Page 1

1

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

Page 2

2

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)

Page 3

3

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.

Page 4

4

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)

Page 5

5

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

Page 6

6

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

Page 7

7

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

Page 8

8

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

Page 9

9

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.

References

1. H. Raether, “Surface plasmons on smooth and rough surfaces and on gratings”, Springer-

Verlag, (Berlin, 1988).

2. S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, Phys. Rev. Lett 89, 186801 (2002).

3. A. Boltasseva, S. I. Bozhevolnyi, , T. Søndergaard, T. Nikolajsen, and L. Kristjan, Optics

Express 13, 4237-4243 (2005).

Page 10

10

4. S. A., Maier, et. al., Nature Materials 2, 229-232 (2003).

5. A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and S. Marin, Phys. Rev. Lett. 95,

063901 (2005).

6. M. Stockman, Phys. Rev. Lett. 93, 137404 (2004).

7. C. Sirtori et.al., Opt. Lett. 23, 1366-1368 (1998).

8. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, J. Nonl. Opt. Phys. Mat. 11, 65-74

(2002).

9. H. Shin, and S. Fan, Phys. Rev. Lett. 96, 073907 (2006).

10. V. M. Shalaev et. al., Opt. Lett. 30, 3356-3358 (2005).

11. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, Science 312, 892-894

(2006).

12. A. Alu, and N. Engheta, J. Opt. Soc. Am. B 23, 571-583 (2006).

13. G. A. Plotz, H. J. Simon, and J. M. Tucciarone, J. Opt. Soc. Am. 69, 419-421 (1979).

14. A. N. Sudarkin and P. A. Demkovich, Sov. Phys. Tech. Phys. 34, 764-766 (1989).

15. I. Avrutsky, Phys. Rev. B 70, 155416 (2004).

16. M. P. Nezhad, K. Tetz and Y. Fainman, Optics Express 12, 4072-4079 (2004).

17. A. A. Govyadinov and V. A. Podolskiy, Phys. Rev. Lett. 97, 223902 (2006).

18. D. Bergman and M. Stockman, Phys. Rev. Lett. 90, 027402 (2003).

19. J. Seidel, S. Grafstroem, and L. Eng, Phys. Rev. Lett. 94, 177401 (2005).

20. N. M. Lawandy, Appl. Phys. Lett. 85, 5040-5042 (2004).

21. M. A. Noginov et. al., Opt. Lett. 31, 3022-3024 (2006).

22. Y. Chen, P. Fisher, and F. W. Wise, Phys. Rev. Lett. 95, 067402 (2005).

23. T. Mackay, and A. Lakhtakia, Phys. Rev. Lett. 96. 159701 (2006)

Page 11

11

24. Y. Chen, P. Fisher and F. W. Wise, Phys. Rev. Lett. 96, 159702 (2006).

25. B. Ya. Kogan, V. M. Volkov, and S. A. Lebedev, JETP Lett. 16, 100-105 (1972).

26. I. Avrutsky, J. Opt. Soc. A 20, 548-556 (2003).

27. X. Ma and C. M. Soukoulis, Physica B, 296, 107-111 (2001).

28. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370-4379 (1972).

29. K. Selanger, A. J. Falnes and T. Sikkeland, J. Phys. Chem. 81, 1960-1963 (1977).

Page 12

12

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 E2 (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

Page 13

13

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.

Page 14

14

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+10-5i.

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).

Page 15

15

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).

Page 16

16

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 E2 (shown in the top inset) along the

propagation in the x direction.

Page 17

17

Fig. 4.

0

0.2

0.4

0.6

0.8

1

6062 64 66687072

Angle (q)

R

0

0.0005

0.001

0.0015

0.002

0.0025

0.0E+00 2.0E-074.0E-076.0E-078.0E-071.0E-06

Time (s)

Reflectivity signal (rel. units)

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.

0

0.2

0.4

0.6

0.8

1

606264666870 72

Angle (q)

R

0

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)

Reflectivity (rel. units)

(a)

(b)

1

2