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Compensation of loss in propagating surface plasmon polariton by gain in adjacent

dielectric medium

M. A. Noginov1*, V. A. Podolskiy2, G. Zhu1, M. Mayy1, M. Bahoura1, J. A. Adegoke1, B. A.

Ritzo1, K. Reynolds1

1 Center for Materials Research, Norfolk State University, Norfolk, VA 23504

* mnoginov@nsu.edu

2 Department of Physics, Oregon State University, Corvallis, OR 97331-6507

Abstract: We report the suppression of loss of surface plasmon polariton propagating at the

interface between silver film and optically pumped polymer with dye. Large magnitude of the

effect enables a variety of applications of ‘active’ nanoplasmonics. The experimental study is

accompanied by the development of the analytical description of the phenomenon and the

solution of the controversy regarding the direction of the wavevector of a wave with a strong

evanescent component in an active medium.

Surface plasmon polaritons (SPPs) – special type of electromagnetic waves coupled to

electron density oscillations – allow nanoscale confinement of electromagnetic radiation [1].

SPPs are broadly used in photonic and optoelectronic devices [1-7], including waveguides,

couplers, splitters, add/drop filters, and quantum cascade lasers. SPP is also the enabling

mechanism for a number of negative refractive index materials (NIMs) [8-12].

Many applications of SPPs suffer from damping caused by absorption in metals. Over the

years, several proposals to compensate loss by incorporating active (gain) media into plasmonic

systems have been made. Theoretically, field-matching approach was employed to calculate the

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reflectivity at surface plasmon excitation [13]; the authors of [14] proposed that the optical gain

in a dielectric medium can elongate the SPP’s propagation length; gain-assisted excitation of

resonant SPPs was predicted in [15]; SPP propagation in active waveguides was studied in [16];

and the group velocity modulation of SPPs in nano-waveguides was discussed in [17]. Excitation

of localized plasmon fields in active nanosystems using surface plasmon amplification by

stimulated emission of radiation (SPASER) was proposed in [18]. Experimentally, the

possibility to influence SPPs by optical gain was demonstrated in Ref. [19], where the effect was

as small as 0.001%.

Here we report conquering the loss of propagating SPPs at the interface between silver film

and optically pumped polymer with dye. The achieved value of gain, ≈ 420 cm-1, is sufficient to

fully compensate the intrinsic SPP loss in high-quality silver films. This, together with the

compensation of loss in localized surface plasmons, predicted in [20] and recently demonstrated

in [21], enables practical applications of a broad range of low-loss and no-loss photonic

metamaterials.

The experimental attenuated total reflection (ATR) setup consisted of a glass prism with the

real dielectric permittivity e0=n02, a layer of metal with the complex dielectric constant e1 and

thickness d1, and a layer of dielectric medium characterized by the permittivity e2, Fig. 1a.

The wave vector of the SPP propagating at the boundary between media 1 and 2, is given by

[1]

†

kx

0=w

c

e1e2

e1+e2

, (1)

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where w is the oscillation frequency and c is the speed of light. SPP can be excited by a p

polarized light falling on the metallic film at the critical angle q0, such that the projection of the

wave vector of the light wave to the axis x,

†

kxq ( )= (w/c)n0sinq0, (2)

is equal to

†

Re(kx

0). At this resonant condition, the energy of incident light is transferred to the

SPP, yielding a minimum (dip) in the angular dependence of the reflectivity R(q) [1]:

†

R q ( )=

r01+ r12exp 2ikz1d1

1+ r01r12exp 2ikz1d1

(

(

)

)

2

, (3)

where

†

rik= kziek- kzkei

() kziek+ kzkei

() and

†

kzi= ± ei

w

c

Ë

Ê

Á

ˆ

¯

˜

2

- kxq ( )

2, i =0,1,2. (4)

The parameter

†

kzc/w defines the field distribution along the z direction. Its real part can be

associated with a tilt of phase-fronts of the waves propagating in the media [16], and is often

discussed in the content of positive vs. negative refractive index materials (see also Refs. [8-12,

22-24]), while its imaginary part defines the wave attenuation or growth. The sign of the square

root in Eq. (4) is selected to enforce the causal energy propagation. For dielectrics excited in total

internal reflection geometry, as well as for metals and other media with

†

Re[kz

2]< 0, which do not

support propagating waves, the imaginary part of the square root should be always positive

regardless of the sign of e”. For other systems, the selection should enforce the wave decay in

systems with loss (e”>0) and the wave growth in materials with gain (e”<0) [23]. This selection

of the sign can be achieved by the cut of the complex plane along the negative imaginary axis.

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Although such cut of the complex plane is different from the commonly accepted cuts along

the positive [22] or negative [8-12,23] real axes, our simulations (Fig.1b) show that this is the

only solution guaranteeing the continuity of measurable parameters (such as reflectivity) under

the transition from a weak loss regime to a weak gain regime. Our sign selection is the only one

consistent with previous results on gain-assisted reflection enhancement, predictions of gain-

assisted SPP behavior [13-15,25], and the experimental data presented below. The implications

of selecting different signs of kz2 are shown in Fig.1b.

Note that active media excited above the angle of total internal reflection, as well as the

materials with e’<0 and e”<0 formally fall under negative index materials category. However,

since

†

|kz"| in this case is greater than

†

|kz'|, the “left handed” wave experiences very large

attenuation (in the presence of gain!!!), which in contrast to claims of Ref. [22], makes the

material unsuitable for superlenses and other proposed applications of NIMs [8-12].

In the limit of small plasmonic loss/gain, when the decay length of SPP, L, is much greater

than 2p/kx0’, and in the vicinity of q0, Eq. (3) can be simplified, revealing the physics behind the

gain-assisted plasmonic loss compensation:

†

R q ( )ª r01

21-

Î

Í

4gigr+d(q)

0-Dkx

(kx- kx

0)2+ gi+ gr

()

2

È

Í

˘

˙

˙ ,

˚

(5)

where

†

x =c e2

'-e1

2w

'

()

e2

Ê

Ë

'+e1

e2

'

'e1

'

Á

ˆ

˜

¯

3/2

,

†

r01

0= r01(q0), and

†

d(q) = 4(kx- kx

0-Dkx

0)Im(r0)Im(ei2kz

0d1) x.

The shape of R(q) is dominated by the Lorentzian term in Eq. (5). Its width is determined by the

propagation length of SPP,

†

L = 2 gi+ gr

(

[

)

]

-1, (6)

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which, in turn, is defined by the sum of the internal (or propagation) loss

†

gi= kx

0"=w

2c

e1

'+e2

'e2

'

e1

'

Ê

Á

Ë

ˆ

˜

¯

3/2

e1

e1

Ë

"

'2+e2

"

e2

'2

Ê

Á

ˆ

˜ .

¯

(7)

and the radiation loss caused by SPP leakage into the prism,

†

gr= Im r01ei2kz

0d1

()/x . (8)

The radiation loss also leads to the shift of the extremum of the Lorentzian profile from its

resonant position kx0 ,

†

Dkx

0= Re r01ei2kz

(

0d1

)/x. (9)

The term d in Eq. (5) results in the asymmetry of R(q).

The excellent agreement between exact Eq. (3), solutions of Maxwell equations using transfer

matrix method [26] and approximate Eqs. (5,6) for the 60 nm silver film are shown in Figs. 1,3.

The gain in the medium reduces internal loss gi of SPP, Eq. 7. In reasonably thick metallic

films (where gi>gr in the absence of gain) the “dip” in the reflectivity profile Rmin is reduced when

gain is first added to the system, reaching Rmin=0 at gi≈gr (Fig. 2a). With further increase of gain,

gi becomes smaller than gp, leading to an increase of Rmin. The resonant value of R is equal to

unity when internal loss is completely compensated by gain (gi=0) at

†

e2"= -e1"e2'2

e1'2

. (10)

In the vicinity of gi=0, the reflectivity profile is dominated by the asymmetric term d. When gain

is increased to even higher values, gi becomes negative and the dip in the reflectivity profile

converts into a peak, consistent with predictions of Refs. [13,14]. The peak has a singularity

when the gain compensates total SPP loss (gi+gr=0). Past the singularity point, the system