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Second harmonic generation with surface plasmons in alkali metals
Volume 13, number 3
OPTICS COMMUNICATIONS
March 1975
SECOND HARMONIC GENERATION WITH SURFACE PLASMONS IN ALKALI METALS H.J. SIMON, D.E. MITCHELL and J.G. WATSON Department of Physics and Astronomy, The University of Toledo, 7bledo, Ohio 43606. USA Received 16 December 1974
We calculate the reflected second harmonic light from alkali metal films with the simultaneous excitation of the surface plasmon mode. The harmonic generation from a sodium film at the ruby laser frequency increases by over two orders of magnitude at the angle for surface plasmon excitation. The harmonic enhancement is closely related to the surface plasmon density and exhibits a strong dependence on the angle of incidence, film thickness, and the linear optical constants of the metal film.
Tile second harmonic generation (SHG) of light from metals has been studied extensively in the past decade, both theoretically and experimentally. Although the reflected SHG of light was first observed in silver films, the complete nonlinear response for the conduction and valence band electrons in the noble metals has not been calculated. For the alkali metals it has been shown that a corrected version of the nonlinear optical response of a free electron gas predicts the observed SHG [1]. Recently the present authors have observed that SHG in a si!ver film is enhanced by over all order of magnitude by coupling to the surface plasmon mode [2]. In this paper we use the free electron model of the nonlinear polarization to calculate the enhancement of SHG in alkali metal fihns due to surface plasmon excitation. Collective oscillations in electron density at the surface of a metal may be described in terms of surface plasmon waves [3]. These waves may also be described as surface polariton modes which are the solutions to the Ewald Oseen extinction theorem with no incident field [4]. Such waves are evanescent since the m o m e n t u m along the surface, h2 s, of a nonradiative surface plasmon of wave vector k s is greater than that of an electromagnetic wave in vacuum of the same angular frequency co. The dispersion of surface plasmons on a semi-infinite dielectric bounded by vacuum is given by =
III
where e = e 1(co) + i e2(co ) is the complex dielectric function of the medium. The coupling of the electromagnetic wave to the surface plasmon is accomplished by the technique of attenuated total reflection [5] (ATR). The component of the photon wave vector parallel to the metal vacuum interface k 11 = (co/c)tl sin 0 is phase matched to the surface plasmon wave vector at the plasmon angle 0p defined by nsin0p=X/~ e e+ I "
(2)
where n is the index of refraction of the glass prism and 0 is the angle of incidence. It has been shown that the model of a free electron gas at a surface describes SHG in the alkali metals. The nonlinear polarization source from this model may be written as: cop2 paNLS(2co) = rE(co) X H(co) + ~3~ {5 lE(co)" V] E(co) + n 6 1 E(co) IV .n 0 E¢co)l }294
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March 1975
with a = ie3no/4m2c6o 3 ,
fJ = e/8nm6o 2 ,
6op2 = 4nnoe2/m
and
Here m and e are the mass and charge of an electron; n o is the number density of free electrons;/~t6o) and H(6o) are the fundamental electric and magnetic field amplitudes. This expression for the nonlinear polarization has been extensively discussed; briefly, the first term is called the "bulk term" since its contribution to SHG comes from within the penetration depth of the electromagnetic field in the metal; the other terms are called "surface terms" since they are proportional to the gradient of the electric field which is large only at the surface due to the discontinuity of the normal component. We note that the contribution from the induced collective oscillation of the electrons at the surface, i.e. the surface plasmon, is implicitly contained in the last term [6]. The reflected harmonic intensity due to the coupling to the surface plasmon mode is calculated using the glassmetal-vacuum geometry shown in the inset in fig. 1. For a p-polarized (polarized in the plane of incidence) incident wave, the contribution of the bulk term may be ignored compared to the surface terms. Since these latter terms are of comparable magnitude we will calculate only the contribution from the electric quadrupole component in the plane of incidence and parallel to the metal surface. This choice will not significantly influence the angular dependence of the surface plasmon resonance and will facilitate comparison of this calculation with those of previous authors. The direction of the nonlinear polarization at the glass-metal interface is reversed with respect to that at the metal-vacuum interface because of the reversal in the electric field gradient. The two surfaces boundary value problem may now be solved for the reflected SHG wave in the glass prism by assuming an evanescent standing wave in the metal film and an evanescent transmitted wave in the vacuum. The details of the solution of Maxwell's equations for the SHG at a vacuum-metal interface have been given previously and the application to the ATR geometry is straightforward [7]. A similar analysis must be done for the fundamental field to determine its amplitude in the metal at the two surfaces [8]. The reflected SHG intensity I(26o) outside the prism may be expressed in terms of the incident fundamental intensity I(6o) as I(26o) = 12(6o) R(26O),
(4)
with the conversion factor for the reflected SHG given by R(26O) = (2rre2/m2 6o2e 3) Fp ,
(5)
where the dimensionless function Fp is further factored into Fp = F(6op, 0)IFNLPI2.
,~6)
Here F(6op, 0) is given by F(6op, 0 ) = (6o4/6o4) sin20 cos40
4[e(6o)-- n121n~G1G2
[ (26o cos0
[2
i56o-3cos +.1c1 2j T,
(7)
with G1 = [e(w ) _ n2sin20] ,/2 ,
G 2 = [e(26o) - n 2 sin021'/2 ,
where e(w) and e(26o) are the complex dielectric constants of the metal at the fundamental and harmonic frequencies respectively, nl, 2 is the index of refraction of the glass prism at the two frequencies, T includes the appropriate transmission Fresnel factors for the front air-prism interface, and 0 is the angle of incidence at the metal-glass interface. The function F(6op, 0) in eq. (7) differs slightly from previous results because of the ATR geometry. The contribution from the surface plasmon mode is contained in the nonlinear plasmon Fresnel factor F NLP which is given by
295
Volume 13, number 3
OPTICS COMMUNICATIONS
March 1975
F N L P = ( I +r12r23e led) 2(I +R12R23e Kd) 1 × ([(I
r23 e 2kd)( 1
R23e
Kd)]
le(co)
11 [e(do)
n~]
1(1
r 3") J e
kale td/2(1 +R23)[e(2cv)l
Lower case letters refer to linear Fresnel reflection amplitude coefficients [8] evaluated at the fundamental frequency, while upper case letters refer to the same quantities evaluated at the harmonic frequency; subscripts 12 and 23 refer to the glass-metal and metal-vacuum interfaces respectively; d is the thickness of the fihn: k and K are the absorption coefficients at nonnormal incidence for the fundamental and harmonic frequencies. When the fundamental surface plasmon mode is excited at the metal to vacuum interface, the internal evanescent fundamental electric field is resonantly enhanced and produces a large nonlinear polarization at this surface which in turn radiates the SHG wave. Since the surface term in the harmonic polarization of eq. (3) is proportional to the surface plasmon density, the enhanced SHG provides a direct observation of the excitation of this collective electron density wave. For an ideal free electron plasma with no absorption, i.e. e2(co) = 0, the Fresnel reflection amplitude coefficient r23 diverges at the plasmon angle [9] and both the surface plasmon density and the SHG increase exponentially with the thickness of the metal fihn. Note that in the limit of no absorption the linear reflectivity remains tmity and no plasmon resonance would be observed. For small absorption in the plasma r23 ~ 2i e I (co)/e2(co) at the plasmon angle and the nonlinear plasmon Fresnel factor F NLP may be more simply
Na
iO 4
/
/
h o>I--"
iO 3
Z
W FZ fJ
/ I
Z 0 :E n., "1Z 0 1,1 ¢/1
i0 =
y~
w
Rb
\
\\
cs
I_J LU (:g
l0
i
m i
i
42 °
I
I
47" ANGLE
OF
INCIDENCE
i
i
52"
e
Fig. 1. Reflected SHG from alkali metal films with surface plasmon excitation versus the angle of incidence. 296
l
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OPTICS COMMUNICATIONS
March 1975
rewritten in the vicinity of the plasmon angle as F NLP = (constant) X (1 - r23)/(1 + r 1 2 r 2 3 e - k d ) 2 .
(9)
The observed angular dependence of SHG with surface plasmon excitation in a silver film has been accurately described by the use of eq. (9). The SHG becomes a maximum at a finite thickness due to the damping of the surface plasmon resonance. Although a resonant enhancement due to the creation of a second harmonic plasmon is predicted by eq. (8), the magnitude of the SHG would be significantly decreased due to the normal attenuation of the fundamental field. Our calculations are carried out for ruby laser (k = 6943 A) radiation incident on alkali metal films at the hypothenuse face of a right crown glass prism. In fig. 1 we plot on a semi-log scale the dimensionless function Fp, calculated from eqs. (6)-(8), versus the angle of incidence in the region of the plasmon resonance. The thicknesses of the metal films are chosen to produce the maximum SHG for each metal. The magnitude of the SHG enhancement due to the surface plasmon excitation can be estimated by noting that for a front vacuum to metal reflection the function F(cop, 0) is always less than 100, Sodium exhibits the strongest surface plasmon resonance among the alkali metals; this comparison is to be expected since sodium is closest to the ideal surface plasmon behavior with the largest value of the ratio e 1 (w)/e2(6o). Although potassium also exhibits a sharp resonant behavior, the magnitude of the enhancement is reduced because e(2~) ~ 0. For the other alkali metals the surface plasmon resonance is heavily damped at this wavelength. In fig. 2 we plot Fp as a function of film thickness for sodium metal. In the region below 200 A. the SHG increases exponentially with thickness as previously discussed and reaches a maximum near 600 A due to the competing surface plasmon damping. For thicker f'dms the SHG decreases to a limiting value that is a factor of three greater than the SHG at a front surface air to metal reflection from a thick sodium film. Enhanced SHG in total internal reflection from noncentrosymmetric media has been previously studied [ 10], In the limit of very thin films of vanishing thickness, F NLP as calculated from eq. (8) does not approach zero; however, the effective nonlinear surface polarization implicitly defined by the SHG boundary conditions vanishes in this limit [6]. The results of these calculations are summarized in table 1. Linear optical data for the alkali metals is taken
--
I
I
t
I
i
I
]
i
I
I
I
>z LU t-
20,000
Z
o_ z
z
IOOO0 b.l
..1
I
i
0
I
I
I
500
FILM
THICKNESS
I
I I000
d
(~)
Fig. 2. Reflected SHG from a sodium film at the angle for surface plasmon excitation versus the film thickness.
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March 1975
Table 1 Parameters for SHG with surface plasmons from alkali metals at ruby-laser wavelength
COp(eV)
el(CO) e2 (co) el(2CO) ez (2CO) RWCB X 101a (W/cm2)-I Rplasmo n x 1018 (W/cmZ)-1 0plasmo n 0max dmax (A)
Na
K
Rb
5.65 -8.71 0.288 -1.49 0.142 6.28 2150 44.4 ° 44.4 ° 610
3.8 3.85 0.188 0.175 0.054 5.85 38 49.8 ° 49° 700
3.4 -2.90 0.371 0.108 0.127 5.15 10 54° 47.8 ° 660
Cs 3.0 2.02 0.768 0.362 0.315 1.92 6.5 63 ° 45.5 °
440
from ref. [ 1 1] and the value of COp is the experimentally determined value from ~5(COp)~ 0. The value of the maximum SHG conversion factor given by the theory of Wang for front surface air to metal reflection is given by RWCB. The peak value of the SHG conversion factor with coupling to the surface plasmon mode is given by Rplasmon. The angle of incidence for creation of the surface plasmon mode defined by eq. (2) is given by 0plasmon while the angle at which Rplasmon is a maximum is given by 0max. The value of fihn thickness at which this maximum occurs is given by dmaxIn conclusion, we calculate that excitation of the surface plasmon mode in alkali metal films produces a sharp increase in SHG which can be greater than two orders of magnitude for a sodium film. The surface term contribution to the optical nonlinearity of a free electron plasma is directly proportional to the surface plasmon electron density. SHG with surface plasmon excitation provides a new and sensitive technique for directly studying surface plasmon phenomena. To observe the full two order of magnitude enhancement in the sodium film it is importanl that the laser source be well collimated and that the fihn be evaporated under conditions of high vacuum since increased divergence of the laser beam and absorptive loss in the metal film will reduce the surface plzsmon resonance [2]. Observation of this effect in the alkali metals is important since it will further demonstrate the applicability of the Drude model for the description of the nonlinear optical properties of these metals.
References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] [ 10] [11]
298
C.S. Wang, J.M. Chen and J.R. Bower, Opt. Commun. 8 (1973) 275. This article contains a complete list of earlier references. H.J. Simon, D.G. Mitchell and J.G. Watson, Phys. Rev. Lett. (to be published). R.H. Ritchie, Surface Sci. 34 (1973) 1, and references cited therein. G.S. Agarwal, Phys. Rev. B8 (1973) 4768. E. Kretschmann and H. Raether, Z. Naturforsch. 23a (1968) 2135. R.H. Ritchie and R.E. Wilems, Phys. Rev. 178 (1969) 372. J.G. Watson, M.S. Thesis, The University of Toledo (unpublished). M. Born and E. Wolf, Principles of Optics, 3rd Rev. Ed. (The MacMillanCompany, New York, 1965) p. 62. M. Cardona, Am. J. Phys. 39 (1971) 1277. N. Bloembergen, H.J. Simon and C.H. Lee, Phys. Rev. 181 (1969) 1261. N.Y. Smith, Phys. Rev. 183 (1969) 634; B2 (1970) 2840.
OPTICS COMMUNICATIONS
March 1975
SECOND HARMONIC GENERATION WITH SURFACE PLASMONS IN ALKALI METALS H.J. SIMON, D.E. MITCHELL and J.G. WATSON Department of Physics and Astronomy, The University of Toledo, 7bledo, Ohio 43606. USA Received 16 December 1974
We calculate the reflected second harmonic light from alkali metal films with the simultaneous excitation of the surface plasmon mode. The harmonic generation from a sodium film at the ruby laser frequency increases by over two orders of magnitude at the angle for surface plasmon excitation. The harmonic enhancement is closely related to the surface plasmon density and exhibits a strong dependence on the angle of incidence, film thickness, and the linear optical constants of the metal film.
Tile second harmonic generation (SHG) of light from metals has been studied extensively in the past decade, both theoretically and experimentally. Although the reflected SHG of light was first observed in silver films, the complete nonlinear response for the conduction and valence band electrons in the noble metals has not been calculated. For the alkali metals it has been shown that a corrected version of the nonlinear optical response of a free electron gas predicts the observed SHG [1]. Recently the present authors have observed that SHG in a si!ver film is enhanced by over all order of magnitude by coupling to the surface plasmon mode [2]. In this paper we use the free electron model of the nonlinear polarization to calculate the enhancement of SHG in alkali metal fihns due to surface plasmon excitation. Collective oscillations in electron density at the surface of a metal may be described in terms of surface plasmon waves [3]. These waves may also be described as surface polariton modes which are the solutions to the Ewald Oseen extinction theorem with no incident field [4]. Such waves are evanescent since the m o m e n t u m along the surface, h2 s, of a nonradiative surface plasmon of wave vector k s is greater than that of an electromagnetic wave in vacuum of the same angular frequency co. The dispersion of surface plasmons on a semi-infinite dielectric bounded by vacuum is given by =
III
where e = e 1(co) + i e2(co ) is the complex dielectric function of the medium. The coupling of the electromagnetic wave to the surface plasmon is accomplished by the technique of attenuated total reflection [5] (ATR). The component of the photon wave vector parallel to the metal vacuum interface k 11 = (co/c)tl sin 0 is phase matched to the surface plasmon wave vector at the plasmon angle 0p defined by nsin0p=X/~ e e+ I "
(2)
where n is the index of refraction of the glass prism and 0 is the angle of incidence. It has been shown that the model of a free electron gas at a surface describes SHG in the alkali metals. The nonlinear polarization source from this model may be written as: cop2 paNLS(2co) = rE(co) X H(co) + ~3~ {5 lE(co)" V] E(co) + n 6 1 E(co) IV .n 0 E¢co)l }294
131
Volume 13, number 3
OPTICS COMMUNICATIONS
March 1975
with a = ie3no/4m2c6o 3 ,
fJ = e/8nm6o 2 ,
6op2 = 4nnoe2/m
and
Here m and e are the mass and charge of an electron; n o is the number density of free electrons;/~t6o) and H(6o) are the fundamental electric and magnetic field amplitudes. This expression for the nonlinear polarization has been extensively discussed; briefly, the first term is called the "bulk term" since its contribution to SHG comes from within the penetration depth of the electromagnetic field in the metal; the other terms are called "surface terms" since they are proportional to the gradient of the electric field which is large only at the surface due to the discontinuity of the normal component. We note that the contribution from the induced collective oscillation of the electrons at the surface, i.e. the surface plasmon, is implicitly contained in the last term [6]. The reflected harmonic intensity due to the coupling to the surface plasmon mode is calculated using the glassmetal-vacuum geometry shown in the inset in fig. 1. For a p-polarized (polarized in the plane of incidence) incident wave, the contribution of the bulk term may be ignored compared to the surface terms. Since these latter terms are of comparable magnitude we will calculate only the contribution from the electric quadrupole component in the plane of incidence and parallel to the metal surface. This choice will not significantly influence the angular dependence of the surface plasmon resonance and will facilitate comparison of this calculation with those of previous authors. The direction of the nonlinear polarization at the glass-metal interface is reversed with respect to that at the metal-vacuum interface because of the reversal in the electric field gradient. The two surfaces boundary value problem may now be solved for the reflected SHG wave in the glass prism by assuming an evanescent standing wave in the metal film and an evanescent transmitted wave in the vacuum. The details of the solution of Maxwell's equations for the SHG at a vacuum-metal interface have been given previously and the application to the ATR geometry is straightforward [7]. A similar analysis must be done for the fundamental field to determine its amplitude in the metal at the two surfaces [8]. The reflected SHG intensity I(26o) outside the prism may be expressed in terms of the incident fundamental intensity I(6o) as I(26o) = 12(6o) R(26O),
(4)
with the conversion factor for the reflected SHG given by R(26O) = (2rre2/m2 6o2e 3) Fp ,
(5)
where the dimensionless function Fp is further factored into Fp = F(6op, 0)IFNLPI2.
,~6)
Here F(6op, 0) is given by F(6op, 0 ) = (6o4/6o4) sin20 cos40
4[e(6o)-- n121n~G1G2
[ (26o cos0
[2
i56o-3cos +.1c1 2j T,
(7)
with G1 = [e(w ) _ n2sin20] ,/2 ,
G 2 = [e(26o) - n 2 sin021'/2 ,
where e(w) and e(26o) are the complex dielectric constants of the metal at the fundamental and harmonic frequencies respectively, nl, 2 is the index of refraction of the glass prism at the two frequencies, T includes the appropriate transmission Fresnel factors for the front air-prism interface, and 0 is the angle of incidence at the metal-glass interface. The function F(6op, 0) in eq. (7) differs slightly from previous results because of the ATR geometry. The contribution from the surface plasmon mode is contained in the nonlinear plasmon Fresnel factor F NLP which is given by
295
Volume 13, number 3
OPTICS COMMUNICATIONS
March 1975
F N L P = ( I +r12r23e led) 2(I +R12R23e Kd) 1 × ([(I
r23 e 2kd)( 1
R23e
Kd)]
le(co)
11 [e(do)
n~]
1(1
r 3") J e
kale td/2(1 +R23)[e(2cv)l
Lower case letters refer to linear Fresnel reflection amplitude coefficients [8] evaluated at the fundamental frequency, while upper case letters refer to the same quantities evaluated at the harmonic frequency; subscripts 12 and 23 refer to the glass-metal and metal-vacuum interfaces respectively; d is the thickness of the fihn: k and K are the absorption coefficients at nonnormal incidence for the fundamental and harmonic frequencies. When the fundamental surface plasmon mode is excited at the metal to vacuum interface, the internal evanescent fundamental electric field is resonantly enhanced and produces a large nonlinear polarization at this surface which in turn radiates the SHG wave. Since the surface term in the harmonic polarization of eq. (3) is proportional to the surface plasmon density, the enhanced SHG provides a direct observation of the excitation of this collective electron density wave. For an ideal free electron plasma with no absorption, i.e. e2(co) = 0, the Fresnel reflection amplitude coefficient r23 diverges at the plasmon angle [9] and both the surface plasmon density and the SHG increase exponentially with the thickness of the metal fihn. Note that in the limit of no absorption the linear reflectivity remains tmity and no plasmon resonance would be observed. For small absorption in the plasma r23 ~ 2i e I (co)/e2(co) at the plasmon angle and the nonlinear plasmon Fresnel factor F NLP may be more simply
Na
iO 4
/
/
h o>I--"
iO 3
Z
W FZ fJ
/ I
Z 0 :E n., "1Z 0 1,1 ¢/1
i0 =
y~
w
Rb
\
\\
cs
I_J LU (:g
l0
i
m i
i
42 °
I
I
47" ANGLE
OF
INCIDENCE
i
i
52"
e
Fig. 1. Reflected SHG from alkali metal films with surface plasmon excitation versus the angle of incidence. 296
l
Volume 13, number 3
OPTICS COMMUNICATIONS
March 1975
rewritten in the vicinity of the plasmon angle as F NLP = (constant) X (1 - r23)/(1 + r 1 2 r 2 3 e - k d ) 2 .
(9)
The observed angular dependence of SHG with surface plasmon excitation in a silver film has been accurately described by the use of eq. (9). The SHG becomes a maximum at a finite thickness due to the damping of the surface plasmon resonance. Although a resonant enhancement due to the creation of a second harmonic plasmon is predicted by eq. (8), the magnitude of the SHG would be significantly decreased due to the normal attenuation of the fundamental field. Our calculations are carried out for ruby laser (k = 6943 A) radiation incident on alkali metal films at the hypothenuse face of a right crown glass prism. In fig. 1 we plot on a semi-log scale the dimensionless function Fp, calculated from eqs. (6)-(8), versus the angle of incidence in the region of the plasmon resonance. The thicknesses of the metal films are chosen to produce the maximum SHG for each metal. The magnitude of the SHG enhancement due to the surface plasmon excitation can be estimated by noting that for a front vacuum to metal reflection the function F(cop, 0) is always less than 100, Sodium exhibits the strongest surface plasmon resonance among the alkali metals; this comparison is to be expected since sodium is closest to the ideal surface plasmon behavior with the largest value of the ratio e 1 (w)/e2(6o). Although potassium also exhibits a sharp resonant behavior, the magnitude of the enhancement is reduced because e(2~) ~ 0. For the other alkali metals the surface plasmon resonance is heavily damped at this wavelength. In fig. 2 we plot Fp as a function of film thickness for sodium metal. In the region below 200 A. the SHG increases exponentially with thickness as previously discussed and reaches a maximum near 600 A due to the competing surface plasmon damping. For thicker f'dms the SHG decreases to a limiting value that is a factor of three greater than the SHG at a front surface air to metal reflection from a thick sodium film. Enhanced SHG in total internal reflection from noncentrosymmetric media has been previously studied [ 10], In the limit of very thin films of vanishing thickness, F NLP as calculated from eq. (8) does not approach zero; however, the effective nonlinear surface polarization implicitly defined by the SHG boundary conditions vanishes in this limit [6]. The results of these calculations are summarized in table 1. Linear optical data for the alkali metals is taken
--
I
I
t
I
i
I
]
i
I
I
I
>z LU t-
20,000
Z
o_ z
z
IOOO0 b.l
..1
I
i
0
I
I
I
500
FILM
THICKNESS
I
I I000
d
(~)
Fig. 2. Reflected SHG from a sodium film at the angle for surface plasmon excitation versus the film thickness.
297
Volume 13, number 3
OPTICS COMMUNICATIONS
March 1975
Table 1 Parameters for SHG with surface plasmons from alkali metals at ruby-laser wavelength
COp(eV)
el(CO) e2 (co) el(2CO) ez (2CO) RWCB X 101a (W/cm2)-I Rplasmo n x 1018 (W/cmZ)-1 0plasmo n 0max dmax (A)
Na
K
Rb
5.65 -8.71 0.288 -1.49 0.142 6.28 2150 44.4 ° 44.4 ° 610
3.8 3.85 0.188 0.175 0.054 5.85 38 49.8 ° 49° 700
3.4 -2.90 0.371 0.108 0.127 5.15 10 54° 47.8 ° 660
Cs 3.0 2.02 0.768 0.362 0.315 1.92 6.5 63 ° 45.5 °
440
from ref. [ 1 1] and the value of COp is the experimentally determined value from ~5(COp)~ 0. The value of the maximum SHG conversion factor given by the theory of Wang for front surface air to metal reflection is given by RWCB. The peak value of the SHG conversion factor with coupling to the surface plasmon mode is given by Rplasmon. The angle of incidence for creation of the surface plasmon mode defined by eq. (2) is given by 0plasmon while the angle at which Rplasmon is a maximum is given by 0max. The value of fihn thickness at which this maximum occurs is given by dmaxIn conclusion, we calculate that excitation of the surface plasmon mode in alkali metal films produces a sharp increase in SHG which can be greater than two orders of magnitude for a sodium film. The surface term contribution to the optical nonlinearity of a free electron plasma is directly proportional to the surface plasmon electron density. SHG with surface plasmon excitation provides a new and sensitive technique for directly studying surface plasmon phenomena. To observe the full two order of magnitude enhancement in the sodium film it is importanl that the laser source be well collimated and that the fihn be evaporated under conditions of high vacuum since increased divergence of the laser beam and absorptive loss in the metal film will reduce the surface plzsmon resonance [2]. Observation of this effect in the alkali metals is important since it will further demonstrate the applicability of the Drude model for the description of the nonlinear optical properties of these metals.
References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] [ 10] [11]
298
C.S. Wang, J.M. Chen and J.R. Bower, Opt. Commun. 8 (1973) 275. This article contains a complete list of earlier references. H.J. Simon, D.G. Mitchell and J.G. Watson, Phys. Rev. Lett. (to be published). R.H. Ritchie, Surface Sci. 34 (1973) 1, and references cited therein. G.S. Agarwal, Phys. Rev. B8 (1973) 4768. E. Kretschmann and H. Raether, Z. Naturforsch. 23a (1968) 2135. R.H. Ritchie and R.E. Wilems, Phys. Rev. 178 (1969) 372. J.G. Watson, M.S. Thesis, The University of Toledo (unpublished). M. Born and E. Wolf, Principles of Optics, 3rd Rev. Ed. (The MacMillanCompany, New York, 1965) p. 62. M. Cardona, Am. J. Phys. 39 (1971) 1277. N. Bloembergen, H.J. Simon and C.H. Lee, Phys. Rev. 181 (1969) 1261. N.Y. Smith, Phys. Rev. 183 (1969) 634; B2 (1970) 2840.