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Second harmonic generation by oppositely travelling long range surface polaritons
Volume 46, number 3,4
OPTICS COMMUNICATIONS
1 July 1983
SECOND HARMONIC GENERATION BY OPPOSITELY TRAVELLING LONG RANGE SURFACE POLARITONS G.I. STEGEMAN, C. LIAO * and C. KARAGULEFF Optical Sciences Center and Arizona Research Laboratories, University of Arizona, Tucson, AZ 85721, USA
Received 5 April 1983
We analyze the nonlinear mixing of two oppositely propagating long range surface plasmon polaritons in an ATR geometry. Typical efficiencies for the second harmonic signal radiated normal to the metal films are calculated.
1. Introduction The nonlinear mixing of two oppositely propagating optical waveforms can be used to achieve picosecond signal processing operations such as convolution etc. This phenomenon has already been demonstrated using waves guided in Ti: LiNbO 3 waveguides with 16 ps pulses [ 1 - 3 ] , as well as with surface plasmons in a Kretschmann geometry [4]. Since the harmonic signal is radiated normal to the surface, it is not phase-matchable and hence maintaining high power densities over millimeter propagation distances is crucial to obtaining useful efficiencies. Any wave phenomenon characterized by strong field confinement and long propagation distances is an interesting candidate for this process. Surface plasmons are electromagnetic waves guided by metal surfaces with their fields highly localized at metal-dielectric boundaries. In the usual prism excitation geometries, i.e. high index prism-metal f i l m dielectric (Kratschmann geometry), the surface plasmon propagation distance is limited to tens of microns in the visible and near infrared. Shen and coworkers [4] have used this geometry to obtain second harmonic generation by mixing oppositely propagating surface plasmons. Their signal levels, although easily measurable, were too small to be useful in practical device * Visiting scholar from Institute of Optics and Electronics, Chinese Academy of Science, Chungdu, China. 0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
applications. However, it has recently been shown that surface plasmon propagation distances can be increased by one to two orders of magnitude by sandwiching a thin metal film between two media of equal [5,6], and almost equal [7] refractive index. For k >/1/am, the propagation distance can be in excess of 1 mm, and it has been shown that the nonlinear interaction between such freely propagating long range polaritons can be quite efficient. Second harmonic generation by surface propagating in the same [5] and opposite directions, degenerate four wave mixing and intensity dependent refractive index phenomena have all been analyzed [5]. For ?~~< 1/am, propagation distances are too small [7] to consider freely propagating waves and the interaction can only ba carried out under a coupler, for example a prism or a grating [8,9]. It is this ATR case that we treat here for oppositely propagating long range surface plasmon polaritons.
2.
Theory
The geometry which we consider is shown in fig. 1. Two external plane wave fields are incident through the prism (nl) at an angle 0 to the surface normal. Surface plasmons are excited in the dielectrics near the n 2 - n 3 and n 4 - n 3 boundaries [10-13]. In this second order nonlinear interactions, a harmonic polarization field is created with zero periodicity parallel to the surfaces [13] and hence it radiates plane waves 253
Volume 46, number 3,4
OPTICS COMMUNICATIONS
t:'(i ~ ( ~ )
1 July 1983
The nonlinear polarization field created by the mixing of oppositely propagating waves is well-known [ 14]. The electric fields associated with a surface polariton are obtained from 1i7× H = ~Dl~t as ----~X
o n4
t
[
E x = -(io~4/6oeon2)A4 ,
Z
Fig. 1. The multilayer ATR geometry investigated. Pl(w) and P2(co) are the incident beams which mix in medium n 4 to produce the harmonic signal P(2co). normal to the boundaries (i.e. along the z-axis in fig. 1) into both the prism, and medium n 4. This requires that either medium n 2 or n 4 be non-centrosymmetric. The pertinent second order susceptibilities are ×(2x)ii and X(2)i/which couple to the x and z electric field' components of the incident surface plasmons. In order to obtain useful signals with long range surface polaritons, the preferred components are -~.x,zz (2) and ,,(2) ,~.y,gg, as will be discussed later. (Note that the x , y , z coordinate system of fig. 1 does not necessarily coincide with the crystal coordinate axes for media n 2 and/or n 4.) The field created near the metal film by a plane wave incident through the prism requires a straightforward application of boundary conditions in a multifilm structure. We write the incident field (beam 1) in terms of its magnetic field as H = ½j A 1 exp[i(cot - kllX - al(Z + d))] + cc.
(1)
In medium n 4 where the nonlinear interaction is assumed to take place
E z = -(kll/Coe0n2)A4 . (5)
Since obtaining long propagation distances with surface plasmons requires operating near mode cut-off, i.e. the fields in media n 2 and n 4 are almost plane waves, kll >)> 0~4 and Ifzl >> IExl. Therefore we retain o n l y E z and write
~x = 2eodx,zzEz(1) f z ( 2 ) ,
9y = 3eody,zzEz(1 ) Ez(:),
where 1 and 2 refer to the incident waves in fig. 1. The calculation of the second harmonic field radiated back into the prism is again a straight-forward application of continuity of tangential boundary conditions to solutions of the wave equation in each medium. Assuming a dx,zz nonlinear coefficient, the radiated field is given by _1
H-~]B
1 exp[2i(cot +nlk(Z + d ) ) ] + c c ,
ico 9xn 3(2w ) B 1 = (a 4 + n4k)(n3(2co)B+ + i n 4 B _ ) ,
B+ = cos(2n2kd ) cos(2n3kh )
Here kll = n 1 sin Ok (k = co/c) where 0 is the angle of incidence measured from the normal to the surfaces, a 2 = n2k 2 - k 2 and a 2 = k 2 - n2k 2. The ratio A I/A4 is given by
+ i(n~/n~) sin(2n2kd) cos(2n3kh ) and
A 11a4 = ½A+ [1 - ~2/~1 ] e x p ( - i a 2 d )
B_ = (n~ln~)
ot4(z - h))] + cc.
+ ~-A_ [1 + ~2/~1] exp(ia2d),
(3)
with
(n2/n 2) sin(2n2kd)
sin(2n3kh ) ,
(9)
cos(2n2kd ) cos(2n3kh )
+ i cos(2n2kd ) sin(2n3kh )
- (n2/n 2) sin(2n2kd ) sin(2n3kh ) . (4)
and ot2 = n2k 2 - k 2, a 2 = n2k 2 - k 2 and ffi = ai/n2" When a surface plasmon-like resonance is excited, the quantity IA4/A 112 exhibits a sharp maximum as 0 is varied. 254
-
+ i(n2/n 2) sin(2n2kd) cos(2n3kh)
A~ = ½[cos(a3h)(1 T- i~4/~2) + sin(c~3h)(~4/~ 3 + i~3/~2) ] ,
(8)
with
+ i(n2/n 2) cos(Zn2ka) sin(2n3kh)
-
(7)
where
(2)
H = ½] A 4 e x p [ i ( w t - kllX
(6)
(10)
(Identical numerical results are obtained ifdy,z z is used instead, as expected.) We have assumed here that nl, n 2 and n 4 are frequency independent since the effects of their dispersion with wavelength on the harmonic signal are small.
1 July 1983
OPTICS COMMUNICATIONS
Volume 46, number 3,4
The cross-section for this interaction is best expressed in terms of the input beam powers P, and P,. For a radiation field in the prism, the Poynting vector gives
with 8 = 0 for the harmonic
fields. Thus DNL
P(2w) =
IB112P1P2
= HL
PIP2
,
where the beam dimensions are H (along the y-axis) and L alongx, and A, is set to unity in eqs.(l)-(10). Since intuitively one expects the signal to be proportional to the illuminated area HL, we write this result in terms of the incident fluxes S, and S, (power/area) as P(2w) = 2~,HL(eo/rro)IB,12~,S2
.
(13)
3. Numerical calculations We now illustrate the efficiency of this process with a detailed calculation. MNA [ 151 was chosen as the nonlinear material (n4 = 1.80, d12 = 3.4 X lo-11 m/V) and rutile as the prism material. Since the refractive index of MNA falls in the range of high index glasses [ 161, we assume that n2 can be chosen equal to the 1.80 7 0.01 which is required for this purpose. The metal (h = 150 A) is silver [ 161. We first numerically varied the incidence angle B until a minimum in the ATR spectrum was obtained for a given film thickness d (medium n2). The result-
FILM THICKNESS (MICRONS)
Fig. 2. The fraction of the incident beam energy absorbed at the minimum of the ATR spectrum as a function of film thickness.
1.5
I
I
J
2.0
2.5
3.0
FILM THICKNESS &lICRONS)
Fig. 3. The fraction of the total incident power (Pt + P-J converted into the harmonic signal (P(2w)) versus the thickness of medium n2.
ing surface polariton fields were then used to evaluate the harmonic signal. Shown in fig. 2 is the fraction of the incident energy (h = 1.06 pm) absorbed which peaks at d = 2.2 pm. Assuming H = L = 1 cm and an incident total power density of 1 MW/cm2, the fraction of the total input power (PI + P2) converted to the harmonic signal is shown in fig. 3. The oscillations are due to multireflection interference effects of the harmonic signal in medium n2 - (the harmonic radiation travels normal to the film interfaces, fig. 1). The maximum conversion efficiency of 1.5 X 1O-4 is significantly better than that reported in Ti:in-diffused LiNbO, waveguides [ 141. The conversion efficiency is very sensitive to metal film thickness. For h = 250 A, the maximum conversion efficiency of 1.7 X 10-S occurs at d = 1.36 pm. In summary, we have analyzed the nonlinear mixing of two oppositely propagating long range surface plasmons in an ATR geometry. Using MNA as the nonlinear material, conversion efficiencies of 1O-4 appear to be feasible. [Just prior to the submission of this paper we learned that a propagation distance enhancement of 20X has been observed for long range surface plasmons by Y. Kuwamura, M. Fukui and 0. Tada, in press, J. Phys. Sot. Japan.] This research was supported by NSF (ECS-8117483) and the Joint Services Optics Program of AR0 and AFOSR.
255
Volume 46, number 3,4
OPTICS COMMUNICATIONS
References [1] R. Normandin and G.I. Stegeman, Appl. Phys. Lett. 36 (1980) 253. [2] R. Normandin and G.I. Stegeman, Appl. Phys. Lett. 40 (1982) 759. [3] G.I. Stegeman and R. Normandin, Proc. SPI, Vol. 321, ed. D.G. Hall (SPIE, Washington, 1982) p. 55. [4] C.K. Chen, A.R.B. de Castro and Y.R. Shen, Optics Lett. 4 (1979) 393. [5] G.I. Stegeman, J.J. Burke and D.G. Hall, Appl. Phys. Lett. 41 (1982) 906. [6] M. Fukui, V. So and R. Normandin, Phys. Status Solidi (b) 91 (1979) K61; M. Fukui, V.C.-Y. So and G.I. Stegeman, Surf. Sci. 85 (1979) 125. [7] D. Sarid, Phys. Rev. Lett. 27 (1981) 1927. [8] R.T. Deck and D. Sarid, J. Opt. Soc. Am. 72 (1982) 1613.
256
1 July 1983
[9] C. Karaguleff and G.I. Stegeman, submitted to Applied Optics. [10[ F. Abeles and T. Lopez-Rios, Optics Comm. 11 (1974) 89. [ 11 ] W.H. Weber and S.L. McCarthy, Appl. Phys. Lett. 25 (1974) 396. [12] A.B. Buckman and C. Kuo, J. Opt. Soc. Am. 69 (1979) 343. [13] D. Sarid, R.T. Deck, A.E. Craig, R.K. Gickernell, R.S. Jameson and J.J. Fasano, Appl. Optics 21 (1982) 3993. [14] R. Normandin and G.I. Stegeman, Optics Lett. 4 (1979) 58. [15] B.F. Levine, C.G. Bethea, C.D. Thurmond, R.T. Lynch and J . g Bernstein, J. Appl. Phys. 50 (1979) 2523. [ 16] Schott Optical Glass Catalogue (Schott, Duryea, PA). [17] M. Dujardin and M. Theye, J. Phys. Chem. Sol. 32 (1971) 2033; T. Hollstein, U. Kreibig and F. Leis, Phys. Stat. Solidi 82b (1977) 545.
OPTICS COMMUNICATIONS
1 July 1983
SECOND HARMONIC GENERATION BY OPPOSITELY TRAVELLING LONG RANGE SURFACE POLARITONS G.I. STEGEMAN, C. LIAO * and C. KARAGULEFF Optical Sciences Center and Arizona Research Laboratories, University of Arizona, Tucson, AZ 85721, USA
Received 5 April 1983
We analyze the nonlinear mixing of two oppositely propagating long range surface plasmon polaritons in an ATR geometry. Typical efficiencies for the second harmonic signal radiated normal to the metal films are calculated.
1. Introduction The nonlinear mixing of two oppositely propagating optical waveforms can be used to achieve picosecond signal processing operations such as convolution etc. This phenomenon has already been demonstrated using waves guided in Ti: LiNbO 3 waveguides with 16 ps pulses [ 1 - 3 ] , as well as with surface plasmons in a Kretschmann geometry [4]. Since the harmonic signal is radiated normal to the surface, it is not phase-matchable and hence maintaining high power densities over millimeter propagation distances is crucial to obtaining useful efficiencies. Any wave phenomenon characterized by strong field confinement and long propagation distances is an interesting candidate for this process. Surface plasmons are electromagnetic waves guided by metal surfaces with their fields highly localized at metal-dielectric boundaries. In the usual prism excitation geometries, i.e. high index prism-metal f i l m dielectric (Kratschmann geometry), the surface plasmon propagation distance is limited to tens of microns in the visible and near infrared. Shen and coworkers [4] have used this geometry to obtain second harmonic generation by mixing oppositely propagating surface plasmons. Their signal levels, although easily measurable, were too small to be useful in practical device * Visiting scholar from Institute of Optics and Electronics, Chinese Academy of Science, Chungdu, China. 0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
applications. However, it has recently been shown that surface plasmon propagation distances can be increased by one to two orders of magnitude by sandwiching a thin metal film between two media of equal [5,6], and almost equal [7] refractive index. For k >/1/am, the propagation distance can be in excess of 1 mm, and it has been shown that the nonlinear interaction between such freely propagating long range polaritons can be quite efficient. Second harmonic generation by surface propagating in the same [5] and opposite directions, degenerate four wave mixing and intensity dependent refractive index phenomena have all been analyzed [5]. For ?~~< 1/am, propagation distances are too small [7] to consider freely propagating waves and the interaction can only ba carried out under a coupler, for example a prism or a grating [8,9]. It is this ATR case that we treat here for oppositely propagating long range surface plasmon polaritons.
2.
Theory
The geometry which we consider is shown in fig. 1. Two external plane wave fields are incident through the prism (nl) at an angle 0 to the surface normal. Surface plasmons are excited in the dielectrics near the n 2 - n 3 and n 4 - n 3 boundaries [10-13]. In this second order nonlinear interactions, a harmonic polarization field is created with zero periodicity parallel to the surfaces [13] and hence it radiates plane waves 253
Volume 46, number 3,4
OPTICS COMMUNICATIONS
t:'(i ~ ( ~ )
1 July 1983
The nonlinear polarization field created by the mixing of oppositely propagating waves is well-known [ 14]. The electric fields associated with a surface polariton are obtained from 1i7× H = ~Dl~t as ----~X
o n4
t
[
E x = -(io~4/6oeon2)A4 ,
Z
Fig. 1. The multilayer ATR geometry investigated. Pl(w) and P2(co) are the incident beams which mix in medium n 4 to produce the harmonic signal P(2co). normal to the boundaries (i.e. along the z-axis in fig. 1) into both the prism, and medium n 4. This requires that either medium n 2 or n 4 be non-centrosymmetric. The pertinent second order susceptibilities are ×(2x)ii and X(2)i/which couple to the x and z electric field' components of the incident surface plasmons. In order to obtain useful signals with long range surface polaritons, the preferred components are -~.x,zz (2) and ,,(2) ,~.y,gg, as will be discussed later. (Note that the x , y , z coordinate system of fig. 1 does not necessarily coincide with the crystal coordinate axes for media n 2 and/or n 4.) The field created near the metal film by a plane wave incident through the prism requires a straightforward application of boundary conditions in a multifilm structure. We write the incident field (beam 1) in terms of its magnetic field as H = ½j A 1 exp[i(cot - kllX - al(Z + d))] + cc.
(1)
In medium n 4 where the nonlinear interaction is assumed to take place
E z = -(kll/Coe0n2)A4 . (5)
Since obtaining long propagation distances with surface plasmons requires operating near mode cut-off, i.e. the fields in media n 2 and n 4 are almost plane waves, kll >)> 0~4 and Ifzl >> IExl. Therefore we retain o n l y E z and write
~x = 2eodx,zzEz(1) f z ( 2 ) ,
9y = 3eody,zzEz(1 ) Ez(:),
where 1 and 2 refer to the incident waves in fig. 1. The calculation of the second harmonic field radiated back into the prism is again a straight-forward application of continuity of tangential boundary conditions to solutions of the wave equation in each medium. Assuming a dx,zz nonlinear coefficient, the radiated field is given by _1
H-~]B
1 exp[2i(cot +nlk(Z + d ) ) ] + c c ,
ico 9xn 3(2w ) B 1 = (a 4 + n4k)(n3(2co)B+ + i n 4 B _ ) ,
B+ = cos(2n2kd ) cos(2n3kh )
Here kll = n 1 sin Ok (k = co/c) where 0 is the angle of incidence measured from the normal to the surfaces, a 2 = n2k 2 - k 2 and a 2 = k 2 - n2k 2. The ratio A I/A4 is given by
+ i(n~/n~) sin(2n2kd) cos(2n3kh ) and
A 11a4 = ½A+ [1 - ~2/~1 ] e x p ( - i a 2 d )
B_ = (n~ln~)
ot4(z - h))] + cc.
+ ~-A_ [1 + ~2/~1] exp(ia2d),
(3)
with
(n2/n 2) sin(2n2kd)
sin(2n3kh ) ,
(9)
cos(2n2kd ) cos(2n3kh )
+ i cos(2n2kd ) sin(2n3kh )
- (n2/n 2) sin(2n2kd ) sin(2n3kh ) . (4)
and ot2 = n2k 2 - k 2, a 2 = n2k 2 - k 2 and ffi = ai/n2" When a surface plasmon-like resonance is excited, the quantity IA4/A 112 exhibits a sharp maximum as 0 is varied. 254
-
+ i(n2/n 2) sin(2n2kd) cos(2n3kh)
A~ = ½[cos(a3h)(1 T- i~4/~2) + sin(c~3h)(~4/~ 3 + i~3/~2) ] ,
(8)
with
+ i(n2/n 2) cos(Zn2ka) sin(2n3kh)
-
(7)
where
(2)
H = ½] A 4 e x p [ i ( w t - kllX
(6)
(10)
(Identical numerical results are obtained ifdy,z z is used instead, as expected.) We have assumed here that nl, n 2 and n 4 are frequency independent since the effects of their dispersion with wavelength on the harmonic signal are small.
1 July 1983
OPTICS COMMUNICATIONS
Volume 46, number 3,4
The cross-section for this interaction is best expressed in terms of the input beam powers P, and P,. For a radiation field in the prism, the Poynting vector gives
with 8 = 0 for the harmonic
fields. Thus DNL
P(2w) =
IB112P1P2
= HL
PIP2
,
where the beam dimensions are H (along the y-axis) and L alongx, and A, is set to unity in eqs.(l)-(10). Since intuitively one expects the signal to be proportional to the illuminated area HL, we write this result in terms of the incident fluxes S, and S, (power/area) as P(2w) = 2~,HL(eo/rro)IB,12~,S2
.
(13)
3. Numerical calculations We now illustrate the efficiency of this process with a detailed calculation. MNA [ 151 was chosen as the nonlinear material (n4 = 1.80, d12 = 3.4 X lo-11 m/V) and rutile as the prism material. Since the refractive index of MNA falls in the range of high index glasses [ 161, we assume that n2 can be chosen equal to the 1.80 7 0.01 which is required for this purpose. The metal (h = 150 A) is silver [ 161. We first numerically varied the incidence angle B until a minimum in the ATR spectrum was obtained for a given film thickness d (medium n2). The result-
FILM THICKNESS (MICRONS)
Fig. 2. The fraction of the incident beam energy absorbed at the minimum of the ATR spectrum as a function of film thickness.
1.5
I
I
J
2.0
2.5
3.0
FILM THICKNESS &lICRONS)
Fig. 3. The fraction of the total incident power (Pt + P-J converted into the harmonic signal (P(2w)) versus the thickness of medium n2.
ing surface polariton fields were then used to evaluate the harmonic signal. Shown in fig. 2 is the fraction of the incident energy (h = 1.06 pm) absorbed which peaks at d = 2.2 pm. Assuming H = L = 1 cm and an incident total power density of 1 MW/cm2, the fraction of the total input power (PI + P2) converted to the harmonic signal is shown in fig. 3. The oscillations are due to multireflection interference effects of the harmonic signal in medium n2 - (the harmonic radiation travels normal to the film interfaces, fig. 1). The maximum conversion efficiency of 1.5 X 1O-4 is significantly better than that reported in Ti:in-diffused LiNbO, waveguides [ 141. The conversion efficiency is very sensitive to metal film thickness. For h = 250 A, the maximum conversion efficiency of 1.7 X 10-S occurs at d = 1.36 pm. In summary, we have analyzed the nonlinear mixing of two oppositely propagating long range surface plasmons in an ATR geometry. Using MNA as the nonlinear material, conversion efficiencies of 1O-4 appear to be feasible. [Just prior to the submission of this paper we learned that a propagation distance enhancement of 20X has been observed for long range surface plasmons by Y. Kuwamura, M. Fukui and 0. Tada, in press, J. Phys. Sot. Japan.] This research was supported by NSF (ECS-8117483) and the Joint Services Optics Program of AR0 and AFOSR.
255
Volume 46, number 3,4
OPTICS COMMUNICATIONS
References [1] R. Normandin and G.I. Stegeman, Appl. Phys. Lett. 36 (1980) 253. [2] R. Normandin and G.I. Stegeman, Appl. Phys. Lett. 40 (1982) 759. [3] G.I. Stegeman and R. Normandin, Proc. SPI, Vol. 321, ed. D.G. Hall (SPIE, Washington, 1982) p. 55. [4] C.K. Chen, A.R.B. de Castro and Y.R. Shen, Optics Lett. 4 (1979) 393. [5] G.I. Stegeman, J.J. Burke and D.G. Hall, Appl. Phys. Lett. 41 (1982) 906. [6] M. Fukui, V. So and R. Normandin, Phys. Status Solidi (b) 91 (1979) K61; M. Fukui, V.C.-Y. So and G.I. Stegeman, Surf. Sci. 85 (1979) 125. [7] D. Sarid, Phys. Rev. Lett. 27 (1981) 1927. [8] R.T. Deck and D. Sarid, J. Opt. Soc. Am. 72 (1982) 1613.
256
1 July 1983
[9] C. Karaguleff and G.I. Stegeman, submitted to Applied Optics. [10[ F. Abeles and T. Lopez-Rios, Optics Comm. 11 (1974) 89. [ 11 ] W.H. Weber and S.L. McCarthy, Appl. Phys. Lett. 25 (1974) 396. [12] A.B. Buckman and C. Kuo, J. Opt. Soc. Am. 69 (1979) 343. [13] D. Sarid, R.T. Deck, A.E. Craig, R.K. Gickernell, R.S. Jameson and J.J. Fasano, Appl. Optics 21 (1982) 3993. [14] R. Normandin and G.I. Stegeman, Optics Lett. 4 (1979) 58. [15] B.F. Levine, C.G. Bethea, C.D. Thurmond, R.T. Lynch and J . g Bernstein, J. Appl. Phys. 50 (1979) 2523. [ 16] Schott Optical Glass Catalogue (Schott, Duryea, PA). [17] M. Dujardin and M. Theye, J. Phys. Chem. Sol. 32 (1971) 2033; T. Hollstein, U. Kreibig and F. Leis, Phys. Stat. Solidi 82b (1977) 545.