFh)/T)]1,
where Ee and Eh are the electron and hole energies measured from the bottom of the conduction band upward and top of the value band downward, respectively, EFe and EFh are the Fermi energies related to electron and hole densities ne and nh and temperature T (in energy units).

In a surface plasmon-induced emission process, an electron in energy state Ee in the conduction band goes to energy state Eh in the valence band, recombining with a hole and producing a plasmon of frequency

(15)

ω=(Ee+Eh+Eg)/,
where Eg is the band gap. The k vector of the plasmon is much smaller than that of the electron before or after the transition, hence in a direct band gap parabolic band semiconductor we may take

(16)

Ee=2k2/2me,Eh=2k2/2  mh.

Equations (15) and (16) give the energy states that participate in the stimulated emission process

Ee=(ωEg)/(1+me/mh),

(17)

Eh=(ωEg)/(1+mh/me).

The rate of e–h recombination per unit volume per second is proportional to spectral energy density uωdω=WEMδ(ωωm)dω, density of states ρe(Ee), the occupation probability fe(Ee) of state Ee, the probability of state Eh being vacant, fh(Eh)

(18)

RE=Bρe(Ee)fe(Ee)fh(Eh)WEMδ(ωωm)dω,
where B is the Einstein’s coefficient of stimulated emission and ωm is the eigenfrequency of the SPW mode. Similarly, the rate of transition of electrons from the valence band to conduction band per unit volume per second is

(19)

Rabs=Bρe(Ee)(1fe(Ee))(1fh(Eh))WEMδ(ωωm)dω.

One may remember that Rabs is proportional to the density of states in the valence band ρh(Eh)=(mh/me)ρe(Ee). The net SPW energy produced per unit volume per second on integrating over ω is

(20)

P=Bρe(Ee)(fe(Ee)+fh(Eh)1)ωWEM,
where we have dropped the subscript m over ω. Integrating it over the volume of the gain medium, we obtain the rate of energy gain by the SPW oscillator

(21)

ddtEEM+2ΓEEM=δ2A12,

(22)

δ2=ϵ0ϵdk||28αI2ΔWLωBe2αId.ρe(Ee)[fe(Ee)+fh(Eh)1].

Using Eqs. (12) and (21) we obtain

A12=A002eγt,

(23)

γ=δ2/δ12Γ=2α1Δe2α1dBρe(Ee)ω[αI3/αII3+(ϵLϵmωp2/ω2ϵd)/ϵd].[fe(Ee)+fh(Eh)1]2Γ.

γ is the growth rate of SPW field energy. The threshold for the SPW growth is given as

(24)

fe+fh1+[αI3/αII3+(ϵLϵmωp2/ω2ϵd)/ϵd]νωp2ϵmϵd2(ϵLϵd+ϵm2)α1Δe2α1dBρe(ϵe)ω.

This determines the electron and hole density threshold for the growth of surface plasmons.

The Einstein’s coefficient B is related to the Einstein’s coefficient for spontaneous emission A as B=πAc3/ω3, hence

Bρe(Ee)ω=14πA(2mec2ω)3/2(Eeω)1/2,
which is of the order of 108A. Typical value of A, the inverse e–h recombination time, is 107  s1. Further, for spaser operating around ωωR , ϵmϵd, αI/αII=1. For silver, gallium nitride case ϵm4, ϵd2.5. Thus, when the occupation probabilities fe(Ee) and fh(Eh) are just greater than 0.5 each the threshold for the spaser is exceeded and the growth rate of the order of 1013  s1 is achieved.

3.

Nanohole-Embedded Metal Film Spaser

Consider a metal film of thickness d (0<z<d) and effective relative permittivity m given above. It comprises a transverse hole of radius a and length d with vacuum inside (cf. Fig. 3).

Fig. 3

Schematic of a cylindrical hole drilled in a metal film deposited on a gain medium.

JNP_10_1_016015_f003.png

The cylindrical vacuum–metal interface supports an SPW with evanescent fields in r and forward backward propagating solutions in z. A forward wave SPW field is

(25)

E=F(r)ei(ωtkzz),
F=A1[z^I0(αIr)r^ikzαII1(αIr)]for  r<a
=A2[z^K0(αIIr)+r^ikzαIIK1(αIIr)]for  r>a,
where αI=(kz2ω2/c2)1/2, αII=(kz2ω2ϵm/c2)1/2 and I0, K0, I1, K1 are the modified Bessel functions. The continuity of Ez, ϵEr at r=a gives the dispersion relation

(26)

I1(αIa)K0(αIIa)I0(αIa)K1(αIIa)=αIαII.

The termination of the cylinder at z=0, d quantizes kz and gives standing wave solution to SPW. One may write

(27)

r<aE=A1[z^I0(αIr)cos(kzz)+r^kzαII1(αIr)sin(kzz)]eiωt,H=φ^ωϵ0iαIA1I1(αIr)cos(kzz)eiωt,

(28)

r>aE=A2[z^K0(αIIr)cos(kzz)r^kzαIIK1(αIIr)sin(kzz)]eiωt,H=ϕ^ωϵ0ϵmiαIIA2K1(αIIr)cos(kzz)eiωt,
with A2=A1I0(αIa)/K0(αIIa) and kz=π/d for the fundamental mode. The dispersion relation, Eq. (26), gives the eigenfrequency and damping rate of the SPW mode.

The energy density of fields of the plasmon nano-oscillator is

(29)

WEM=ϵ0A124α12[I12(αIr)(ω2c2cos2(kzz)+kz2sin2(kzz))+αI2I02(αIr)cos2(kzz))]for  r<a=ϵ0A224αII2[K02(αIIr)αII2(ϵL+ωp2ω2)cos2(kzz)+K12(αIIr){kz2(ϵL+ωp2ω2)sin2(kzz)+ω2c2ϵmcos2(kzz)}]for  r>a.
The total energy stored in the SPW is
EEM=δ1A12,δ1=ϵ0d8αI2[G1+G2αI2I02(αIa)αII2K02(αIIa)]
G1=0αIa[I02(ξ)+1αI2(ω2c2+π2d2)I12(ξ)]ξdξ

(30)

G2=αIIa[(ϵL+ωp2ω2)K02(ξ)+K12(ξ)(kz2(ϵL+ωp2ω2)+ω2c2ϵm)]ξdξ.

3.1.

Optical Gain

We allow a thin layer of gain medium of thickness Δ placed at z=0. The net SPW energy produced per unit volume per second, P, is given by Eq. (20). The rate of energy gain of the SPW oscillator is given by Eq. (21) with

(31)

δ2=Bρe(Ee)[fe(Ee)+fh(Eh)1]ωϵ0Δ4αI2.[G3+G4αI2I02(αIa)αII2K02(αIIa)],
G3=0αIa[I02(ξ)+(ω2c2αI2)I12(ξ)]ξdξ
G4=αIIa[(ϵL+ωp2ω2)K02(ξ)+(ω2ϵmc2)K12(ξ)]ξdξ,

The linear damping rate Γ is of the order of the one given by Eq. (10). Equation (21) gives A12=A002exp(γt) with growth rate

(32)

γ=δ2/δ12Γ=2ΔωdBρe(Ee)[fe(Ee)+fh(Eh)1].G3+G4αI2I02(αIa)αII2K02(αIIa)G1+G2αI2I02(αIa)αII2K02(αIIa)2Γ.

The threshold value of fe(Ee)+fh(Eh) above which the SPW grows is obtained by putting γ=0 in Eq. (32). Above the threshold the growth rate is comparable to the earlier case.

4.

Nanoparticle Spaser

Consider a metallic nanoparticle of radius rc, free electron density n00, electron effective mass m, and lattice permittivity ϵL. It is surrounded by a layer of gain medium of thickness Δ and dielectric constant ϵd (cf. Fig. 4).

Fig. 4

A nanoparticle of radius rc covered by a thin layer of gain medium of thickness Δ.

JNP_10_1_016015_f004.png

For a moment, let us take ϵd=1 and give a displacement δ to free electrons of the nanoparticles. This creates a space charge field Es=(n00e/(ϵ0(ϵL+2)))δ in the overlap region of the free electron sphere and ion sphere. Each electron thus experiences a restoration force (mωp2/(ϵL+2))δ, where ωp=(n00e2/ϵ0m)1/2. The momentum loss per electron per second via collisions is mνν where ν is the collision frequency and ν is the drift velocity of electrons. Thus the equation of motion for an electron is18

(33)

d2dt2δ+νdδdt+ωp2(ϵL+2)δ=0.
Presuming a solution δ=A1exp(iωt) we get

(34)

ω2+iνωωp2(ϵL+2)=0,
giving ω=ωriΓ,
ωr=ωp/(ϵL+2),

(35)

Γν/2,
where we have assumed ν2ω2. Thus the nanoparticle is a natural oscillator of frequency ωp/(ϵL+2) and quality factor Q=ωp/(ν(ϵL+2)).

When we allow ϵd to have arbitrary value, the Poisson’s equation, governing the space charge field .ϵϕ=0, on writing E=ϕ, gives the solution (for E||z^)

ϕ=C1rcosθfor  r<rc,ϕ=C2cosθ/r2for  r>rc.

Demanding the continuity of ϕ, ϵϕ/r at r=rc we obtain

ϵm+2ϵd=0.

Using the value of ϵm given below Eq. (2), we obtain the frequency given by Eq. (35) with ϵL+2 replaced by ϵL+2ϵd. This is the same result obtained by Bergman and Stockman19 and Noginov et al.20 for the fundamental mode.

The field of the nanoparticle [of dipole moment eδ(4πn00rc3/3)] as seen by the gain layer is

(36)

E=n00erc33ϵ0ϵdr3(δ3δ.rr2r).

Outside the gain layer (r>r0+Δ) the field is given by the above expression with ϵd replaced by 1.

The energy stored in the oscillator is

(37)

EEM=ϵ04ω(ϵω)|E|2dV=δ1A12,
δ1=4π3n00rc3mωp218(1+2/ϵL),
where the volume integration has been carried out from inside the particle to the entire outside.

The oscillator field induces e–h recombination producing electromagnetic energy per second as given by Eq. (21) with Γ given by Eq. (35) and

(38)

δ2=2π9n00mωp2Δrc3ωBρe(Ee)·(fe(Ee)+fh(Eh)1).

The growth rate of the SPW turns out to be

(39)

γ=δ2/δ12Γ=3ΔωBrc(1+2/ϵL)ρe(Ee)[fe(Ee)+fh(Eh)1]2Γ.

For Bρe(Ee)ω1017  s1, Δ/rc102, and fe(Ee), fh(Eh). One percent above the 50% occupation probability, the growth rate γ1013  s1.

5.

Discussion

The surface plasmon eigenmode in the vicinity of SPW resonance, ωωR is strongly localized near the metal–dielectric interface with αI/αII1 and has resonantly short wavelength. The damping rate of the mode, however, shows no resonant enhancement. A gain medium, within an SPW wavelength from the boundary sustaining SPW, excites the SPW eigenmode. Usually one employs optical pumping to achieve fe(Ee)+fh(Eh)>1, the condition equivalent of population inversion. For electron and hole occupation probabilities of relevant energy states in the conduction and valence bands exceeding 0.5 each by 1% the growth rate is of the order of 1013  s1.

In the case of metal film loaded with a gain medium nanostrip, frequency can be tuned by varying the permittivity and width of the strip. A drastic reduction in operating frequency can be achieved by shrinking the thickness of the metal film that lowers the SPW resonance frequency.21 However, the present analysis is not valid in that case.

The nanohole oscillator has a gain comparable to the above oscillator at the same level of pumping power flux density. The SPW eigenmode acts as an oscillating dipole and emits far-field radiation. An array of such holes would give a well-collimated beam.

The oscillator comprising a nanoparticle coated with an optically pumped gain medium has operating frequency independent of the radius of the particle. It depends only on its free carrier density, lattice permittivity, and dielectric constant of the surrounding medium. Noginov et al.20 have reported compensation of loss in metal nanoparticles oscillator by gain in interfacing rhodemine 6G dye. With emission cross section of R6G q4×1016  cm2 and density of active molecules ns=2×1018  cm3 (corresponding to one molecule per nanoparticles of radius 5 nm) the local gain turns out to be ki103  cm1, corresponding to temporal growth kic/η1013  s1 which is comparable to our case where e–h recombination leads to growth of plasmons when fe0.5, i.e., ne1018  cm3. The quality factor Q of the oscillators estimated here due to collisional losses appears to be higher than that reported experimentally. Photoabsorption via interband transitions appears to dominate collisional losses. These losses are equivalent to enhanced collision frequency of electrons.

Acknowledgments

VKT is grateful to University of Macao for facilitating the collaborative work.

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Biography

Chuan S. Liu is a leading plasma physicist, credited with groundbreaking contributions to theory of laser plasma interaction and promoting international collaboration in higher education. He joined the University of Maryland in 1975 as professor of physics and served twice as department chair. He also served as vice president for research and dean of the graduate school. From 2003 to 2006, he was president of National Central University, Taiwan. Currently, he is the founding master of the Chao Kuang Piu College.

Vipin K. Tripathi received his master’s in physics from Agra University in 1967 and a PhD from IIT Delhi in 1971. In 1972, he joined the IIT faculty. In 1976, he moved to the University of Maryland and worked in the area of thermonuclear fusion, developing nonlinear theories of RF heating and current drive in tokamak. In 1983, he joined IIT Delhi as professor of physics. He established a leading group in free electron laser and laser plasma interaction.

Chuan Sheng Liu, Vipin K. Tripathi, "Optical gain in surface plasmon nanocavity oscillators," Journal of Nanophotonics 10(1), 016015 (14 March 2016). http://dx.doi.org/10.1117/1.JNP.10.016015
Submission: Received ; Accepted
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KEYWORDS
Oscillators

Surface plasmons

Metals

Nanoparticles

Plasmons

Dielectrics

Plasmonic nanolaser

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