Abstract

We have derived analytic expressions for the angular distribution of light emitted by an atomic system, e.g., a fluorescent molecule, located in a dielectric medium 1 at distance z0 from the interface to a different dielectric medium 2. The theory is rigorously valid for electric and magnetic dipole transitions with arbitrary orientation of the dipole transition moment. (In Paper II [ J. Opt. Soc. Am. 57, 1615– 1619 ( 1977)], only the special case of dipoles oriented perpendicular to the interface had been treated.) The radiation patterns of dipoles located on the interface (z0 = 0) and oriented parallel to it, and ensembles of such dipoles radiating incoherently with randomly oriented dipole moments were examined in particular. They differ greatly from the corresponding well-known dipole radiation patterns in an unbounded medium 1, due to the wide-angle interferences of emitted plane waves, and radiation from evanescent waves in the dipole’s near field into medium 2, if this is denser than medium 1.

© 1979 Optical Society of America

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References

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  1. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977).
    [CrossRef]
  2. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977).
    [CrossRef]
  3. References 1 and 2 contain lists of references to related experimental and theoretical work.
  4. We emphasize that the following of our results remain valid with appropriate reflection coefficients r1,2(s,p)even if medium 2 is absorbing and/or is not homogeneous but planar stratified in the z direction: in Paper (I) Eqs. (3.12)–(3.17) for the normalized total radiated power and, in this Paper expressions (3.21) and (3.22) for the radiation patterns in medium 1.

1977 (2)

J. Opt. Soc. Am. (2)

Other (2)

References 1 and 2 contain lists of references to related experimental and theoretical work.

We emphasize that the following of our results remain valid with appropriate reflection coefficients r1,2(s,p)even if medium 2 is absorbing and/or is not homogeneous but planar stratified in the z direction: in Paper (I) Eqs. (3.12)–(3.17) for the normalized total radiated power and, in this Paper expressions (3.21) and (3.22) for the radiation patterns in medium 1.

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Figures (8)

FIG. 1
FIG. 1

D dipole located in medium 1 at distance z0 from the interface to medium 2. The unit vector n ˆ in direction of the dipole moment lies in the xz plane.

FIG. 2
FIG. 2

Unit vector k ˆ in the direction of observation, α is the angle between k ˆ and the −z axis. The x ˜ z plane of emission is defined by k ˆ and the z axis. The directions of the electric fields of an s- and p-polarized wave emitted into direction k ˆ are indicated.

FIG. 3
FIG. 3

Wide-angle interferences of plane waves (I and II) emitted by dipole D. At the location of the dipole D the electric fields E of the s-polarlzed waves I and II are (a) in phase for the perpendicular dipole (Ө = 0), but (b) 180° out of phase for the parallel dipole (Ө = 90°) oriented in the x direction, while (c) the magnetic fields H of the p-polarized waves I and II emitted by the parallel (Ө = 90°) dipole oriented in the y direction are in phase.

FIG. 4
FIG. 4

Angular distribution of power P(s,p)(α) versus angle α, emitted by an ensemble of magnetic (m, ||) and electric (e, ||) dipoles with random orientations parallel to the interface. Dipoles are located in medium 1 on the interface (z0 = 0) to medium 2. (---): s-polarized; (·····) p-polarized, (—) sum of (s) and (p) polarized light. Relative refractive index n = n 2 / n 1 = ( 1 / 2 ) 2.

FIG. 5
FIG. 5

Radiation patterns as in Fig. 4 but for n = 2.

FIG. 6
FIG. 6

Radiation patterns as in Fig. 4 but for n = (1/2).

FIG. 7
FIG. 7

Radiation patterns as in Fig. 4 but for n = 2.

FIG. 8
FIG. 8

Radiation patterns of (e,||) dipoles as in Fig. 4 but for n = 0.99 and n = 1.01. Interchange of polarizations (s) ↔ (p) yields the radiation patterns of (m,||) dipoles.

Equations (84)

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ϕ ( E , H ) ( x ) = ϕ ˜ ( E , H ) + ( k x , k y ) × exp { i [ k x x + k y y + k z ( z z 0 ) ] } d k x d k y ,
k z = { + k z 1 k z , 1 for z z 0 z z 0 ,
k z , 1 = { + ( k 1 2 k x 2 k y 2 ) 1 / 2 if k x 2 + k y 2 k 1 2 + i ( k x 2 k y 2 k 1 2 ) 1 / 2 if k x 2 + k y 2 > k 1 2 ,
ϕ ˜ ( H ) ( k x , k y ) = μ 0 ω p 0 sin Ө k y ( k x 2 + k y 2 ) 1 ϕ ˜ ( k x , k y ) ,
ϕ ˜ ( E ) ( k x , k y ) = p 0 [ cos Ө k x k z ( k x 2 + k y 2 ) 1 sin Ө ] × ϕ ˜ ( k x , k y ) ,
ϕ ˜ t ( E , H ) ( k x , k y ) = t 1 , 2 ( p , s ) ( k x , k y ) ϕ ˜ ( E , H ) ( k x , k y ) exp ( i k z , 1 z 0 ) ;
ϕ ˜ 1 ( E , H ) ( k x , k y ) = r 1 , 2 ( p , s ) ( k x , k y ) ϕ ˜ ( E , H ) ( k x , k y ) exp ( i k z , 1 z 0 ) + ϕ ˜ ( E , H ) ( k x , k y ) exp ( i k z , 1 z 0 ) .
( 4 π ) [ P ( s ) ( a , φ ) + P ( p ) ( α , φ ) ] d Ω = L ( z 0 ) / L .
L P ( s , p ) ( α , φ ) d Ω = lim R ( A ) k · S ( x ) d σ .
L P ( p ) ( α , φ ) = ( 1 2 ) π 2 μ 0 ( ω k j ) 3 sin 2 2 α | ϕ ˜ j ( E ) ( k x , k y ) | 2 ,
L P ( s ) ( α , φ ) = ( 1 2 ) π 2 0 j ( ω k j ) 3 sin 2 2 α | ϕ ˜ j ( H ) ( k x , k y ) | 2 ,
k x = k j sin α cos φ , k y = k j sin α sin φ , ( k j = n j ω / c )
L = ( n 1 ) q L vac ,
( L vac ) m = m 0 2 ω 4 / 12 π μ 0 c 3 ,
( L vac ) e = p 0 2 ω 4 / 12 π 0 c 3 ,
P ( s , p ) ( α 2 , φ ) = P ( s , p ) ( α 1 , φ ) T 1 , 2 ( s , p ) ( α 1 ) d Ω 1 / d Ω 2 ,
d Ω 1 / d Ω 2 = n 2 cos α 2 / cos α 1 ,
P ( s ) ( α , φ ) = 3 8 π [ cos Ө sin α + sin Ө cos φ cos α ] 2 ,
P ( p ) ( α , φ ) = 3 8 π sin 2 Ө sin 2 φ ,
P m ( s ) ( α 2 , φ ) = 3 2 π n 3 cos 2 α 2 ( cos Ө sin α 1 + sin Ө cos φ cos α 1 ) 2 ( cos α 1 + n cos α 2 ) 2 ,
P m ( p ) ( α 2 , φ ) = 3 2 π n 3 cos 2 α 2 sin 2 Ө sin 2 φ ( n cos α 1 + cos α 2 ) 2 ,
P e ( p ) ( α 2 , φ ) = 3 2 π n 3 cos 2 α 2 ( cos Ө sin α 1 + sin Ө cos φ cos α 1 ) 2 ( n cos α 1 + cos α 2 ) 2 ,
P e ( s ) ( α 2 , φ ) = 3 2 π n 3 cos 2 α 2 sin 2 Ө sin 2 φ ( cos α 1 + n cos α 2 ) 2 .
P m ( s ) ( α 2 , φ ) = f ( s ) ( α 2 ) [ n 2 cos 2 Ө sin 2 α 2 + sin 2 Ө cos 2 φ ( n 2 sin 2 α 2 1 ) ] ,
P m ( p ) ( α 2 , φ ) = f ( p ) ( α 2 ) sin 2 Ө sin 2 φ ,
P e ( p ) ( α 2 , φ ) = f ( p ) ( α 2 ) [ n 2 cos 2 Ө sin 2 α 2 + sin 2 Ө cos 2 φ ( n 2 sin 2 α 2 1 ) ] ,
P e ( s ) ( α 2 , φ ) = f ( s ) ( α 2 ) sin 2 Ө sin 2 φ ,
f ( s ) ( α 2 ) = 3 2 π n 3 n 2 1 cos 2 α 2 exp [ 2 z 0 / Δ z ( α 2 ) ] ,
f ( p ) ( α 2 ) = f ( s ) ( α 2 ) [ ( n 2 + 1 ) sin 2 α 2 1 ] 1 ,
Δ z ( α 2 ) = ( λ 1 / 2 π ) ( n 2 sin 2 α 2 1 ) 1 / 2 .
P e ( s , p ) ( α 2 = 90 ° , φ ) = P m ( s , p ) ( α 2 = 90 ° , φ ) = 0 .
P m ( s ) ( α 2 = α 2 , c , φ ) = n 2 P e ( p ) ( α 2 = α 2 , c , φ ) = 3 2 π n cos Ө .
P m ( p ) ( α 2 = α 2 , c , φ ) = n 2 P e ( s ) ( α 2 = α 2 , c , φ ) = 3 2 π n 3 sin 2 Ө sin 2 φ .
P m ( s ) ( α 2 = 0 , φ ) = P e ( p ) ( α 2 = 0 , φ ) = 3 2 π n 3 ( n + 1 ) 2 sin 2 Ө cos 2 φ ,
P m ( p ) ( α 2 = 0 , φ ) = P e ( s ) ( α 2 = 0 , φ ) = 3 2 π n 3 ( n + 1 ) 2 sin 2 Ө sin 2 φ .
P m ( s ) ( α 1 , φ ) = 3 8 π ( cos 2 Ө sin 2 α 1 { 1 + | r 1 , 2 ( s ) ( α 1 ) | 2 + 2 | r 1 , 2 ( s ) ( α 1 ) | cos [ 2 k z , 1 z 0 + δ 1 , 2 ( s ) ( α 1 ) ] } + sin 2 Ө cos 2 φ cos 2 α 1 { 1 + | r 1 , 2 ( s ) ( α 1 ) | 2 2 | r 1 , 2 ( s ) ( α 1 ) | cos [ 2 k z , 1 z 0 + δ 1 , 2 ( s ) ( α 1 ) } ( 1 / 2 ) sin 2 Ө cos φ sin 2 α 1 [ 1 | r 1 , 2 ( s ) ( α 1 ) | 2 ] ) ,
P m ( p ) ( α 1 , φ ) = 3 2 π sin 2 Ө sin 2 φ { 1 + | r 1 , 2 ( p ) ( α 1 ) | 2 + 2 | r 1 , 2 ( p ) ( α 1 ) | cos [ 2 k z , 1 z 0 + δ 1 , 2 ( p ) ( α 1 ) ] } ,
r 1 , 2 ( s , p ) ( α 1 ) = | r 1 , 2 ( s , p ) ( α 1 ) | exp [ i δ 1 , 2 ( s , p ) α 1 ) ] .
P m , e ( s , p ) ( α 1 = 90 ° , φ ) = 0 ,
P m ( s ) ( α 1 , φ ) = 3 2 π cos 2 α 1 ( cos Ө sin α 1 n sin Ө cos φ cos α 2 ) 2 ( cos α 1 + n cos α 2 ) 2 ,
P m ( p ) ( α 1 , φ ) = 3 2 π n 2 cos 2 α 1 sin 2 Ө sin 2 φ ( n cos α 1 + cos α 2 ) 2 ,
P e ( p ) ( α 1 , φ ) = 3 2 π cos 2 α 1 ( n cos Ө sin α 1 sin Ө cos φ cos α 2 ) 2 ( n cos α 1 + cos α 2 ) 2 ,
P e ( s ) ( α 1 , φ ) = 3 2 π cos 2 α 1 sin 2 Ө sin 2 φ ( cos α 1 + n cos α 2 ) 2 .
P m ( s ) ( α 1 , φ ) = [ cos 2 Ө sin 2 α 1 + sin 2 Ө cos 2 φ ( sin 2 α 1 n 2 ) ] g ( s ) ( α 1 ) ,
P m ( p ) ( α 1 , φ ) = sin 2 Ө sin 2 φ g ( p ) ( α 1 ) ,
P e ( p ) ( α 1 , φ ) = [ cos 2 Ө sin 2 α 1 + n 4 sin 2 Ө cos 2 φ ( sin 2 α 1 n 2 ) ] g ( p ) ( α 1 ) ,
P e ( s ) ( α 1 , φ ) = sin 2 Ө sin 2 φ g ( s ) ( α 1 ) ,
g ( s ) ( α 1 ) = 3 2 π ( 1 n 2 ) 1 cos 2 α 1
g ( p ) ( α 1 ) = n 2 [ ( 1 + n 2 ) sin 2 α 1 1 ] 1 g ( s ) ( α 1 ) .
P m ( s ) ( α 1 = α 1 , c , φ ) = P e ( p ) ( α 1 = α 1 , c , φ ) = ( 3 / 2 π ) n 2 cos 2 Ө ,
P m ( p ) ( α 1 = α 1 , c , φ ) = P e ( s ) ( α 1 = α 1 , c , φ ) = ( 3 / 2 π ) sin 2 Ө sin 2 φ
P m ( s ) ( α 1 = 0 , φ ) = n 2 P e ( p ) ( α 1 = 0 , φ ) = 3 2 π n 2 sin 2 Ө cos 2 φ ( 1 + n ) 2 ,
P m ( p ) ( α 1 = 0 , φ ) = n 2 P e ( s ) ( α 1 = 0 , φ ) = 3 2 π n 2 sin 2 Ө sin 2 φ ( 1 + n ) 2 .
P ( s , p ) ( α , φ ) = P ( s , p ) ( α , φ ) .
P m ( s ) ( α , φ ) = 2 cos 2 φ P m , ( s ) ( α ) ,
P m ( p ) ( α , φ ) = 2 sin 2 φ P m , ( p ) ( α ) .
P m , ( s , p ) ( α ) = 1 2 π 0 2 π P m ( s , p ) ( α , φ φ 0 ) d φ 0 = 1 2 [ P m , x ( s , p ) ( α , φ ) + P m , y ( s , p ) ( α , φ ) ] .
P m , ( s ) ( α 2 ) = 3 2 π n 3 cos 2 α 1 cos 2 α 2 ( cos α 1 + n cos α 2 ) 2 ,
P m , ( p ) ( α 2 ) = 3 4 π n 3 cos 2 α 2 ( n cos α 1 + cos α 2 ) 2 ,
P e , ( p ) ( α 2 ) = 3 4 π n 3 cos 2 α 1 cos 2 α 2 ( n cos α 1 + cos α 2 ) 2 ,
P e , ( s ) ( α 2 ) = 3 4 π n 3 cos 2 α 2 ( cos α 1 + n cos α 2 ) 2 ,
P m , ( s ) ( α 2 ) = 3 4 π n 3 n 2 1 cos 2 α 2 ( n 2 sin 2 α 2 1 ) ,
P m , ( p ) ( α 2 ) = 3 4 π n 3 n 2 1 cos 2 α 2 ( n 2 + 1 ) sin 2 α 2 1 ,
P e , ( p ) ( α 2 ) = 3 4 π n 3 n 2 1 cos 2 α 2 ( n 2 sin 2 α 2 1 ) ( n 2 + 1 ) sin 2 α 2 1 ,
P e , ( s ) ( α 2 ) = 3 4 π n 3 n 2 1 cos 2 α 2 ;
P m , ( s ) ( α 1 ) = 3 4 π n 2 cos 2 α 1 cos 2 α 2 ( cos α 1 + n cos α 2 ) 2 ,
P m , ( p ) ( α 1 ) = 3 4 π n 2 cos 2 α 1 ( n cos α 1 + cos α 2 ) 2 ,
P e , ( p ) ( α 1 ) = 3 4 π cos 2 α 1 cos 2 α 2 ( n cos α 1 + cos α 2 ) 2 ,
P e , ( s ) ( α 1 ) = 3 4 π cos 2 α 1 ( cos α 1 + n cos α 2 ) 2 ;
P m , ( s ) ( α 1 ) = 3 4 π ( 1 n 2 ) 1 cos 2 α 1 ( sin 2 α 1 n 2 ) ,
P m , ( p ) ( α 1 ) = 3 4 π n 2 1 n 2 cos 2 α 1 ( 1 + n 2 ) sin 2 α 1 1 ,
P e , ( p ) ( α 1 ) = 3 4 π n 2 1 n 2 cos 2 α 1 ( sin 2 α 1 n 2 ) ( 1 + n 2 ) sin 2 α 1 1 ,
P e , ( s ) ( α 1 ) = 3 4 π ( 1 n 2 ) 1 cos 2 α 1 .
P m , ( s ) ( α 1 = α 1 , c ) = P e , ( p ) ( α 1 = α 1 , c ) = 0 ,
P m , ( p ) ( α 1 = α 1 , c ) = P e , ( s ) ( α 1 = α 1 , c ) = 3 4 π ,
P m , ( s ) ( α 2 = α 2 , c ) = 0 , P m , ( p ) ( α 2 = α 2 , c ) = 3 4 π n 3 .
P e , ( p ) ( α 2 = α 2 , c ) = 0 , P e , ( s ) ( α 2 = α 2 , c ) = 3 4 π n ,
P m , ( s ) ( α ) = P e , ( p ) ( α ) = 3 16 π cos 2 α ,
P m , ( p ) ( α ) = P e , ( s ) ( α ) = 3 16 π .
P m , ( p ) ( α j , c ) = P e , ( s ) ( α j , c ) 3 4 π ( j = 1 or 2 )
P m , ( p ) ( α = 90 ° ) = P e , ( s ) ( α = 90 ° ) = 0 .
P ( s , p ) ( α , φ ; n ) = n γ P ( s , p ) ( 180 ° α , φ ; 1 / n ) ,
P m ( s , p ) ( α , φ ; θ , n ) = n 3 P m ( s , p ) ( 180 ° α , φ ; 180 ° Ө , 1 / n ) ,
P 1 , 0 , 2 ( s , p ) ( α ) = ( n 1 / n 0 ) γ P ( s , p ) ( α ) ,