Compensation of loss in propagating surface plasmon polariton by gain in
  adjacent dielectric medium
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
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
                                                               
e1 + e2
,                      (1)
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,
                                                              
kx q( ) = (w /c)n0 sinq0 ,                (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 + r12 exp 2ikz1d1( )
1+ r01r12 exp 2ikz1d1( )
,               (3)
where 
rik = kziek - kzkei( ) kziek + kzkei( ) and
                                                       
kzi = ± ei
- kx q( )
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.
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
4g ig r + d(q)
(kx - kx
0 - Dkx
0)2 + g i + g r( )
,     (5)
where 
' -e1
' + e1
3 / 2
0 = r01(q0) , and 
d(q) = 4(kx - kx
0 - Dkx
0)Im(r0 )Im(e
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 g i + g r( )[ ]
,                       (6)
which, in turn, is defined by the sum of the internal (or propagation) loss
                                                 
g i = kx
' + e2
3 / 2
˜ .            (7)
and the radiation loss caused by SPP leakage into the prism,
                                                               
g r = Im r01e
0d1( ) /x .                     (8)
The radiation loss also leads to the shift of the extremum of the Lorentzian profile from its
resonant position kx0 ,
                                                              
0 = Re r01e
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 .                         (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
becomes unstable and cannot be described by stationary Eqs. (3-5) [27]. Instead, one should
consider the rate equations describing populations of energy states of dye as well as a coupling
between excited molecules and the SPP field. In thin metallic films (when gi<gr at e2”=0), the
resonant value of R monotonically grows with the increase of gain, Fig. 2b.
Experimentally, SPPs were studied in the attenuated total internal reflection setup of Fig. 1a.
The 90o degree prism was made of glass with the index of refraction n0=1.784. Metallic films
were produced by evaporating 99.99% pure silver.
Rhodamine 6G dye (R6G) and polymethyl methacrylate (PMMA) were dissolved in
dichloromethane. The solutions were deposited to the surface of silver and dried to a film. In the
majority of experiments, the concentration of dye in dry PMMA was equal to 10 g/l (2.1x10-2 M)
and the thickness of the polymer film was of the order of 10 mm.
The prism was mounted on a motorized goniometer. The reflectivity R was probed with p
polarized He-Ne laser beam at l=594 nm. The reflected light was detected by a photodiode or a
photomultiplier tube (PMT) connected to the integrating sphere, which was moved during the
scan to follow the walk of the beam.
The permittivity of metallic film was determined by fitting the experimental reflectivity
profile R(q) of not pumped system with Eq.(3), inset of Fig. 4a. As a rule, experimental values
e1’ and e1” did not coincide with the commonly used data of Ref. [28].
In the measurements with optical gain, the R6G/PMMA film was pumped from the back side
of the prism (Fig. 1a) with Q-switched pulses of the frequency doubled Nd:YAG laser (l=532
nm, tpulse=10 ns, repetition rate 10 Hz). The pumped spot, with the diameter of ~3 mm,
completely overlapped the smaller spot of the He-Ne probe beam. Reflected He-Ne laser light
was directed to the entrance slit of the monochromator, set at l=594 nm, with PMT attached to
the monochromator’s exit slit. Experimentally, we recorded reflectivity kinetics R(q,t) under
short-pulsed pumping at different incidence angles (Fig. 4b).
In samples with relatively thin (≈ 40 nm) metallic films, strong emission signal from the R6G-
PMMA film was observed in the absence of He-Ne probe beam. We therefore performed two
measurements of kinetics for each data point: one in the absence of the probe beam, and one in
the presence of the beam. We then subtracted “emission background” (measured without He-Ne
laser) from the combined reflectivity and emission signal. The kinetics measurements had a
relatively large data scatter, which was partially due to the instability of the Nd:YAG laser.
The results of the reflectivity measurements in the 39 nm silver film are summarized in Fig.
4a. Two sets of data points correspond to the reflectivity without pumping (measured in flat parts
of the kinetics before the laser pulse) and with pumping (measured in the peaks of the kinetics).
By dividing the values of R measured in the presence of gain by those without gain, we
calculated the relative enhancement of the reflectivity signal to be as high as 280% – a
significant improvement in comparison to Ref. [19], where the change of the reflectivity in the
presence of gain did not exceed 0.001%. Fitting both reflectivity curves with Eq. (3) and known
e'1=-15, e"2=0.85 and e2’=  n22=2.25, yields e2”≈-0.006. For l =594 nm, this corresponds to
optical gain of g=420 cm-1.
In thicker silver films, calculations predict initial reduction of the minimal reflectivity R(q) at
small values of gain followed by its increase (after passing the minimum point R=0) at larger
gains, Fig. 2a. The predicted reduction of R was experimentally observed in the 90 nm thick film,
where instead of a peak in the reflectivity kinetics, we observed a dip, inset of Fig. 4b.
For the silver film parameters measured in our experiment, Eq. (5) predicts complete
compensation of intrinsic SPP loss at optical gain of 1310 cm-1. For a better quality silver
characterized by the dielectric constant of Ref. [28], the critical gain is smaller, equal to 600
cm-1. In addition, if a solution of R6G in methanol (n=1.329) is used instead of the R6G/PMMA
film, then the critical value of gain is further reduced to 420 cm-1. This is the value of gain
achieved in our experiment. Thus, in principle, at the available gain, one can fully compensate
the intrinsic SPP loss in silver.
For complete compensation of plasmonic loss in the system, one must also compensate
radiation losses. The huge gain equal to 4090 cm-1 is required to completely compensate
attenuation of SPP in the 39 nm thick film used in our experiment. This value is dramatically
reduced in thicker metallic films, since radiative loss strongly depends on the film thickness. For
relatively thick (≥100 nm) metallic films, the total loss is almost identical to the internal loss.
In the experiment described above, the concentration of R6G molecules in the PMMA film
was equal to 1.26x1019 cm-3 (2.1x10-2M). Using the spectroscopic parameters known for the
solution of R6G dye in methanol and neglecting any stimulated emission effects, one can
estimate that 18 mJ laser pulses used in the experiment should excite more than 95% of all dye
molecules. At the emission cross section equal to 2.7x10-16 cm2 at l=594 nm, this concentration
of excited molecules corresponds to the gain of 3220 cm-1. Nearly eight-fold difference between
this value and the one obtained in our experiment is probably due to the combined effects of
luminescence quenching of R6G due to dimerization of rhodamine 6G molecules occurring at
high concentration of dye [29], and amplified spontaneous emission (ASE). While the detailed
study of the ASE-induced effects in the R6G/PMMA-silver systems is beyond the scope of this
work, we note that at the value of gain equal to 420 cm-1 and the diameter of the pumped spot
equal to 3 mm, the optical amplification is enormously large. Obviously, these giant values of the
amplification and the gain cannot be maintained in a cw regime, and ASE appears to be a
detrimental factor controlling the gain in the pulsed regime. Correspondingly, the choice of a
more efficient amplifying medium (as was proposed in Ref. [19]) may not help in compensating
the SPP loss by gain.
To summarize, in our study of the propagating surface plasmon polariton in the attenuated
total reflection setup, we have established the relationship between (i) the gain in the dielectric
adjacent to the metallic film, (ii) the internal, radiative and total losses, (iii) the propagation
length of the SPP, and (iv) the shape of the experimentally measured reflectivity profile R(q). We
have experimentally demonstrated the optical gain in the dielectric (PMMA film with R6G dye)
equal to 420 cm-1, yielding nearly threefold increase of the resonant value of the reflectivity. In
the case of thick low-loss silver film [28] and low index dielectric, the demonstrated value of
gain is sufficient for compensation of the total loss hindering the propagation of surface plasmon
polariton.
The work was supported by the NSF PREM grant # DMR 0611430, the NSF CREST grant #
HRD 0317722, the NSF NCN grant # EEC-0228390, the NASA URC grant # NCC3-1035, and
the Petroleum Research Fund. The authors cordially thank Vladimir M. Shalaev for useful
discussions.
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Figure captions
Fig.1  (a)  Schematic of SPP excitation in ATR geometry.  (b)  Reflectivity R as a function of
angle q.  Traces – solutions of exact Eq. (3). Dots – solution of approximate Eq. (6). For all data
sets: e1=-15.584+0.424i, d1=60 nm. Trace 1: dielectric with very small loss, e2=2.25+10-5i.
Traces 2-4: dielectric with very small gain, e2=2.25-10-5i. Trace 2 and dots: complex cut along
negative imaginary axis (correct; nearly overlaps with trace 1; no discontinuity at the transition
from small loss to small gain). Trace 3: complex cut along positive real axis (yields incorrect
predictions for incident angles below total internal reflection). Trace 4: complex cut along
negative real axis (yields incorrect predictions for incident angles above total internal reflection).
Fig.2.  Reflectivity R [Eq.(5)] of the three-layer system depicted in Fig. 1a as a function of angle
q  and pumping (given by imaginary part of e2); panel (a) illustrates the evolution of reflectivity
in a relatively thick metallic film (d1=70nm); panel (b) corresponds to a thin film (d2=39 nm).
Fig. 3. Inverse propagation length of SPP, L-1, in the system depicted in Fig.1a as a function of
gain in dielectric, e2”. Solid line – solution of Eq. (11), dots – exact numerical solution of
Maxwell equations. Top inset: intensity distribution across the system. Bottom inset: Exponential
decay of the SPP wave intensity E2 (shown in the top inset) along the propagation in the x
direction.
Fig.  4. (a) Reflectivity R(q) measured without (diamonds) and with (circles) optical pumping in
the glass-silver-R6G/PMMA system. Dashed lines – guides for eye. Solid lines – fitting with
Eq. (3) at e0’=n02=1.7842=3.183, e0”=0, e1’=-15, e1”=0.85, d1=39 nm, e2’=  n22=1.52=2.25,
e2”≈0 (trace 1) and e2”≈-0.006 (trace 2). Inset: Angular reflectivity profile R(q) recorded in the
same system without pumping (dots) and its fitting with Eq. (3) (solid line). (b) Reflectivity
kinetics recorded in the glass-silver-R6G/PMMA structure under pumping. The angle q
corresponds to the minimum of the reflectivity; d1=39 nm. Inset: Reflectivity kinetics recorded
in a thick film (d1=90 nm) shows a ‘dip’ at small values of gain.
Fig. 1
Fig.1  (a)  Schematic of SPP excitation in ATR geometry.  (b)  Reflectivity R as a function of
angle q.  Traces – solutions of exact Eq. (3). Dots – solution of approximate Eq. (6). For all data
sets: e1=-15.584+0.424i, d1=60 nm. Trace 1: dielectric with very small loss, e2=2.25+10
Traces 2-4: dielectric with very small gain, e2=2.25-10
-5i. Trace 2 and dots: complex cut along
negative imaginary axis (correct; nearly overlaps with trace 1; no discontinuity at the transition
from small loss to small gain). Trace 3: complex cut along positive real axis (yields incorrect
predictions for incident angles below total internal reflection). Trace 4: complex cut along
negative real axis (yields incorrect predictions for incident angles above total internal
reflection).
Fig. 2.
Fig.2.  Reflectivity R [Eq.(5)] of the three-layer system depicted in Fig. 1a as a function of angle
q and pumping (given by imaginary part of e2); panel (a) illustrates the evolution of reflectivity
in a relatively thick metallic film (d1=70nm); panel (b) corresponds to a thin film (d2=39 nm).
Fig. 3.
Fig. 3. Inverse propagation length of SPP, L-1, in the system depicted in Fig.1a as a function of
gain in dielectric, e2”. Solid line – solution of Eq. (11), dots – exact numerical solution of
Maxwell equations. Top inset: intensity distribution across the system. Bottom inset:
Exponential decay of the SPP wave intensity E2 (shown in the top inset) along the
propagation in the x direction.
Fig. 4.
60 62 64 66 68 70 72
Angle (q)
0.0005
0.001
0.0015
0.002
0.0025
0.0E+00 2.0E-07 4.0E-07 6.0E-07 8.0E-07 1.0E-06
Time (s)
Fig.  4. (a) Reflectivity R(q) measured without (diamonds) and with (circles) optical pumping in
the glass-silver-R6G/PMMA system. Dashed lines – guides for eye. Solid lines – fitting with
Eq. (3) at e0’=n02=1.7842=3.183, e0”=0, e1’=-15, e1”=0.85, d1=39 nm, e2’=  n22=1.52=2.25,
e2”≈0 (trace 1) and e2”≈-0.006 (trace 2). Inset: Angular reflectivity profile R(q) recorded in the
same system without pumping (dots) and its fitting with Eq. (3) (solid line). (b) Reflectivity
kinetics recorded in the glass-silver-R6G/PMMA structure under pumping. The angle q
corresponds to the minimum of the reflectivity; d1=39 nm. Inset: Reflectivity kinetics recorded
in a thick film (d1=90 nm) shows a ‘dip’ at small values of gain.
60 62 64 66 68 70 72
Angle (q)
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.0E+00 2.5E-07 5.0E-07 7.5E-07 1.0E-06
Time (s)
