Abstract

We have derived analytical expressions for the angular distribution P(α) of the power radiated by magnetic or electric dipoles located at distance z0 from a dielectric interface and oriented perpendicular to it. For dipoles in the rarer medium very close to the interface, evanescent waves in the dipoles’ near field give rise to strong radiation into the denser medium. The resulting large maximum of P(α) shows characteristic differences for magnetic and electric dipoles when the relative refractive index of the two dielectrics is greater than √2. For dipoles lying on the interface a symmetry relation is established connecting the power distributions P(α) for the values n and 1/n of the relative refractive index.

© 1977 Optical Society of America

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Corrections

W. Lukosz and R. E. Kunz, "Errata: Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation pattern of perpendicular oriented dipoles," J. Opt. Soc. Am. 68, 1155_2-1155 (1978)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-68-8-1155_2

References

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  1. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles. I. Total radiated power,” J. Opt. Soc. Am. 67, xxxx–xxxx (1977) (preceding paper).
    [CrossRef]
  2. K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Progress in Optics, Vol. XII (North-Holland, Amsterdam, 1974), p. 165–232.
  3. With a telescope a sharp boundary between light and dark is seen, which is the well-known principle of some refractometers.
  4. P. Selenyi, “Sur l’existence et 1’observation des ondes lumineuses sphériques inhomogènes,” Compt. Rend. 157, 1408–1410 (1913).
  5. P. Fröhlich, “Die gültigkeitsgrenze des geometrischen gesetzes der lichtbrechung,” Ann. Phys. (Leip.) 65, 577–592 (1921).
    [CrossRef]
  6. C. K. Carniglia, L. Mandel, and K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
    [CrossRef]
  7. C. Carniglia and L. Mandel, “Quantization of evanescent electromagnetic waves,” Phys. Rev. D 3, 280–296 (1971).
    [CrossRef]
  8. W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977).
    [CrossRef]

1977 (2)

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles. I. Total radiated power,” J. Opt. Soc. Am. 67, xxxx–xxxx (1977) (preceding paper).
[CrossRef]

W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977).
[CrossRef]

1972 (1)

1971 (1)

C. Carniglia and L. Mandel, “Quantization of evanescent electromagnetic waves,” Phys. Rev. D 3, 280–296 (1971).
[CrossRef]

1921 (1)

P. Fröhlich, “Die gültigkeitsgrenze des geometrischen gesetzes der lichtbrechung,” Ann. Phys. (Leip.) 65, 577–592 (1921).
[CrossRef]

1913 (1)

P. Selenyi, “Sur l’existence et 1’observation des ondes lumineuses sphériques inhomogènes,” Compt. Rend. 157, 1408–1410 (1913).

Carniglia, C.

C. Carniglia and L. Mandel, “Quantization of evanescent electromagnetic waves,” Phys. Rev. D 3, 280–296 (1971).
[CrossRef]

Carniglia, C. K.

Drexhage, K. H.

C. K. Carniglia, L. Mandel, and K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[CrossRef]

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Progress in Optics, Vol. XII (North-Holland, Amsterdam, 1974), p. 165–232.

Fröhlich, P.

P. Fröhlich, “Die gültigkeitsgrenze des geometrischen gesetzes der lichtbrechung,” Ann. Phys. (Leip.) 65, 577–592 (1921).
[CrossRef]

Kunz, R. E.

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles. I. Total radiated power,” J. Opt. Soc. Am. 67, xxxx–xxxx (1977) (preceding paper).
[CrossRef]

W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977).
[CrossRef]

Lukosz, W.

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles. I. Total radiated power,” J. Opt. Soc. Am. 67, xxxx–xxxx (1977) (preceding paper).
[CrossRef]

W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977).
[CrossRef]

Mandel, L.

C. K. Carniglia, L. Mandel, and K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[CrossRef]

C. Carniglia and L. Mandel, “Quantization of evanescent electromagnetic waves,” Phys. Rev. D 3, 280–296 (1971).
[CrossRef]

Selenyi, P.

P. Selenyi, “Sur l’existence et 1’observation des ondes lumineuses sphériques inhomogènes,” Compt. Rend. 157, 1408–1410 (1913).

Ann. Phys. (Leip.) (1)

P. Fröhlich, “Die gültigkeitsgrenze des geometrischen gesetzes der lichtbrechung,” Ann. Phys. (Leip.) 65, 577–592 (1921).
[CrossRef]

Compt. Rend. (1)

P. Selenyi, “Sur l’existence et 1’observation des ondes lumineuses sphériques inhomogènes,” Compt. Rend. 157, 1408–1410 (1913).

J. Opt. Soc. Am. (2)

C. K. Carniglia, L. Mandel, and K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[CrossRef]

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles. I. Total radiated power,” J. Opt. Soc. Am. 67, xxxx–xxxx (1977) (preceding paper).
[CrossRef]

Opt. Commun. (1)

W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977).
[CrossRef]

Phys. Rev. D (1)

C. Carniglia and L. Mandel, “Quantization of evanescent electromagnetic waves,” Phys. Rev. D 3, 280–296 (1971).
[CrossRef]

Other (2)

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Progress in Optics, Vol. XII (North-Holland, Amsterdam, 1974), p. 165–232.

With a telescope a sharp boundary between light and dark is seen, which is the well-known principle of some refractometers.

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Figures (5)

FIG. 1
FIG. 1

Perpendicular dipole D located in medium 1 at distance z0 from the interface. Unit vector û points into direction of observation.

FIG. 2
FIG. 2

Angular distribution P(α) of emitted power vs angle α for electric (e) and magnetic (m) dipoles located on the interface (z0 = 0) for relative refractive indices n = 1. 01 (—), n = 2 ( ) , and n = 2 (· · ·). Curves (– · – · –) give P(α) for dipoles in an unbounded medium 1 (n = 1).

FIG. 3
FIG. 3

Radiation patterns as in Fig. 2 but for n = 1/1. 01 (—), n = 1 / 2 ( ) , and n = 1 2 ( ) .

FIG. 4
FIG. 4

Angular power distributions P(α) for electric (e) and magnetic (m) dipoles at distances z0 = 0 (—), z0 = λ1/4π (– – –), and z0 = λ1 (· · ·) from the interface. Relative refractive index n = 2 .

FIG. 5
FIG. 5

Polar diagrams of radiation patterns P(α) of electric (e) and magnetic (m) dipoles for n = 1. 5. Distance z0 = 0 (—), z0 = λ1 (– – –). Inset shows radiation pattern of dipoles in unbounded medium 1 on same scale.

Equations (41)

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Ω P ( α ) d Ω = 2 π 0 π P ( α ) sin α d α = L ( z 0 ) / L ,
P ( α ) = ( 3 / 8 π ) sin 2 α
P ( α 2 ) = T 1 , 2 ( α 1 ) P ( α 1 ) d Ω 1 / d Ω 2 .
sin α 1 = n sin α 2 ,
d Ω 1 / d Ω 2 = n 2 cos α 2 / cos α 1 .
T 1 , 2 ( s ) ( α 1 ) = 4 n cos α 1 cos α 2 ( cos α 1 + n cos α 2 ) 2 ,
T 1 , 2 ( p ) ( α 1 ) = 4 n cos α 1 cos α 2 ( n cos α 1 + cos α 2 ) 2 ,
P m ( α 2 ) = 3 8 π n 5 sin 2 2 α 2 ( cos α 1 + n cos α 2 ) 2 ,
P e ( α 2 ) = 3 8 π n 5 sin 2 2 α 2 ( n cos α 1 + cos α 2 ) 2 ,
P e ( α 2 ) = [ cos ( α 1 α 2 ) ] 2 P m ( α 2 ) ,
P e ( α 2 ) P m ( α 2 ) .
k 1 < ( k x 2 + k y 2 ) 1 / 2 = k 2 sin α 2 k 2
L ( z 0 ) L = 3 2 0 ( n 2 1 ) 1 / 2 Im { r 1 , 2 } ( 1 + w 2 ) × exp ( 2 k 1 z 0 w ) d w ,
w k z , 1 / i k 1 = ( n 2 sin 2 α 2 1 ) 1 / 2 .
Im { r 1 , 2 ( s ) } = 2 ( n 2 1 ) 1 ( n 2 sin 2 α 2 1 ) 1 / 2 n cos α 2 ,
Im { r 1 , 2 ( p ) } = [ ( n 2 + 1 ) sin 2 α 2 1 ] 1 Im { r 1 , 2 ( s ) } .
P m ( α 2 ) = 3 8 π n 5 n 2 1 sin 2 2 α 2 × exp [ z ¯ 0 ( n 2 sin 2 α 2 1 ) 1 / 2 ] ,
P e ( α 2 ) = [ ( n 2 + 1 ) sin 2 α 2 1 ] 1 P m ( α 2 ) .
P m ( α 2 , c ) = ( 3 / 2 π ) n ,
P e ( α 2 , c ) = ( 3 / 2 π ) n 3 .
P m ( 45 ° ) = ( 3 / 8 π ) n 5 / ( n 2 1 )
P m ( α 2 = 90 ° ) = P e ( α 2 = 90 ° ) = 0 .
Δ z = ( 1 / 2 π ) λ 1 ( n 2 sin 2 α 2 1 ) 1 / 2 ,
P ( α 1 ) = P ( α 1 ) { 1 + | r 1 , 2 | 2 + 2 | r 1 , 2 | cos ( z ¯ 0 cos α 1 + δ 1 , 2 ) } ,
r 1 , 2 ( α 1 ) = | r 1 , 2 ( α 1 ) | exp [ i δ 1 , 2 ( α 1 ) ]
P m ( α 1 = 90 ° ) = P e ( α 1 = 90 ° ) = 0 ,
r 1 , 2 ( s , p ) ( α 1 = 90 ° ) = 1 .
r 1 , 2 ( s ) ( α 1 ) = cos α 1 n cos α 2 cos α 1 + n cos α 2 ,
r 1 , 2 ( p ) ( α 1 ) = n cos α 1 cos α 2 n cos α 1 + cos α 2 ,
P m ( α 1 ) = 3 8 π sin 2 2 α 1 ( cos α 1 + n cos α 1 ) 2 ,
P e ( α 1 ) = 3 8 π n 2 sin 2 2 α 1 ( n cos α 1 + cos α 2 ) 2 .
P e ( α 1 ) = n 2 [ cos ( α 1 α 2 ) ] 2 P m ( α 1 ) ,
P e ( α 1 ) n 2 P m ( α 1 ) .
| r 1 , 2 ( s , p ) | = 1 ,
cos δ 1 , 2 ( s ) = ( 1 n 2 ) 1 / 2 cos α 1 ,
cos δ 1 , 2 ( p ) = n [ ( 1 + n 2 ) sin 2 α 1 1 ] 1 / 2 cos δ 1 , 2 ( s ) .
P m ( α 1 ) = 3 8 π ( 1 n 2 ) 1 sin 2 2 α 1 ,
P e ( α 1 ) = n 2 [ ( 1 + n 2 ) sin 2 α 1 1 ] 1 P m ( α 1 ) .
P m ( α 1 , c ) = P e ( α 1 , c ) = 4 P ( α 1 , c ) = ( 3 / 2 π ) n 2 .
P m ( 45 ° ) = ( 3 / 8 π ) ( 1 n 2 ) 1 .
P ( α ; n ) = n γ P ( 180 ° α ; 1 / n ) ,