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Nonlinear s-polarized surface plasmon polaritons
Solid State Communications, Printed in Great Britain.
Vo1.52,No.3,
UOULIYKAE
S-POLAUZED
George I. Stegenan, Optical
pp.293-297,
SOILFACE PLASLIOU POLARIIOUS
Jesus
Sciences Center University of
D. Valera,
of
and
Colin
T. Seaton
and Arizona Research Laboratories Arizona, Tucson, AZ 85721 U.S.A. John
University
003%1098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
Department Toronto,
Sipe
of Physics Toronto, Ontario, Canada
M5S IA7
and
University
of
Alexel A. Waradudin Department of Physics California - Irvine, Irvine, U.S.A.
(Received:
June
29
1984, by
CA 92717
R. H. Silsbee)
We have
examined the dispersion relations for s-polarized surface plasmon polaritons, guided by (a) the interface between a semi-infinite metal and dielectric medium, and (b) a metal film bounded by semiinfinite dielectric media for situations in which one or more of the dielectric media are characterized by an intensify-dependent refractive index. We found that s-polarized waves satisfy the dispersion relations for very thin metal films bounded by nonlinear dielectric media. These waves exist only for power levels above a threshold that depends on the We also comment on the experimental feasibility of material parameters.
observing
these
waves.
respectively
Iwrx0DlIcTI0II
The variety of waves guided by single or multiple interfaces changes when one or more of the guiding media are characterized by an intensity-dependent refractive index.“** The boundary between two linear dielectric media cannot support an s-polarized surface polariton. However, if one or both of the media exhibit an intensity-dependent refractive index, it has been predicted that an s-polarized surface polariton should exist at high powers.“’ It has also been shovn that new s- and p-polarized waves exhibiting power thresholds should occur for dielectric films bounded by nonlinear media.‘,‘,‘,“** For metal films, new waves have also been predicted” for the ppolarized case. However, the case of s-polarized surface plasmon polaritons guided by virtue of nonlinear media has not yet been reported. The results of a cursory search involving large values of In this paper we examine this 1c,l proved negative.” problem in more detail and find solutions that correspond to s-polarized waves over restricted ranges of material parameters and geometries. DISPSDSIOU gDLATIOtFS The dispersion relations for waves guided by an arbitrary film bounded on one or both sides by nonlinear media has been reported by us before.‘,‘” The field distributions in the cladding, film, and substrate are proportional to exp[i(Bk,x-wt)] and are given
293
Ecy(x,d
by
=
+ Efy(O,z) x
[
=
‘=c
- wt)
cosh[k,q(z,-z)] (la)
C.C.
Ecy(O,O)
cosh(k,xz)
Esy(O,s)
_i(k,Bx
q
3
+ +
tanh(k,qz,)
sinh(k,rz)
1
(lb)
cosh[k,s(d-x,)1 =
Efy(G,d)
(lc)
cosh[k,s(d-2,-z)] 9
‘s
cosh[k,s(d-z,)]
’
(Id)
for a film of thickness d and dielectric constant rm < 0 bounded by semi-infinite media with refractive index given by n = nc+n2$ and n - ns+n2sS for the cladding (upper medium) and substrate (lower medium) respectively, and with si = nizcEon2i > 0. The dispersion relation is xlq tanh(k,qz,) + s tanh(k,ss,)] tanh(k,xd) __c1 - sq tanh(k,qz ,) tanh(k,ss,) ’ (2) Here q’ is the
= B’-ncz, sz = B’-ns2 and x1 = B*-cm where Bk, The guided wave wavevector (k, - w/c).
NONLINEAR
294
S-POLARIZED
SIJRFAGE PLASMON POLAEITONS
Vol. 52, No. 3
are summarked in Figs. 1 through 4. The variation in guided wave power with effective index 6 is shown in Fig. 1 for a few metal film thicknesses and for identical nonlinear media bounding'the metal film (sm - -10). There is a definite power threshold at which the solutions first appear and that increases with film thickness. Note that for reasonable metal film thicknesses of 0.01 urn or more, the threshold value of 6 corresponds to a maximum index change in f 1 arc cash 21 the bounding media In excess of 0.1 and is therefore k,q The field beyond the available material limits. (3) distributions, Fig. Za, exhibit symmetric maxima in both bounding media, and these maxima move toward the The power per unit distance along the wavefront metal film with increasing power. carried by the waves is given by" The solutions are sensitive to the value of cm. As 0, decreases, that is 1emI increases, both the p, .& [I - tanMk,qzt)] threshold power and minimum effective index increase rapidly. For example, for cm = -40, these values are (4a) 374 W/m and 2.05 respectively, which should be to 88.6 W/m and 1.61 for em - -10 compared (d = 0.005 urn). This explains why the initial search" 'f = e cosh'fk,qr,] at cm - -1000 failed. Unfortunately, for most real metals, this range of useful values for cm is usually x [d(l-$tanh'(k,qr,)) characterized by imaginary components that result in high propagation losses. sinh(2k,rd) I Wave solutions also exist when the bounding media + (l+$tanh'(k,qs,)) have different optical properties. For example, when Zk,c the value of one of the refractive indices of the bounding media is changed, the solutions persist and + * tanh(k,qz,)(cosh(Zk,rd)-l)] the field distributions become asymmetric. Two . (4b) examples are shown in Figs. 2b and 2c. Note that for large enough In,-n,l, fields are obtained with maxima in only one medium. - tanh(k,sz,)] . We also investigated the effect of n2= + n2s. (4c) The results for the guided wave power versus B are shown in Fig. 3 and the corresponding field distributions in Fig 4. As the nonlinearity in the WnERKAL PgsuLrs substrate is reduced, the threshold guided wave power increases correspondingly. Furthermore, the' field Our numerical calculations were based in a distributions become distorted with progressively more general way on the liquid crystal MBBA (n = 1.55, n2 - lo-' ml/W in the blue-green region of the visible of the power carried by the medium with the smaller nonlinearity. This material is of particular interest spectrum). These last results have bearing on the case when because Ansat, the maximum change in refractive index which can be produced optically, is large, of the order n2i - 0 in one of the bounding media. From Figs. 3 and 4, the threshold power diverges in the limit nZs + 0 of 0.1. and the fields are localized in the weakly nonlinear S-polarized solutions were found for very thin medium, that is, there are no physically realistic metal films bounded on both sides by nonlinear media This was verified numerically with a solutions. characterized by positive values of n2i. The results parameter 2, ia related to s, by Eq. (Id) and z, can be vritten either in terms of the total guided wave power or the Poynting vector at the cladding-film interface. In terms of the iterative calculations necessary for solving the dispersion relations, it is more convenient to define x1 in terms of the Poynting vector S, at t-0 as
a
ps =
..’
/./ .
......
...“-’
E
_....A
Fig. 1: Guided wave power (mW/mm) VerSUS effective index 6 for nc - ns = 1.55, cm - -10, and "2c = "2s - 10-Y ma/W for d - 0.001 ym (solid line), d - 0.005um (dashed line), d - 0.010 nm (dotted line) and d = .015 urn (dash-dotted line).
NONLINEAR S-POLARIZED SURFACE PLASPIONPOLARITONS
Vol. 52, No. 3
Pig:
;:>l?ield
295
distributions with increasing B for m’/W, Lm = -10 and n2s = 10” film thickness scale is expanded by that of the bounding media. (a) - 1.40. (c) ns = 1.25.
nc *2c d = 0.005’nm. The Xl00 relative to (b) n, ns = 1.55.
separate computer program in which “2s - 0 was explicitly assumed. It was also verified that, if one of the media is characterized by an QI
-
f
.
(5)
Because 6’ - q’ + nc* = C* + Em for this case, Cqs’, r/q is always larger than unity and Eq. (5) can never Therefore s-polarized waves are not be satisfied. This helps explain why spossible in this case. polarized polaritons in the thin metal film geometry exist only for reasonably small 1cm1 and d. If the effective skin depth of the metal is on the order of d, the geometry essentially separates into two independent metal-dielectric interfaces at which spolarized polaritons do not exist even at high powers. It is of course precisely because n-polarized surface polaritons do exist at such independent interfaces
296
NONLINEAR
S-POLARIZED
SURFACE PLASMON
POLARITONS
Vol. 52, No. 3
that p-polarized polaritons in the thin metal film geometry exist over much larger ranges of 1o,l and d. In summary, we found that s-polarized surface polaritons can exist for thin metal films bounded by nonlinear media with positive nonlinearlties, but not for any other planar geometry involving metals. These waves all have power thresholds. very metal thickness required, with surface plasmons for small lsml, and the large changes in refractive index required will probably make such waves difficult to observed experimentally.
................ _......... .*
.“.
/...‘.. . . ...** *....*
\
-+
1.6
1.8
B 0)
2.0
B
Fig. 3: Guided wave power (mW/mm) versus effective index 6 for nc = ns - 1.55, d - 0.005 urn,sm = -10 and "2c = lo-' m'/W with nzs =lO'LO ml/W (solid line), "2s = lo-" ma/W (dashed line) and n2s = 10-l' m'/w (dotted line).
>
This research was supported by NSF (KS-8117483 and -8304749) and the Joint Services Optics Program of the Army Research Office and the Air Force Office Of Scientific Research.
Fig. 4: Field distributions with increasing 6 for nc = ns = 1.55, d - 0.005 arm, = -10 and for (a) n2s ='?O-, ml/W, (b) "2c = lo-' m*/W nzs = lo-" m'/W and (c) n2s - lo-" m'/W.
1. TOMLINSON, W.J., Opt. Lett. 5, 323 (1980). 2. MABADUDIN, A.A., Zeit. Phys. B. 41, 341 (1981). 3. MARADIJDW, A.A., in Proceedings of the Second International School on Condensed Matter Physics, Varna, Bulgaria (World Scientific Publishers, Singapore, 1983) in press.
4. AGRANOVICH, V.M., BABICHENRO, V.S., and CHERNYAE, V. YA., Sov. Phys. JETP Let. 32, 512 (1981). 5. AEHMEDIEV. N.N., Sov. Phys. JETP 56, 299 (1982). 6. ROBBINS, DJ., Opt. Comm. 47, 309 (1983). 7. LANGBEIN, U, LEDERER, F., and PCNATH, H.E., Opt. Comm. 46, 167 (1983).
Vol.
52,
No.
3
NONLINEAR S-POLARIZED
297
SLJRPACE PLASMON POLARITONS
8. BOARDXAN, A& and EGAN, P, J. Physique Colloq., In press (1984)‘. 9. BOARDMAN, A., and EGAN, P, prlvate communication, 1984. 10. STEGEMAN, G.I.. SEATON, C.T.. CHILWELL, J, and SMITH, S.D., Appl. Phys. Lett. 44, 830 (1984).
11.
STBGEMAN, G.I.. and SEATON, C.T., Opt. (1984). 12. VACH, H.. STEGEMAN. GJ., SEATON, C.T., Opt. Lett. 9, 238 (1984).
Lett. and
9,
235
KHOO, IX.,
Vo1.52,No.3,
UOULIYKAE
S-POLAUZED
George I. Stegenan, Optical
pp.293-297,
SOILFACE PLASLIOU POLARIIOUS
Jesus
Sciences Center University of
D. Valera,
of
and
Colin
T. Seaton
and Arizona Research Laboratories Arizona, Tucson, AZ 85721 U.S.A. John
University
003%1098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
Department Toronto,
Sipe
of Physics Toronto, Ontario, Canada
M5S IA7
and
University
of
Alexel A. Waradudin Department of Physics California - Irvine, Irvine, U.S.A.
(Received:
June
29
1984, by
CA 92717
R. H. Silsbee)
We have
examined the dispersion relations for s-polarized surface plasmon polaritons, guided by (a) the interface between a semi-infinite metal and dielectric medium, and (b) a metal film bounded by semiinfinite dielectric media for situations in which one or more of the dielectric media are characterized by an intensify-dependent refractive index. We found that s-polarized waves satisfy the dispersion relations for very thin metal films bounded by nonlinear dielectric media. These waves exist only for power levels above a threshold that depends on the We also comment on the experimental feasibility of material parameters.
observing
these
waves.
respectively
Iwrx0DlIcTI0II
The variety of waves guided by single or multiple interfaces changes when one or more of the guiding media are characterized by an intensity-dependent refractive index.“** The boundary between two linear dielectric media cannot support an s-polarized surface polariton. However, if one or both of the media exhibit an intensity-dependent refractive index, it has been predicted that an s-polarized surface polariton should exist at high powers.“’ It has also been shovn that new s- and p-polarized waves exhibiting power thresholds should occur for dielectric films bounded by nonlinear media.‘,‘,‘,“** For metal films, new waves have also been predicted” for the ppolarized case. However, the case of s-polarized surface plasmon polaritons guided by virtue of nonlinear media has not yet been reported. The results of a cursory search involving large values of In this paper we examine this 1c,l proved negative.” problem in more detail and find solutions that correspond to s-polarized waves over restricted ranges of material parameters and geometries. DISPSDSIOU gDLATIOtFS The dispersion relations for waves guided by an arbitrary film bounded on one or both sides by nonlinear media has been reported by us before.‘,‘” The field distributions in the cladding, film, and substrate are proportional to exp[i(Bk,x-wt)] and are given
293
Ecy(x,d
by
=
+ Efy(O,z) x
[
=
‘=c
- wt)
cosh[k,q(z,-z)] (la)
C.C.
Ecy(O,O)
cosh(k,xz)
Esy(O,s)
_i(k,Bx
q
3
+ +
tanh(k,qz,)
sinh(k,rz)
1
(lb)
cosh[k,s(d-x,)1 =
Efy(G,d)
(lc)
cosh[k,s(d-2,-z)] 9
‘s
cosh[k,s(d-z,)]
’
(Id)
for a film of thickness d and dielectric constant rm < 0 bounded by semi-infinite media with refractive index given by n = nc+n2$ and n - ns+n2sS for the cladding (upper medium) and substrate (lower medium) respectively, and with si = nizcEon2i > 0. The dispersion relation is xlq tanh(k,qz,) + s tanh(k,ss,)] tanh(k,xd) __c1 - sq tanh(k,qz ,) tanh(k,ss,) ’ (2) Here q’ is the
= B’-ncz, sz = B’-ns2 and x1 = B*-cm where Bk, The guided wave wavevector (k, - w/c).
NONLINEAR
294
S-POLARIZED
SIJRFAGE PLASMON POLAEITONS
Vol. 52, No. 3
are summarked in Figs. 1 through 4. The variation in guided wave power with effective index 6 is shown in Fig. 1 for a few metal film thicknesses and for identical nonlinear media bounding'the metal film (sm - -10). There is a definite power threshold at which the solutions first appear and that increases with film thickness. Note that for reasonable metal film thicknesses of 0.01 urn or more, the threshold value of 6 corresponds to a maximum index change in f 1 arc cash 21 the bounding media In excess of 0.1 and is therefore k,q The field beyond the available material limits. (3) distributions, Fig. Za, exhibit symmetric maxima in both bounding media, and these maxima move toward the The power per unit distance along the wavefront metal film with increasing power. carried by the waves is given by" The solutions are sensitive to the value of cm. As 0, decreases, that is 1emI increases, both the p, .& [I - tanMk,qzt)] threshold power and minimum effective index increase rapidly. For example, for cm = -40, these values are (4a) 374 W/m and 2.05 respectively, which should be to 88.6 W/m and 1.61 for em - -10 compared (d = 0.005 urn). This explains why the initial search" 'f = e cosh'fk,qr,] at cm - -1000 failed. Unfortunately, for most real metals, this range of useful values for cm is usually x [d(l-$tanh'(k,qr,)) characterized by imaginary components that result in high propagation losses. sinh(2k,rd) I Wave solutions also exist when the bounding media + (l+$tanh'(k,qs,)) have different optical properties. For example, when Zk,c the value of one of the refractive indices of the bounding media is changed, the solutions persist and + * tanh(k,qz,)(cosh(Zk,rd)-l)] the field distributions become asymmetric. Two . (4b) examples are shown in Figs. 2b and 2c. Note that for large enough In,-n,l, fields are obtained with maxima in only one medium. - tanh(k,sz,)] . We also investigated the effect of n2= + n2s. (4c) The results for the guided wave power versus B are shown in Fig. 3 and the corresponding field distributions in Fig 4. As the nonlinearity in the WnERKAL PgsuLrs substrate is reduced, the threshold guided wave power increases correspondingly. Furthermore, the' field Our numerical calculations were based in a distributions become distorted with progressively more general way on the liquid crystal MBBA (n = 1.55, n2 - lo-' ml/W in the blue-green region of the visible of the power carried by the medium with the smaller nonlinearity. This material is of particular interest spectrum). These last results have bearing on the case when because Ansat, the maximum change in refractive index which can be produced optically, is large, of the order n2i - 0 in one of the bounding media. From Figs. 3 and 4, the threshold power diverges in the limit nZs + 0 of 0.1. and the fields are localized in the weakly nonlinear S-polarized solutions were found for very thin medium, that is, there are no physically realistic metal films bounded on both sides by nonlinear media This was verified numerically with a solutions. characterized by positive values of n2i. The results parameter 2, ia related to s, by Eq. (Id) and z, can be vritten either in terms of the total guided wave power or the Poynting vector at the cladding-film interface. In terms of the iterative calculations necessary for solving the dispersion relations, it is more convenient to define x1 in terms of the Poynting vector S, at t-0 as
a
ps =
..’
/./ .
......
...“-’
E
_....A
Fig. 1: Guided wave power (mW/mm) VerSUS effective index 6 for nc - ns = 1.55, cm - -10, and "2c = "2s - 10-Y ma/W for d - 0.001 ym (solid line), d - 0.005um (dashed line), d - 0.010 nm (dotted line) and d = .015 urn (dash-dotted line).
NONLINEAR S-POLARIZED SURFACE PLASPIONPOLARITONS
Vol. 52, No. 3
Pig:
;:>l?ield
295
distributions with increasing B for m’/W, Lm = -10 and n2s = 10” film thickness scale is expanded by that of the bounding media. (a) - 1.40. (c) ns = 1.25.
nc *2c d = 0.005’nm. The Xl00 relative to (b) n, ns = 1.55.
separate computer program in which “2s - 0 was explicitly assumed. It was also verified that, if one of the media is characterized by an QI
-
f
.
(5)
Because 6’ - q’ + nc* = C* + Em for this case, Cqs’, r/q is always larger than unity and Eq. (5) can never Therefore s-polarized waves are not be satisfied. This helps explain why spossible in this case. polarized polaritons in the thin metal film geometry exist only for reasonably small 1cm1 and d. If the effective skin depth of the metal is on the order of d, the geometry essentially separates into two independent metal-dielectric interfaces at which spolarized polaritons do not exist even at high powers. It is of course precisely because n-polarized surface polaritons do exist at such independent interfaces
296
NONLINEAR
S-POLARIZED
SURFACE PLASMON
POLARITONS
Vol. 52, No. 3
that p-polarized polaritons in the thin metal film geometry exist over much larger ranges of 1o,l and d. In summary, we found that s-polarized surface polaritons can exist for thin metal films bounded by nonlinear media with positive nonlinearlties, but not for any other planar geometry involving metals. These waves all have power thresholds. very metal thickness required, with surface plasmons for small lsml, and the large changes in refractive index required will probably make such waves difficult to observed experimentally.
................ _......... .*
.“.
/...‘.. . . ...** *....*
\
-+
1.6
1.8
B 0)
2.0
B
Fig. 3: Guided wave power (mW/mm) versus effective index 6 for nc = ns - 1.55, d - 0.005 urn,sm = -10 and "2c = lo-' m'/W with nzs =lO'LO ml/W (solid line), "2s = lo-" ma/W (dashed line) and n2s = 10-l' m'/w (dotted line).
>
This research was supported by NSF (KS-8117483 and -8304749) and the Joint Services Optics Program of the Army Research Office and the Air Force Office Of Scientific Research.
Fig. 4: Field distributions with increasing 6 for nc = ns = 1.55, d - 0.005 arm, = -10 and for (a) n2s ='?O-, ml/W, (b) "2c = lo-' m*/W nzs = lo-" m'/W and (c) n2s - lo-" m'/W.
1. TOMLINSON, W.J., Opt. Lett. 5, 323 (1980). 2. MABADUDIN, A.A., Zeit. Phys. B. 41, 341 (1981). 3. MARADIJDW, A.A., in Proceedings of the Second International School on Condensed Matter Physics, Varna, Bulgaria (World Scientific Publishers, Singapore, 1983) in press.
4. AGRANOVICH, V.M., BABICHENRO, V.S., and CHERNYAE, V. YA., Sov. Phys. JETP Let. 32, 512 (1981). 5. AEHMEDIEV. N.N., Sov. Phys. JETP 56, 299 (1982). 6. ROBBINS, DJ., Opt. Comm. 47, 309 (1983). 7. LANGBEIN, U, LEDERER, F., and PCNATH, H.E., Opt. Comm. 46, 167 (1983).
Vol.
52,
No.
3
NONLINEAR S-POLARIZED
297
SLJRPACE PLASMON POLARITONS
8. BOARDXAN, A& and EGAN, P, J. Physique Colloq., In press (1984)‘. 9. BOARDMAN, A., and EGAN, P, prlvate communication, 1984. 10. STEGEMAN, G.I.. SEATON, C.T.. CHILWELL, J, and SMITH, S.D., Appl. Phys. Lett. 44, 830 (1984).
11.
STBGEMAN, G.I.. and SEATON, C.T., Opt. (1984). 12. VACH, H.. STEGEMAN. GJ., SEATON, C.T., Opt. Lett. 9, 238 (1984).
Lett. and
9,
235
KHOO, IX.,