Long-range surface plasmon electrooptic modulator

Jay S. Schildkraut

An electrooptic modulator based on electrically varying the degree of coupling of a light beam to a long-rangesurface plasmon excitation is modeled. A specific embodiment of a modulator is shown to be capable ofchanging its reflectance from 0.00 to 0.84 on the application of 100 V across a 1-,am thick electrooptic film witha second-order susceptibility of 2 X 10-7 esu. The affect that the width of the long-range surface plasmonexcitation resonance has on the performance of the modulator is considered. Plane-wave calculations showthat decreasing the width of the resonance increases the amount of modulation possible for a given voltagechange. More realistic calculations which take into account diffraction of a beam of finite width show that adecrease in resonance width can in some cases degrade device performance.

1. Introduction

Surface plasmons are electromagnetic waves guidedalong a metal-dielectric interface. 1 In the case of athin metal film bordered on both sides by dielectricswith identical 2 or close3 indices of refraction, surfaceplasmon waves on opposite faces of the metal filmbecome coupled to form a mode in which the electricfield is symmetric and a mode in which it is antisym-metric. The antisymmetric mode carries most of itsintensity in the dielectrics, which results in a decreasein damping of the wave by the metal film and conse-quently an increase in propagation length. Thesewaves are referred to as long-range surface plasmons(LRSPs). LRSPs have been considered as a means forthe enhancement of nonlinear optical processes suchas second harmonic generation, degenerate four-wavemixing, and intensity-dependent refractive-indexphenomena. 4 ' 5

An electrooptic modulator based on electricallyvarying the efficiency of coupling a collimated mono-chromatic beam into surface plasmons has been de-scribed by Sincerbox and Gordon.6 In this paper wemodel an electrooptic modulator based on varying theefficiency of coupling light into LRSPs. 7 This deviceis shown to allow a greater degree of modulation for agiven voltage because of the narrower excitation reso-nance for LRSP compared to surface plasmons. We

The author is with Eastman Kodak Company, Corporate Re-search Laboratories, Rochester, New York 14652.

Received 28 December 1987.0003-6935/88/214587-04$02.00/0.©) 1988 Optical Society of America.

show how performance can be optimized by maximiz-ing the fraction of guided wave intensity propagatingin the dielectric layer across which a voltage is applied.Also, we consider the effect that diffraction of a lightbeam of finite width has on modulator operation bycalculating the reflectivity of a beam with a Gaussianintensity profile of various widths. It is shown thatdecreasing the width of the excitation resonance bydecreasing the thickness of the metal film will increasethe degree of modulation of a plane wave but can bedetrimental in the case of a beam of finite width.

II. Plasma-Wave Calculations

A LRSP electrooptic modulator is shown in Fig. 1.A collimated beam of 632.8-nm p-polarized light isincident on the base of a SF- flint prism with index ofrefraction no = 1.72 at an angle 0. Below the prism is adielectric film with index of refraction nj and thicknessd1. Next is a thin silver film with refractive index n2 =

0.085 + 4.22i8 and thickness d2. An electrooptic filmwith index of refraction n3 and thickness d3 is sand-wiched between the thin silver film and an electrodewith index of refraction n4. A voltage supply is con-nected to the thin silver film and the electrode to applya voltage V across the electrooptic film.

The electrooptic film is modeled as a noncentrosym-metric organic film.9 Such a film will generally haveuniaxial symmetry about the film normal, but since theelectric field of a surface plasmon is mostly normal tothe metal film,' it is possible to treat the film as iso-tropic as long as the optical constants normal to the filmare used in calculations. The electrooptic film is as-sumed to have n3 = 1.54, d3 = 1 m, and x(2) = 2 X 10-7esu.

The reflectance Iut/in of the four-layer system10 asa function of 0 for the case of n = n3, n4 = n2, d2 = 20

1 November 1988 / Vol. 27, No. 21 / APPLIED OPTICS 4587

z

Lx'in lout

Fig. 1. Long-range surface plasmon electrooptic modulator. Acollimated beam of p-polarized light is incident on the base of aprism with index of refraction no at an angle 0. Below the prism is adielectric film with index of refraction ni and thickness dl. Next is athin metal film with refractive index n2 and thickness d2. Anelectrooptic film with index of refraction n and thickness d3 issandwiched between the thin metal film and an electrode with indexof refraction n4. A voltage supply is connected to the thin film and

the electrode to apply a voltage V across the electrooptic film.

1.2

1.0 6

a) 0.8 ifC

00.6 I

ci~ ~~~ncdn angl

a)~~~~~Irr~~~~~~I

0.4

0.2-

0640 645 65.0 65.5 66.0

Incident angle

Fig. 2. Reflectance vs incident angle: Voltage equals: 0 V, solidline; 50 V, dashed line; 100 V, dotted line.

nm, and d1 = 950 nm is shown in Fig. 2. The sharp dipin reflectance at 65.0400 is due to the excitation of aLRSP. This occurs when the x-component of thewave vector of light in the prism equals the wave-number of a LRSP.

When a voltage is applied across the electroopticfilm, its index of refraction will change by an amount6n 3 because of the Pockels effect arising from the film'shigh x(. We now derive a relationship between n3and x(, which is appropriate for the modulator. Asmentioned above, it is only necessary to consider thenormal-to-film index of refraction, and, therefore, weneed only be concerned with the z-component of thenonlinear polarization. The interaction of the dc elec-tric field in the z-direction and the z-component of the

) 0.8

oO

00.6

0)0.2 /_

0 20 40 60 80 100Voltage

Fig. 3. Reflectance vs voltage. Long-range surface plasmon modu-lator: n = 1.54, solid line; n = 1.53, dotted line. Surface plasmon

modulator, dashed line.

p-polarized optical electric field gives rise to a polar-ization which is given in esu units by"1

Pz = x(2)E()E(), (1)

where Ez(O) and Ez(w) are the dc and optical electricfield amplitudes, respectively. There is no contribu-tion to P, from the x-component of the optical fieldbecause xZ~Z = Xz22X = 0 in the case of a uniaxial film.1 2

The change in the dielectric constant normal to thefilm due to the electrooptic effect bEn is related to Pz by

Ez(w)(2)

Usingthe relation e = n2 we obtain In = be/2n, which incombination with Eqs. (1) and (2) leads to

27rX(2) E (0)6% =

n3(3)

Equation (3) is more conveniently written in terms ofthe voltage applied across the film as

2r10 5 X(2) V

3n3 d3(4)

where x(2) is expressed in esu, d3 in nanometers, and Vin volts.

A change in n3 causes the wavenumber of the LRSPto shift and consequently causes a shift in the dip inreflectance (see Fig. 2). Figure 3 shows a plot of reflec-tance vs voltage for the LRSP modulator. The reflec-tance changes from 0.00 to 0.84 when the voltage isswitched from 0 to 100 V. (Calculations were carriedout only up to V = 100 V, since that is typical of thebreakdown voltage of a 1l-m thick organic film.) Forcomparison, a similar plot was made for an electroopticmodulator based on surface plasmons. This modula-tor is as shown in Fig. 1 except with d1 set to zero and d2equal to 50 nm. Figure 3 shows that the reflectivityonly changes from 0.00 to 0.12 when 100 V are appliedacross the electrooptic film.

The degree of modulation is expected to increase asmore of the LRSP electric field is guided inside the

4588 APPLIED OPTICS / Vol. 27, No. 21 / 1 November 1988

electrooptic film. The amount of power guided in thisfilm can be increased by lowering n1 .3 As n, is de-creased, the z-component of the wave vector in theelectrooptic film k3, which is a positive imaginary num-ber, will also decrease. Since the electric field decaysas exp(ik'z) away from the metal film, this results inslower field fall-off and, therefore, in an increase inpower guided in the electrooptic layer. If the elec-trooptic material were semi-infinite, it would be desir-able to operate near a stop point ( = 0) where thefield changes from evanescent to harmonic. Since theelectrooptic material is a film, it is not necessary todesign the modulator with k 3 0 because this willextend the field into the electrode without appreciablyincreasing the power guided in the film. Figure 3shows a curve of reflectance vs voltage when n, = 1.53.The response of the modulator is slightly increasedcompared with n = 1.54. Choosing n, closer to thestop point, which is between 1.50 and 1.51, did notincrease the response further.

Ill. Gaussian Beam Calculations

The plane-wave calculations in Sec. II serve to illus-trate the concept of a LRSP electrooptic modulator,but since the width of the reflectance dip (see Fig. 2)can be of the same magnitude as the spread in incidentangle due to diffraction of a light beam of finite width,a realistic calculation must include finite beam-widtheffects. In this section modulator operation is calcu-lated assuming an incident beam with a Gaussian in-tensity profile.

The coupling of a beam with a step-function intensi-ty profile to LRSP has been analyzed by Deck et al. 13

and Booman et al.14 The theory of the excitation of asurface plasmon by a beam with a Gaussian intensityprofile has been outlined by Chuang.15 Here we derivethe equations needed to calculate the reflectance of aGaussian beam from the LRSP electrooptic modula-tor.

The magnetic field of the incident beam at the baseof the prism is assumed to have the form15

Hi= exp[-(x/WX) 2 ] exp(ik'x), (5)

where 5 is the unit vector in the y-direction, k is thecentral x-component of the wave vector of the beamwhich is related to the incident angle by

k = (/c)no sin0, (6)

and Wx is the projection of the beamwidth Won the x-axis and is given by

W = W/coso. (7)

The field is decomposed into plane waves by finding itsFourier transform H(kx), which is given by

h(kX) = a'_ exp[-(kx - k) 2( W/2) 2] . (8)

The ratio of the reflected field to the incident field isproportional to the inverse Fourier transform of h(kx)multiplied by the plane-wave reflection coefficient forthe four-layer structure r(kx) and is given by

a)a)

CO1Za,

cc

CO0)

Cc

Voltage0 20 40 60 80

VoltageDO

Fig. 4. Reflectance vs voltage. d 2 equals: 20 nm, solid line; 15 nm,dashed line; 10 mn, dotted line. Plane-wave calculation (a). Gauss-ian beam calculation with W equal to 500 Acm (b), 100 Aim (c), 1 0 Am

(d).

G(X) = J r(k,)h(k 0) ep(ikxx)dkx. (9)

The reflectance R is equal to the square of the magni-tude of G(x) integrated over x divided by a normaliza-tion factor, which is obtained by evaluating G(x) withr(kx) equal to one:

R = X2 W A ' IG(x)I2d. (10)

In practice, the integrals in Eqs. (9) and (10) must benumerically calculated with finite limits. It was foundto be sufficient to use integration limits of +5/Wx and±5WX in Eqs. (9) and (10), respectively.

The width of the reflectance dip may be decreasedby decreasing the thickness of the silver film d2, whichlessens the damping of the LRSP. We now investigatethe effect that this has on the response of the modula-tor. The width of the dip in reflectance at half-mini-mum for d2 = 20, 15, and 10 nm is found to be 0.1, 0.05,and 0.03°, respectively. (d1was set equal to 950, 1250,and 1700 nm, respectively, so that the minimum valueof reflectance is zero.) Figure 4(a) shows reflectancevs voltage curves for a plane wave. The response isseen to increase greatly with decreasing d2. Reflec-tance vs voltage curves for Gaussian beams with W =500, 100, and 10 gum that were calculated using Eqs.(5)-(10) are shown in Figs. 4(b), (c), and (d), respec-tively. When W equals 500 ,vm, the response is almostthe same as for a plane wave except that the reflectanceis no longer zero when V = 0 V. This effect is mostpronounced when d2 = 10 nm. When W equals 100gtm, the response decreases with decreasing d2 becausethe resonance becomes too sharp with respect to the

1 November 1988 / Vol. 27, No. 21 / APPLIED OPTICS 4589

1.0(d) W=10)u

.................................................................

U..

0.6-

04-

0.2-

0 I I I 1

I1.

spread in incident angle of the beam. Finally, when Wequals 10 ,gm, the modulator is no longer functional atthe values of d2 we have considered.

The above calculations illustrate an importanttrade-off in the design of a LRSP electrooptic modula-tor. The response of the modulator is optimized withan excitation resonance, which is as narrow as possiblewithout degrading its performance due to the finitewidth of the light beam to be modulated. Control ofthe thickness of the metal film is a possible way todesign a modulator with the desired excitation reso-nance width.

IV. Conclusion

We have modeled a new type of electrooptic modula-tor, which is based on electrically varying the efficiencyof coupling a light beam to LRSP. It is shown that thenarrow width of the LRSP excitation greatly increasesthe response over that of a modulator based on cou-pling light to surface plasmons. Calculations per-formed for a finite-width beam with a Gaussian inten-sity profile illustrate the effect that the spread inincident angle due to diffraction of a finite-width beamhas on modulator performance.

References1. H. Raether, "Surface Plasma Oscillations and Their Applica-

tions," in Physics of Thin Films, Vol. 9 (Academic, New York,1977), p. 145.

2. D. Sarid, "Long-Range Surface-Plasma Waves on Very ThinMetal Films," Phys. Rev. Lett. 47, 1927 (1981).

3. L. Wendler and R. Haupt, "Long-Range Surface Plasmon-Po-laritons in Asymmetric Layered Structures," J. Appl. Phys. 59,3289 (1986).

4. G. I. Stegeman, J. J. Burke, and D. G. Hall, "Nonlinear Optics ofLong Range Surface Plasmons," Appl. Phys. Lett. 41, 906(1982).

5. J. C. Quail, J. G. Rako, H. J. Simon, and R. T. Deck, "OpticalSecond-Harmonic Generation with Long-Range Surface Plas-mons," Phys. Rev. Lett. 50, 1987 (1983).

6. G. T. Sincerbox and J. C. Gordon II, "Small Fast Large-ApertureLight Modulator Using Attenuated Total Reflection," ApplOpt. 20, 1491 (1981).

7. While this work was in progress an electrooptic modulator usinglong-range surface plasmons with a different configuration thanthe one discussed in this paper was proposed in C. Plumereau,A.-M. Bouchoux, and A. Cachard, "Electrooptic Light Modula-tor Using Long-Range Surface Plasmons," Proc. Soc. Photo-Opt. Instrum. Eng. 800, 79 (1987).

8. H. Gugger, M. Jurich, J. D. Swalen, and A. J. Sievers, "Observa-tion of an Index of Refraction Induced Change in the DrudeParameters of Ag Films," Phys. Rev. B 30, 4189 (1984).

9. J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, andR. Lytel, "Poled Electro-optic Waveguide Formation in Thin-Film Organic Media," Appl. Phys. Lett. 52, 1031 (1988).

10. 0. S. Heavens, Optical Properties of Thin Solid Films (Butter-worth, London, 1955), p. 62.

11. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, NewYork, 1984), pp. 38-39.

12. B. Dick, A. Gierulski, G. Marowsky, and G. A. Reider, "Determi-nation of the Nonlinear Optical Susceptibility x(2) of SurfaceLayers by Sum and Difference Frequency Generation in Reflec-tion and Transmission," Appl. Phys. B 38, 107 (1985).

13. R. T. Deck, D. Sarid, G. A. Olson, and J. M. Elson, "CouplingBetween Finite Electromagnetic Beam and Long-Range Sur-face-Plasmon Mode," Appl. Opt. 22, 3397 (1983).

14. R. A. Booman, G. A. Olson, and D. Sarid, "Determination of LossCoefficients of Long-Range Surface Plasmons," Appl. Opt. 25,2729 (1986).

15. S. L. Chuang, "Lateral Shift of an Optical Beam due to LeakySurface-Plasmon Excitations," J. Opt. Soc. Am. A 3, 593 (1986).

Of Optics continued from page 4519

This summer, several years behind schedule, the PatentOffice completed a six-month test with a small group ofpatent examiners. The results have not yet been fully dis-closed, but officials acknowledge that several problems be-came apparent. According to the outside review team, ex-aminers complained that the system was "unnecessarilycomplex and difficult to use." It was also erratic, ofteninforming the examiners that certain documents known to bestored in the system were not there. The computer searchesactually proved slower than manual searches, officials say."We really get no short-term increases in productivity," saidMr. Giammo. "The real gain we're seeing now is a substan-tial improvement in the quality of the work being done."

Critics in private industry remain worried, however. TheAmerican Intellectual Property Law Association, a Wash-ington trade association for patent attorneys, has estimatedthat the automation program could increase the cost of thepatent system by $50 million a year, or as much as $100 anhour in computer time. "If you can get a search but it's goingto cost $10,000, it's going to put a lot of inventors out of thefield," said Joseph DeGrandi, president of the association.

Some observers suggest that the Patent Office should havelooked at less-advanced technologies as a way to solve itsproblems. "They started out trying to build the space shut-

tle instead of starting out with disposable rockets," re-marked Mr. Armitage of the Upjohn Company. An alterna-tive, he suggested, would have been to examine modern waysof preserving paper files while using less advanced comput-ers for more limited purposes. "Fundamentally, what Iwould do is shut down the whole system and buy desk-topcomputers with modems for the examiners," said HaroldWegner, a Washington lawyer. He suggests that the PatentOffice improve its paper files simply by hiring a small army ofadditional clerical workers to maintain them.

Mr. Giammo said the office would spend more time testingthe system. After that, he plans to expand the system to onesmall group of examiners at a time. As each group becomesautomated, he plans to assess the results. "The design isvery modular," he explained. The purpose of this is to letthe agency change key components without disrupting theentire system.

All of this will add several years to the development time,but many observers welcome the cautious approach. "Hadthey not decided to delay, I would have had grave doubts,"Mr. Armitage said.

Mr. Giammo, meanwhile, insists that the Government'sfundamental design appears to make sense. "At this point,based on what we know, it appears easily worthwhile tocontinue," he said.

4590 APPLIED OPTICS / Vol. 27, No. 21 / 1 November 1988