- Email: [email protected]
Optical dipole radiation inside nanometric planar two-surface systems
15 December 1998
Optics Communications 158 Ž1998. 11–17
Optical dipole radiation inside nanometric planar two-surface systems Mufei Xiao a
a,)
, Xin Chen
b
Centro de Ciencias de la Materia Condensada, UNAM, Apartado Postal 2681, Ensenada CP 22800, Baja California, Mexico b Department of Physics, UniÕersity of Manchester, Institute Sciences and Technology, Manchester M60 1QD, UK Received 28 May 1998; revised 21 September 1998; accepted 23 September 1998
Abstract In classic optics, we propose a calculation for the electromagnetic radiation stemming from a point dipole located between two planar interfaces with arbitrary separations. We calculate the field distribution inside the planar two-surface system, particularly near the dipole source. First, the modified dipole strength due to the cavity environments is self-consistently calculated as a function of the dipole position, and second, the field distribution inside the planar cavity is obtained with an appropriate consideration of the multireflection contributions. The multireflections and the s- and p-polarized reflection coefficients are treated in the two-dimensional Fourier plane wave representation. The treatment enables us to accurately calculate all the field components, the evanescent fields as well as the propagating waves, for arbitrary cavity sizes. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Dipole; Two-surface system; Field distribution
1. Introduction Electromagnetic dipole radiation in the presence of microscopic environments has been the subject of intense study for a long time. The studied microscopic environments include metallic and dielectric planar surfaces w1–6x, and multilayer structures w7–9x. In general, the studies are concentrated on two different problems. One is to study how the dipole source is modified by the microscopic environments, for example, the modified fluorescent single molecule radiation w10–12x. The other is to study the radiation patterns in the presence of the microscopic environments, for example, the radiation of single w1,13x and many near-surface emitters w14,15x 1. The problems can be
)
Corresponding author. Correspondence address: CCMCUNAM, P.O. Box 439036, San Ysidro, CA 92143, USA. E-mail: [email protected] 1 In Ref. w13x, there are two typos in the article: there should be a Žy. sign in front of I zprop in Eq. Ž9. and Izevan in Eq. Ž12.. x x
generally described in the framework of the classic theories, such as the electromagnetic antenna theory. However, quantum descriptions are often needed when the dipole sources are atom-like systems that carry intrinsic modes due to the interstate transitions Žsee, e.g., Ref. w16x for some related discussions.. Our interest in the problem has been concentrated on the description of the optical radiation of microscopic emitters on surfaces. Particularly we are interested in an accurate description of the radiation near the source region: the near field zone. In our work, we describe the dipole sources as homogeneous small particles, which means the dipole sources are not necessarily associated with specific intrinsic modes. In our previous work, we proposed an exact formalism for the dipole field radiation in free space w17x and in nanometric dipole–surface systems w13x. Incorporating the exact formalism for single dipole radiation into the previous established microscopic theory w14,15,18,19x, we can now calculate accurately the field distribution for many scatterers.
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 5 2 1 - 5
12
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
In the present work, we consider the cases where the scatterers are located between two surfaces. It is of current interest to study the local field inside ultrathin films. Since ultrathin layer systems are often island-like structures, it is interesting to calculate the dipole radiation inside the multilayer structures. In order to use the microscopic theory to describe the self-consistent local field inside the layered structures and the modified radiation of the emitters, one needs an accurate formalism to study the radiation of both the evanescent and propagating modes. In the present paper we provide such a formalism. In the theory, the evanescent and propagating waves can be exactly computed everywhere inside the two planar surfaces. This is made possible by describing the multireflections as well as the direct propagation in the two-dimensional spectrum representation. The formalism is particularly adequate for calculating the dipole radiation near the dipole source within several wavelengths of the stimulating light source, though, apart from computational difficulties, there is no restriction to use the formalism for radiation fields beyond this region, none there is any restriction for the materials of the inside and outside regions as far as the materials can be described with their dielectric functions. The theory is however not convenient for calculating the field far away from the dipole source because of computational limit. Hence, to study the far field, it would be more convenient to adopt traditional methods in the literature that study the resonance couplings between the cavity propagating modes and the source intrinsic modes, i.e., the so-called cavity optics, which is not included in the present context. Readers are referred to Refs. w8,9x for studies on modification of spontaneous emission in multilayer structures. The paper is organized as follows. In Section 1 we present the self-consistent formalism for the modified local
field at the dipole site when the dipole is located between two planar surfaces, and the formalism for the dipole radiation inside the two-surface system. In Section 2, we present numerical results from the proposed calculations and finally we discuss the results.
2. The theory In Fig. 1 we describe the two-surface system schematically. The space is divided into three regions of different materials with their associated dielectric functions e 1Ž v ., e Ž v ., and e 2 Ž v .. A point dipole is located inside the two interfaces at a distance d from surface I. The separation of the two interfaces is a nanometric Žor arbitrary. distance D. We assume that the point dipole can be represented as a mesoscopic sphere of radius a and dielectric function edŽ v .. The dipolar polarizability may be calculated from the sphere radius a and the dielectric function edŽ v . Žsee, e.g., Ref. w13x.. Once the sphere is stimulated with an external light source, a dipole moment is established. The dipole strength is modified due to the presence of the two interfaces. The modified local field at the dipole site has to be calculated self-consistently, and the final dipole strength is obtained as a function of its position ™ rd s Ž0,0,d.. The established dipole radiates electromagnetically to the space. The field radiation inside the region between the two surfaces can be finally calculated by collecting the contributions from both the direct propagation and the multireflections from the dipole source to the observation point ™ ro s Ž x,0, z .. In the following we assume e Ž v . s 1, just to simplify the notation.
Fig. 1. A scheme for a point dipole in a planar two-surface system. The dipole radiation travels directly and indirectly Žmultireflections. towards the observation point. Only one reflection is plotted, which represents all possible multireflections.
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
Let us start with the general expression for the radiation of a dipole inside two surfaces, as ™
l
l
™ ™ ™ E Ž™ r o . s ym 0 v 2 D Ž ™ rd ,r o . q I Ž rd ,ro . ™
l = a rd . Ž v . E local Ž™
Ž1.
where m 0 is the permeability in vacuum and v angular l l™ ™ ™ . Ž . frequency. Tensors DŽ™ rd ,r o and I rd ,ro are the so-called direct and reflection Žindirect. propagators, which describe the field propagation from the source point to the observation point directly and indirectly Žvia the surface multireflections., respectively. In Eq. Ž1., the dipole is represented by the term below ™
™
l pŽ v . s a rd . , Ž v . E local Ž™
Ž2.
l
where a Ž v . is the dipolar polarizability, which can be obtained from the parameters a and edŽ v ., as l a Ž v . s 4pe 0 a3
ed Ž v . y 1 ed Ž v . q 2
,
Ž3.
tions, namely the evanescent and propagating waves by dividing the integration limit for the whole spectra 0 ; q ; v into the two ranges: the propagating range 0 ; q ; vrc Žwhich corresponds to vrc ; q z ; 0. and the evanescent range vrc ; q ; ` Žwhich corresponds to i0 ; q z ; i`.. For the direct field propagator, we shall use the analytical results already obtained in our previous work in Ref. w17x. The direct propagator for the whole field is l
D Ž™ r. s
1
ž
4p
q
ž
l™ ™
l E local Ž ™ rd . s U q m 0 v 2 a Ž v . I Ž rd ,rd . l
y1 ™ E0 ™ rd
Ž . , Ž4.
™ ™
where U is a unit tensor, EoŽ rd . is the incident field at the l dipole site, and the tensor I Ž d,d . represents all multireflections of the dipole radiation to the dipole itself. In the following we calculate the field propagators. The key in our theory is to represent the field propagation and reflections in the angular spectrum Ž ™ q, which is parallel to the surfaces. representation. In this case, for a surface reflection, it is readily possible to exactly match the boundary conditions to obtain the s- and p-polarized reflection coefficients r s Ž q . and r p Ž q .. The results are as follows w13x rsŽ q. s
qz y k z qz q k z
, and r p Ž q . s
qz es y k z qz es q k z
,
Ž5.
where e s represents the dielectric function of the outside region of the interfaces, and q z2 s Ž vrc . 2 y q 2 , and k z2 s Ž vrc . 2e s y q 2 Ž c is the light speed in vacuum., with q z and k z representing the vertical component of the wave vector in air Žas we have assumed the dipole being located in the region e s 1. and in solids Žthe two regions outside the interfaces., respectively. The essential advantages to work in the angular spectrum representation are as follows. Firstly, with respect to the surfaces, the reflections become simply the cases of plane waves in vertical incidence, which enables us to avoid including the incidence angles. Secondly, the reflection coefficients are now exactly matching the wave number q, which enables us to accurately calculate the field components for all the spectra 0 ; q ; `. Thirdly, with an inverse Fourier transformation, one is able to exactly separate the two important contribu-
vr
r
vr
2
q
v2r3
3c 2
3ic q
r
c2
ic y
1
l
Devan Ž™ r. s
1 4p
™
l
1 y
y 2
v2r3
/
/
l
U
™™
nn e i v r r c ,
Ž6.
where r s <™ ro y™ rd < and ™ n s™ rrr. The above direct propagator can be divided into its evanescent and propagating parts, where the evanescent field is
and the local field E localŽ™ rd . at the dipole site has to be calculated self-consistently as w20x ™
13
ž
ž
c2
1 y
q 2r
3c 2
1
q y
y 2r
v2r3
v2r3
/
/
l
U
™™
nn ,
Ž7.
and the propagating field propagator can be obtained by l l l Dprop s D y Devan . In the following, we shall calculate the reflection field propagators for the multireflections. To start, let us write down in the angular spectrum representation the reflection l ™ . propagator I Ž™ rd ,r o for only one reflection at the interface I Žsee Fig. 1. w13x, l
I Ž Z,q . s
exp w iq z Z x 2 iq z
r 1s Ž q . ™ es ™ es q r1p Ž q . ™ er ™ ei ,
Ž8.
where the unit vector ™ es s Ž0,1,0. is the s-field direction Žin y-axis in Fig. 1., whereas ™ e r s Ž crv .Žyq z ,0,y q . and ™ e i s Ž crv .Ž q z ,0,y q . are unit vectors for reflection and incidence p-polarized light, respectively. The distance Z in Eq. Ž8. is the distance the above plane wave has travelled from the dipole to the observation point, in the case of one reflection at interface I, Z s d q z. Then we consider the reflection at interface II. In this case, the expression in Eq. Ž8. needs to be modified as follows. The reflection coefficients are now r 2s and r 2p, the value Z becomes Z s 2 D y Ž d q z ., and the term ™ er ™ e i becomes ™ ei™ e r . Finally we consider the case of multireflections, where the light goes from the dipole to the observation point via more than one reflection at the interfaces. The change of the value Z can be easily obtained by summing up the whole trip, for example, Z s 2 D q d q z is for the trip of touching twice at interface I and once at interface II. Using the same example, the reflection coefficients become r s s r1s r 2s r 1s and r p s r 1p r 2p r 1p. In all cases, the term ™ es ™ es does not ™™ change, but the term e r e i may have four possibilities such as Ž1. ™ er ™ e i ; Ž2. ™ ei™ e r ; Ž3. ™ er ™ e r ; and Ž4. ™ ei™ ei.
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
14
The reflection propagator in spatial domain is the Fourier inverse transform of the above propagators for 0 ; q ; `. We stress that in the range 0 ; q ; vrc Žwhich corresponds to vrc ; q z ; 0. the field is homogeneous Žpropagating., whereas in the range vrc ; q ; ` Žwhich corresponds to i0 ; q z ; i`. the field is inhomogeneous Ževanescent.. The inverse Fourier transform can be written as w13x, 1
l
I Ž x , y,Z . s
`l
y1 l
Ž 2p .
2
l
™™ I Ž Z,q . S exp Ž iq r . d 2 q,
H0 S
yc 2 I yevan y s
`
8pv 2
H0 exp wyq Z x z
2
v
yc 2 4pv
2
`
H0 exp wyq Z x r z
yc 2
l
Ss
1 q
yq y
qx
0 .
0
0
q
Ž 10 .
l
l
I Ž x , y,Z . s Iprop Ž x , y,Z . q Ievan Ž x , y,Z . ,
Ž 11.
where always I x yŽ y x . s I y zŽ z y . s 0, and I xprop x s
c2
vrc
i8pv 2
H0
v
2
c
I yprop y s
c2
H0
v
2
c
I zprop z s
c2
exp w iq z Z x
r s y q z2 r p J0 Ž k .
/
c
½ž
H0
exp w iq z Z x r p
yc 2
(ž
4pv 2
c
/
2
v
ž / c
y q z2
exp w iq z Z x r p J1 Ž k . q z
y q z2 d q z ,
Ž 15.
and, yc 2 I xevan x s
`
8pv 2
H0 exp wyq Z x
v =
ž / c
z
2
½ž
2
v c
5
q q z2 J0 Ž k . d q z
where
(
4pv 2
z
p
J1 Ž k . q z
2
v c
`
H0 exp wyq Z x r
/
q q z2 d q z ,
Ž 19.
JnŽ k . are nth order Bessel functions, and prop
(
2
, k s r Ž vr2 . y q z2 , while for I evan , k s 2
r Ž vrc . q q z2 . The above expressions are for the case ™ er ™ e i . For the case ™ ei™ e r , since ™ ei™ e r s Ž™ er ™ e i . T, the elements I x z and I z x reverse the sign wchanging from Žq or y. to Žy or q.x. For the cases ™ er ™ e r and ™ ei™ e i , the elements I x z s I z x and the elements I x xŽ y y . have a reverse sign for the terms containing r p. By using the above field propagators, one is now able to compute the field distribution inside the two-layer systems. As mentioned in the introduction, the formulae are particularly convenient for the near field zones, because one can reduce the number of multireflections when higher numbers do not bring out significant contributions.
3. Some numerical results and remarks
2
v
=
vrc
c
2
Ž 13.
Ž 14 .
H0
v
ž /
r s y q z2 r p J0 Ž k .
/
c
=J0 Ž k . d q z , prop I xprop z s yI z x
Ž 12.
2
v
5
vrc
i4pv 2
½ž
r s q q z2 r p J2 Ž k . d q z ,
ž /
y
2
v
5
vrc
2
i8pv
exp w iq z Z x
r s q q z2 r p J2 Ž k . d q z ,
ž /
q
(ž
for I
In polar coordinate system Ž q, u ., after integration over u , the propagator can be written in terms of the two propagating and evanescent parts, as follows. l
evan I xevan z s yI z x s
=
0
p
Ž 17.
Ž 18.
l
qy
r s q q z2 r p J0 Ž k .
Ž9.
™ ™ where ™ r s xe x q ye y is in x–y plane and S is the orthogonal tensor to rotate the coordinate system, which can be written as
qx
/
5
c
I zevan z s
c
r s y q z2 r p J2 Ž k . d q z
ž /
y
½ž
2
v
/
r s q q z2 r p J0 Ž k .
r s y q z2 r p J2 Ž k . d q z
Ž 16.
In this section, we present numerical results from the proposed calculations. We assume the light wavelength l s 633 nm. We select e 1 s e 2 s Ž0.06 q i4.15. 2 for silver. The strength of the polarizability of the point dipole is calculated with the expression in Eq. Ž3. for parameters a s 50 nm and ed s Ž0.21 q i3.27. 2 Žgold.. We stress that in the developed formulae there is no restriction on selection of the material Žindeed the dielectric function for a given frequency. for the dipole sphere and the surfaces. The separation of the two interfaces is chosen D s 200 nm. The intensity of the fields is calculated for a length of 2000 nm with the dipole at the center. To avoid the divergence at the dipole site, the field at the dipole site is excluded. To obtain visible images we have to plot them logarithmically, and the images are all for the x–z plane. We select the dipole position at the center of the two planar interfaces d s 100 nm. The calculated field distributions are shown in Fig. 2, which contains three group of images. In Fig. 2a the
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
images are for the whole field radiation, whereas in Fig. 2b and 2c the propagating and evanescent components are shown respectively. In each group, the images are for
15
different polarizations of the stimulating light source, from up down the three images are for x, y and z polarizations, respectively. As already discussed in Refs. w13,17x, we
Fig. 2. Field intensity distributions in the system shown in Fig. 1, where l s 633 nm, d s 100 nm, D s 200 nm, e 1 s e 2 s Ž0.06 q i4.15. 2 , and the strength of the dipolar polarizability calculated with the expression in Eq. Ž3. with parameters a s 50 nm and ed s Ž0.21 q i3.27. 2 . Ža. Total field, Žb. propagating waves, and Žc. evanescent fields. In each group of Ža., Žb. and Žc., the three images are for x, y and z polarizations in up down order. All the images are in the x–z plane for x s Žy1000 nm, 1000 nm. and z s Ž0, 200 nm.. The intensity is scaled logarithmically with the field at the dipole point excluded.
16
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
Fig. 2 Žcontinued..
point out that the total intensity of the field does not equal to the summation of the evanescent and propagating components. There exist interference terms between the two components. The images in Fig. 2 can be compared with the cases where the two surfaces are absent in Ref. w17x and the cases where only one surface is present in Ref. w13x. In the following we take several notes on the images in Fig. 2. The influence of the two surfaces is clearly shown in the images. Particularly there is a concentrated zone in between the two planar interfaces when the light is in x polarization Žsee the first image in Fig. 2a. and the concentration is not present for other polarizations. The concentration should be the combined contribution stemming from both the propagating and evanescent components. The fact that the concentrations cannot be seen in the separated propagating and evanescent intensities in Fig. 2b and 2c implies that the polarization states of the two components are very different in the region near the interfaces. Furthermore, based on the images in Fig. 2b, one notes that there exit dark zones about one Žor integer times of. wavelength from the dipole, which can be explained as the diffraction patterns of the propagating waves Žsee Refs. w13,17x for some discussions.. An application of the developed formulae we have in mind is to calculate the scattering of internal microconcentrations in ultrathin films. The present formulae will be useful when the density of the microscatterers becomes so high that the interactions between the neighboring particles
become of great importance and the local field inside the films changes dramatically. In order to know the field propagation from one particle to another, one needs an accurate field propagator including the multireflections. The formulae in the present paper provide such a propagator. In the formulae, the contributions from the propagating and the evanescent waves are accurately separated, which will be useful in the microscopic theory in near field optics. In conclusion we have demonstrated that dipole radiation in nanometric two-surface systems can be exactly computed with all the field components Žpropagating and evanescent. and the full retardation effects being rigorously included. With this propagator one expects to more accurately describe the N-dipole interaction and radiation in two-surface systems.
References w1x w2x w3x w4x w5x w6x w7x
W. Lukosz, R.E. Kunz, J. Opt. Soc. Am. 67 Ž1977. 1607. S. Efrima, H. Metiu, J. Chem. Phys. 70 Ž1979. 1602. H. Kuhn, J. Chem. Phys. 53 Ž1970. 101. R.R. Chance, A. Prock, R. Silbey, J. Chem. Phys. 60 Ž1974. 2744. H. Morawitz, M.R. Philpott, Phys. Rev. B 10 Ž1974. 4863. R. Fuchs, R.G. Barrera, Phys. Rev. B 24 Ž1981. 2940. W. Lukosz, J. Opt. Soc. Am. 71 Ž1981. 744.
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17 w8x H. Rigneault, S. Monneret, Quantum Semiclass. Opt. 9 Ž1997. 1017, and references therein. w9x H. Benisty, R. Stanley, M. Mayer, J. Opt. Soc. Am. A 15 Ž1998. 1192, and references therein. w10x W.E. Moerner, Science 265 Ž1994. 46. w11x W.P. Ambrose, P.M. Goodwin, J.C. Martin, R.A. Keller, Phys. Rev. Lett. 72 Ž1994. 160. w12x M. Xiao, X. Chen, J. Phys. Soc. Jpn. 67 Ž1998. 351. w13x M. Xiao, Opt. Commun. 143 Ž1997. 11.
17
w14x M. Xiao, A. Zayats, J. Siqueiros, Phys. Rev. B 55 Ž1997. 1824. w15x M. Xiao, J. Opt. Soc. Am. A 14 Ž1997. 2977. w16x O. Keller, Ultramicroscopy 71 Ž1998. 1. w17x M. Xiao, J. Mod. Opt. 44 Ž1997. 327. w18x M. Xiao, S. Bozhevolnyi, O. Keller, Appl. Phys. A 62 Ž1996. 115. w19x M. Xiao, S. Bozhevolnyi, Opt. Commun. 130 Ž1996. 337. w20x M. Xiao, Chem. Phys. Lett. 270 Ž1997. 169.
Optics Communications 158 Ž1998. 11–17
Optical dipole radiation inside nanometric planar two-surface systems Mufei Xiao a
a,)
, Xin Chen
b
Centro de Ciencias de la Materia Condensada, UNAM, Apartado Postal 2681, Ensenada CP 22800, Baja California, Mexico b Department of Physics, UniÕersity of Manchester, Institute Sciences and Technology, Manchester M60 1QD, UK Received 28 May 1998; revised 21 September 1998; accepted 23 September 1998
Abstract In classic optics, we propose a calculation for the electromagnetic radiation stemming from a point dipole located between two planar interfaces with arbitrary separations. We calculate the field distribution inside the planar two-surface system, particularly near the dipole source. First, the modified dipole strength due to the cavity environments is self-consistently calculated as a function of the dipole position, and second, the field distribution inside the planar cavity is obtained with an appropriate consideration of the multireflection contributions. The multireflections and the s- and p-polarized reflection coefficients are treated in the two-dimensional Fourier plane wave representation. The treatment enables us to accurately calculate all the field components, the evanescent fields as well as the propagating waves, for arbitrary cavity sizes. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Dipole; Two-surface system; Field distribution
1. Introduction Electromagnetic dipole radiation in the presence of microscopic environments has been the subject of intense study for a long time. The studied microscopic environments include metallic and dielectric planar surfaces w1–6x, and multilayer structures w7–9x. In general, the studies are concentrated on two different problems. One is to study how the dipole source is modified by the microscopic environments, for example, the modified fluorescent single molecule radiation w10–12x. The other is to study the radiation patterns in the presence of the microscopic environments, for example, the radiation of single w1,13x and many near-surface emitters w14,15x 1. The problems can be
)
Corresponding author. Correspondence address: CCMCUNAM, P.O. Box 439036, San Ysidro, CA 92143, USA. E-mail: [email protected] 1 In Ref. w13x, there are two typos in the article: there should be a Žy. sign in front of I zprop in Eq. Ž9. and Izevan in Eq. Ž12.. x x
generally described in the framework of the classic theories, such as the electromagnetic antenna theory. However, quantum descriptions are often needed when the dipole sources are atom-like systems that carry intrinsic modes due to the interstate transitions Žsee, e.g., Ref. w16x for some related discussions.. Our interest in the problem has been concentrated on the description of the optical radiation of microscopic emitters on surfaces. Particularly we are interested in an accurate description of the radiation near the source region: the near field zone. In our work, we describe the dipole sources as homogeneous small particles, which means the dipole sources are not necessarily associated with specific intrinsic modes. In our previous work, we proposed an exact formalism for the dipole field radiation in free space w17x and in nanometric dipole–surface systems w13x. Incorporating the exact formalism for single dipole radiation into the previous established microscopic theory w14,15,18,19x, we can now calculate accurately the field distribution for many scatterers.
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 5 2 1 - 5
12
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
In the present work, we consider the cases where the scatterers are located between two surfaces. It is of current interest to study the local field inside ultrathin films. Since ultrathin layer systems are often island-like structures, it is interesting to calculate the dipole radiation inside the multilayer structures. In order to use the microscopic theory to describe the self-consistent local field inside the layered structures and the modified radiation of the emitters, one needs an accurate formalism to study the radiation of both the evanescent and propagating modes. In the present paper we provide such a formalism. In the theory, the evanescent and propagating waves can be exactly computed everywhere inside the two planar surfaces. This is made possible by describing the multireflections as well as the direct propagation in the two-dimensional spectrum representation. The formalism is particularly adequate for calculating the dipole radiation near the dipole source within several wavelengths of the stimulating light source, though, apart from computational difficulties, there is no restriction to use the formalism for radiation fields beyond this region, none there is any restriction for the materials of the inside and outside regions as far as the materials can be described with their dielectric functions. The theory is however not convenient for calculating the field far away from the dipole source because of computational limit. Hence, to study the far field, it would be more convenient to adopt traditional methods in the literature that study the resonance couplings between the cavity propagating modes and the source intrinsic modes, i.e., the so-called cavity optics, which is not included in the present context. Readers are referred to Refs. w8,9x for studies on modification of spontaneous emission in multilayer structures. The paper is organized as follows. In Section 1 we present the self-consistent formalism for the modified local
field at the dipole site when the dipole is located between two planar surfaces, and the formalism for the dipole radiation inside the two-surface system. In Section 2, we present numerical results from the proposed calculations and finally we discuss the results.
2. The theory In Fig. 1 we describe the two-surface system schematically. The space is divided into three regions of different materials with their associated dielectric functions e 1Ž v ., e Ž v ., and e 2 Ž v .. A point dipole is located inside the two interfaces at a distance d from surface I. The separation of the two interfaces is a nanometric Žor arbitrary. distance D. We assume that the point dipole can be represented as a mesoscopic sphere of radius a and dielectric function edŽ v .. The dipolar polarizability may be calculated from the sphere radius a and the dielectric function edŽ v . Žsee, e.g., Ref. w13x.. Once the sphere is stimulated with an external light source, a dipole moment is established. The dipole strength is modified due to the presence of the two interfaces. The modified local field at the dipole site has to be calculated self-consistently, and the final dipole strength is obtained as a function of its position ™ rd s Ž0,0,d.. The established dipole radiates electromagnetically to the space. The field radiation inside the region between the two surfaces can be finally calculated by collecting the contributions from both the direct propagation and the multireflections from the dipole source to the observation point ™ ro s Ž x,0, z .. In the following we assume e Ž v . s 1, just to simplify the notation.
Fig. 1. A scheme for a point dipole in a planar two-surface system. The dipole radiation travels directly and indirectly Žmultireflections. towards the observation point. Only one reflection is plotted, which represents all possible multireflections.
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
Let us start with the general expression for the radiation of a dipole inside two surfaces, as ™
l
l
™ ™ ™ E Ž™ r o . s ym 0 v 2 D Ž ™ rd ,r o . q I Ž rd ,ro . ™
l = a rd . Ž v . E local Ž™
Ž1.
where m 0 is the permeability in vacuum and v angular l l™ ™ ™ . Ž . frequency. Tensors DŽ™ rd ,r o and I rd ,ro are the so-called direct and reflection Žindirect. propagators, which describe the field propagation from the source point to the observation point directly and indirectly Žvia the surface multireflections., respectively. In Eq. Ž1., the dipole is represented by the term below ™
™
l pŽ v . s a rd . , Ž v . E local Ž™
Ž2.
l
where a Ž v . is the dipolar polarizability, which can be obtained from the parameters a and edŽ v ., as l a Ž v . s 4pe 0 a3
ed Ž v . y 1 ed Ž v . q 2
,
Ž3.
tions, namely the evanescent and propagating waves by dividing the integration limit for the whole spectra 0 ; q ; v into the two ranges: the propagating range 0 ; q ; vrc Žwhich corresponds to vrc ; q z ; 0. and the evanescent range vrc ; q ; ` Žwhich corresponds to i0 ; q z ; i`.. For the direct field propagator, we shall use the analytical results already obtained in our previous work in Ref. w17x. The direct propagator for the whole field is l
D Ž™ r. s
1
ž
4p
q
ž
l™ ™
l E local Ž ™ rd . s U q m 0 v 2 a Ž v . I Ž rd ,rd . l
y1 ™ E0 ™ rd
Ž . , Ž4.
™ ™
where U is a unit tensor, EoŽ rd . is the incident field at the l dipole site, and the tensor I Ž d,d . represents all multireflections of the dipole radiation to the dipole itself. In the following we calculate the field propagators. The key in our theory is to represent the field propagation and reflections in the angular spectrum Ž ™ q, which is parallel to the surfaces. representation. In this case, for a surface reflection, it is readily possible to exactly match the boundary conditions to obtain the s- and p-polarized reflection coefficients r s Ž q . and r p Ž q .. The results are as follows w13x rsŽ q. s
qz y k z qz q k z
, and r p Ž q . s
qz es y k z qz es q k z
,
Ž5.
where e s represents the dielectric function of the outside region of the interfaces, and q z2 s Ž vrc . 2 y q 2 , and k z2 s Ž vrc . 2e s y q 2 Ž c is the light speed in vacuum., with q z and k z representing the vertical component of the wave vector in air Žas we have assumed the dipole being located in the region e s 1. and in solids Žthe two regions outside the interfaces., respectively. The essential advantages to work in the angular spectrum representation are as follows. Firstly, with respect to the surfaces, the reflections become simply the cases of plane waves in vertical incidence, which enables us to avoid including the incidence angles. Secondly, the reflection coefficients are now exactly matching the wave number q, which enables us to accurately calculate the field components for all the spectra 0 ; q ; `. Thirdly, with an inverse Fourier transformation, one is able to exactly separate the two important contribu-
vr
r
vr
2
q
v2r3
3c 2
3ic q
r
c2
ic y
1
l
Devan Ž™ r. s
1 4p
™
l
1 y
y 2
v2r3
/
/
l
U
™™
nn e i v r r c ,
Ž6.
where r s <™ ro y™ rd < and ™ n s™ rrr. The above direct propagator can be divided into its evanescent and propagating parts, where the evanescent field is
and the local field E localŽ™ rd . at the dipole site has to be calculated self-consistently as w20x ™
13
ž
ž
c2
1 y
q 2r
3c 2
1
q y
y 2r
v2r3
v2r3
/
/
l
U
™™
nn ,
Ž7.
and the propagating field propagator can be obtained by l l l Dprop s D y Devan . In the following, we shall calculate the reflection field propagators for the multireflections. To start, let us write down in the angular spectrum representation the reflection l ™ . propagator I Ž™ rd ,r o for only one reflection at the interface I Žsee Fig. 1. w13x, l
I Ž Z,q . s
exp w iq z Z x 2 iq z
r 1s Ž q . ™ es ™ es q r1p Ž q . ™ er ™ ei ,
Ž8.
where the unit vector ™ es s Ž0,1,0. is the s-field direction Žin y-axis in Fig. 1., whereas ™ e r s Ž crv .Žyq z ,0,y q . and ™ e i s Ž crv .Ž q z ,0,y q . are unit vectors for reflection and incidence p-polarized light, respectively. The distance Z in Eq. Ž8. is the distance the above plane wave has travelled from the dipole to the observation point, in the case of one reflection at interface I, Z s d q z. Then we consider the reflection at interface II. In this case, the expression in Eq. Ž8. needs to be modified as follows. The reflection coefficients are now r 2s and r 2p, the value Z becomes Z s 2 D y Ž d q z ., and the term ™ er ™ e i becomes ™ ei™ e r . Finally we consider the case of multireflections, where the light goes from the dipole to the observation point via more than one reflection at the interfaces. The change of the value Z can be easily obtained by summing up the whole trip, for example, Z s 2 D q d q z is for the trip of touching twice at interface I and once at interface II. Using the same example, the reflection coefficients become r s s r1s r 2s r 1s and r p s r 1p r 2p r 1p. In all cases, the term ™ es ™ es does not ™™ change, but the term e r e i may have four possibilities such as Ž1. ™ er ™ e i ; Ž2. ™ ei™ e r ; Ž3. ™ er ™ e r ; and Ž4. ™ ei™ ei.
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
14
The reflection propagator in spatial domain is the Fourier inverse transform of the above propagators for 0 ; q ; `. We stress that in the range 0 ; q ; vrc Žwhich corresponds to vrc ; q z ; 0. the field is homogeneous Žpropagating., whereas in the range vrc ; q ; ` Žwhich corresponds to i0 ; q z ; i`. the field is inhomogeneous Ževanescent.. The inverse Fourier transform can be written as w13x, 1
l
I Ž x , y,Z . s
`l
y1 l
Ž 2p .
2
l
™™ I Ž Z,q . S exp Ž iq r . d 2 q,
H0 S
yc 2 I yevan y s
`
8pv 2
H0 exp wyq Z x z
2
v
yc 2 4pv
2
`
H0 exp wyq Z x r z
yc 2
l
Ss
1 q
yq y
qx
0 .
0
0
q
Ž 10 .
l
l
I Ž x , y,Z . s Iprop Ž x , y,Z . q Ievan Ž x , y,Z . ,
Ž 11.
where always I x yŽ y x . s I y zŽ z y . s 0, and I xprop x s
c2
vrc
i8pv 2
H0
v
2
c
I yprop y s
c2
H0
v
2
c
I zprop z s
c2
exp w iq z Z x
r s y q z2 r p J0 Ž k .
/
c
½ž
H0
exp w iq z Z x r p
yc 2
(ž
4pv 2
c
/
2
v
ž / c
y q z2
exp w iq z Z x r p J1 Ž k . q z
y q z2 d q z ,
Ž 15.
and, yc 2 I xevan x s
`
8pv 2
H0 exp wyq Z x
v =
ž / c
z
2
½ž
2
v c
5
q q z2 J0 Ž k . d q z
where
(
4pv 2
z
p
J1 Ž k . q z
2
v c
`
H0 exp wyq Z x r
/
q q z2 d q z ,
Ž 19.
JnŽ k . are nth order Bessel functions, and prop
(
2
, k s r Ž vr2 . y q z2 , while for I evan , k s 2
r Ž vrc . q q z2 . The above expressions are for the case ™ er ™ e i . For the case ™ ei™ e r , since ™ ei™ e r s Ž™ er ™ e i . T, the elements I x z and I z x reverse the sign wchanging from Žq or y. to Žy or q.x. For the cases ™ er ™ e r and ™ ei™ e i , the elements I x z s I z x and the elements I x xŽ y y . have a reverse sign for the terms containing r p. By using the above field propagators, one is now able to compute the field distribution inside the two-layer systems. As mentioned in the introduction, the formulae are particularly convenient for the near field zones, because one can reduce the number of multireflections when higher numbers do not bring out significant contributions.
3. Some numerical results and remarks
2
v
=
vrc
c
2
Ž 13.
Ž 14 .
H0
v
ž /
r s y q z2 r p J0 Ž k .
/
c
=J0 Ž k . d q z , prop I xprop z s yI z x
Ž 12.
2
v
5
vrc
i4pv 2
½ž
r s q q z2 r p J2 Ž k . d q z ,
ž /
y
2
v
5
vrc
2
i8pv
exp w iq z Z x
r s q q z2 r p J2 Ž k . d q z ,
ž /
q
(ž
for I
In polar coordinate system Ž q, u ., after integration over u , the propagator can be written in terms of the two propagating and evanescent parts, as follows. l
evan I xevan z s yI z x s
=
0
p
Ž 17.
Ž 18.
l
qy
r s q q z2 r p J0 Ž k .
Ž9.
™ ™ where ™ r s xe x q ye y is in x–y plane and S is the orthogonal tensor to rotate the coordinate system, which can be written as
qx
/
5
c
I zevan z s
c
r s y q z2 r p J2 Ž k . d q z
ž /
y
½ž
2
v
/
r s q q z2 r p J0 Ž k .
r s y q z2 r p J2 Ž k . d q z
Ž 16.
In this section, we present numerical results from the proposed calculations. We assume the light wavelength l s 633 nm. We select e 1 s e 2 s Ž0.06 q i4.15. 2 for silver. The strength of the polarizability of the point dipole is calculated with the expression in Eq. Ž3. for parameters a s 50 nm and ed s Ž0.21 q i3.27. 2 Žgold.. We stress that in the developed formulae there is no restriction on selection of the material Žindeed the dielectric function for a given frequency. for the dipole sphere and the surfaces. The separation of the two interfaces is chosen D s 200 nm. The intensity of the fields is calculated for a length of 2000 nm with the dipole at the center. To avoid the divergence at the dipole site, the field at the dipole site is excluded. To obtain visible images we have to plot them logarithmically, and the images are all for the x–z plane. We select the dipole position at the center of the two planar interfaces d s 100 nm. The calculated field distributions are shown in Fig. 2, which contains three group of images. In Fig. 2a the
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
images are for the whole field radiation, whereas in Fig. 2b and 2c the propagating and evanescent components are shown respectively. In each group, the images are for
15
different polarizations of the stimulating light source, from up down the three images are for x, y and z polarizations, respectively. As already discussed in Refs. w13,17x, we
Fig. 2. Field intensity distributions in the system shown in Fig. 1, where l s 633 nm, d s 100 nm, D s 200 nm, e 1 s e 2 s Ž0.06 q i4.15. 2 , and the strength of the dipolar polarizability calculated with the expression in Eq. Ž3. with parameters a s 50 nm and ed s Ž0.21 q i3.27. 2 . Ža. Total field, Žb. propagating waves, and Žc. evanescent fields. In each group of Ža., Žb. and Žc., the three images are for x, y and z polarizations in up down order. All the images are in the x–z plane for x s Žy1000 nm, 1000 nm. and z s Ž0, 200 nm.. The intensity is scaled logarithmically with the field at the dipole point excluded.
16
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17
Fig. 2 Žcontinued..
point out that the total intensity of the field does not equal to the summation of the evanescent and propagating components. There exist interference terms between the two components. The images in Fig. 2 can be compared with the cases where the two surfaces are absent in Ref. w17x and the cases where only one surface is present in Ref. w13x. In the following we take several notes on the images in Fig. 2. The influence of the two surfaces is clearly shown in the images. Particularly there is a concentrated zone in between the two planar interfaces when the light is in x polarization Žsee the first image in Fig. 2a. and the concentration is not present for other polarizations. The concentration should be the combined contribution stemming from both the propagating and evanescent components. The fact that the concentrations cannot be seen in the separated propagating and evanescent intensities in Fig. 2b and 2c implies that the polarization states of the two components are very different in the region near the interfaces. Furthermore, based on the images in Fig. 2b, one notes that there exit dark zones about one Žor integer times of. wavelength from the dipole, which can be explained as the diffraction patterns of the propagating waves Žsee Refs. w13,17x for some discussions.. An application of the developed formulae we have in mind is to calculate the scattering of internal microconcentrations in ultrathin films. The present formulae will be useful when the density of the microscatterers becomes so high that the interactions between the neighboring particles
become of great importance and the local field inside the films changes dramatically. In order to know the field propagation from one particle to another, one needs an accurate field propagator including the multireflections. The formulae in the present paper provide such a propagator. In the formulae, the contributions from the propagating and the evanescent waves are accurately separated, which will be useful in the microscopic theory in near field optics. In conclusion we have demonstrated that dipole radiation in nanometric two-surface systems can be exactly computed with all the field components Žpropagating and evanescent. and the full retardation effects being rigorously included. With this propagator one expects to more accurately describe the N-dipole interaction and radiation in two-surface systems.
References w1x w2x w3x w4x w5x w6x w7x
W. Lukosz, R.E. Kunz, J. Opt. Soc. Am. 67 Ž1977. 1607. S. Efrima, H. Metiu, J. Chem. Phys. 70 Ž1979. 1602. H. Kuhn, J. Chem. Phys. 53 Ž1970. 101. R.R. Chance, A. Prock, R. Silbey, J. Chem. Phys. 60 Ž1974. 2744. H. Morawitz, M.R. Philpott, Phys. Rev. B 10 Ž1974. 4863. R. Fuchs, R.G. Barrera, Phys. Rev. B 24 Ž1981. 2940. W. Lukosz, J. Opt. Soc. Am. 71 Ž1981. 744.
M. Xiao, X. Chen r Optics Communications 158 (1998) 11–17 w8x H. Rigneault, S. Monneret, Quantum Semiclass. Opt. 9 Ž1997. 1017, and references therein. w9x H. Benisty, R. Stanley, M. Mayer, J. Opt. Soc. Am. A 15 Ž1998. 1192, and references therein. w10x W.E. Moerner, Science 265 Ž1994. 46. w11x W.P. Ambrose, P.M. Goodwin, J.C. Martin, R.A. Keller, Phys. Rev. Lett. 72 Ž1994. 160. w12x M. Xiao, X. Chen, J. Phys. Soc. Jpn. 67 Ž1998. 351. w13x M. Xiao, Opt. Commun. 143 Ž1997. 11.
17
w14x M. Xiao, A. Zayats, J. Siqueiros, Phys. Rev. B 55 Ž1997. 1824. w15x M. Xiao, J. Opt. Soc. Am. A 14 Ž1997. 2977. w16x O. Keller, Ultramicroscopy 71 Ž1998. 1. w17x M. Xiao, J. Mod. Opt. 44 Ž1997. 327. w18x M. Xiao, S. Bozhevolnyi, O. Keller, Appl. Phys. A 62 Ž1996. 115. w19x M. Xiao, S. Bozhevolnyi, Opt. Commun. 130 Ž1996. 337. w20x M. Xiao, Chem. Phys. Lett. 270 Ž1997. 169.