1. Introduction: Maxwell's Equations and Plane Waves

(1)

1. Introduction: Maxwell's Equations and Plane Waves

In what follows, we will consider the electrodynamic interaction of fluorescing molecules with an inhomogeneous environment. Although the absorption and emission of light is a thoroughly quantum-mechanical process, many aspects of this interaction can be studied within the so called semi-classical framework, which is based on Maxwell’s electrodynamics. The core idea of this semi-classical approach is to handle the molecule as a classical oscillating electric dipole, which is an excellent approximation for most molecules of practical interest (corresponding to the fact the most electronic transitions involved in fluorescence are dipole transitions). All calculations of light absorption and emission are calculated by solving Maxwell’s equations for the electromagnetic field in the presence of such a dipole. The term “semi” refers to the fact that the results of these calculations are then interpreted from a quantum-mechanical point of view.

Thus, from a technical point of view, what we have do deal with are Maxwell’s equations, which read (in cgs units, which will be used throughout this tutorial)

( )

^ε^E ⁼⁴^πρ

div

=0 B div

t c ∂

− ∂

= B

E

rot 1

( )

^E _j

rot B

c t c

+ π

∂ ε

= ∂



 µ

4 1

where E and B are the electric and magnetic field, r and j are the electric charge and current density, ε and µ are the dielectric susceptibility and magnetic permeability of the medium, and c is the speed of light. For a singular dipole at the center of the coordinate system r=0 and oscillating with frequency ω, one has

=0

ρ and ^j⁼⁻ⁱ^ω^p^δ

( ) (

^r ^exp ⁻ⁱ^ω^t

)

^,

with ^δ

( )

^r Dirac’s delta function in three dimensions, and p the dipole amplitude vec- tor. However, before we consider in detail the solution of Maxwell’s equations for the

(2)

currents), which are plane electromagnetic waves, and study the interaction of these plane waves with a planar discontinuity of the dielectric susceptibility of the medium.

With other words, we study the interaction of plane electromagnetic waves with an interface. The results of these studies will be of fundamental importance for the subsequent considerations of the oscillating dipole and its interaction with such an interface.

We seek the solution for the electromagnetic field within a source-free medium in the form of plane waves, ^E^,^B^∝^exp

(

ⁱ^k^⋅^r⁻ⁱ^ω^t

)

^{, where}k and ω are the wave vector and oscillation frequency of the waves. For simplicity, we will always consider media with unity magnetic permeability, µ=1, which is in excellent approximation for most mate- rials of practical interest. By direct insertion into Maxwell’s equations a presented above, we find the relations

=0

⋅ ε

= εE k E

div i

B k B div = ⋅

B B E

k E

rot c

i t

i c = ω

∂

− ∂

=

×

= 1

E E B

k B

rot c

i t

i c =− εω

∂

= ε

×

=

These relations show, first at all, that E , B , and k are mutually orthogonal one to another. Next, applying the rotor a second time to the third equation and using the first and last equation leads to

(

^k ^E

)

^E ^rot^B ^E

k E rot

rot ₂

2 2

c c

k =iω =εω

=

×

−

=

thus defining the amplitude of the wave vector as k ≡k = εω c≡nω c (which is also called the dispersion relation), and relating the amplitude of the electric and magnetic field B n= E. Here, the so called index of refraction n= ε was introduced.

Thus, plane waves are indeed solutions of Maxwell’s equations, as long as the relations B

E⊥ , E⊥k, B⊥k, B n= E and k =nω c hold. As free variables we still have the electric field amplitude, E, the oscillation frequency ω, and the propagation direc-

(3)

tion given by the unit vector kˆ =k k (throughout the tutorial, we will use a hat for symbolizing unit vectors).

An important point is that the found plane wave solutions form a complete set of solutions: any solution of the electromagnetic field within a source-free homogeneous (ε=const.) medium can be represented by a superposition of these plane waves. In subsequent chapters, we will learn of alternative complete systems of solutions that are more adapted to problems with cylindrical or spherical symmetry. The plane wave solutions, however, are perfect when studying problems involving only planar parallel interfaces, and the simplest problem that we will consider next is the interaction of a plane wave with a planar interface.

(4)

2. Plane Waves and Planar Boundaries: Fresnel’s Relations

Let us consider the most general case of such plane waves at a planar interface dividing two media with index of refraction n and _j n . There are up to four interconnected _j₊₁ plane waves (see figure below), denoted by E and ^±_j E , where the subscript refers to ^±_j₊₁ the medium, and the ±-sign to whether the wave travels into a positive direction (from medium j towards medium j+1) or into a negative direction (from j+1 towards j). If the electric field vectors are within the plane of the directions of wave propagation, the wave is called transversal electric (TE) or p-wave; if the electric field vectors are parallel to the boundary, the wave is called transversal magnetic (TM) or s-wave.

−

kj +

j

Ep,

− j

Ep ,

−p,j+1

E

+p,j+1

E

+j

Es ,

−j

Es,

+,j+1

Es

−,j+1

Es

+

kj

−

kj

+j+1

− k

+1

kj

To get a connection between these four plane waves, we have to consider the boundary conditions. First of all, the plane waves represent periodic structures of the electric field in space. Along the boundary, the periodicity of the electric field on both sides of the boundary has to be the same. The periodicity is given by the projections of the wave vectors k^±_j_,_j₊₁ onto the boundary. Let us denote these components of the k^±_j_,_j₊₁ as q , and the modulus of the component perpendicular to the boundary as w_j_,_j₊₁ (see figure below). Pay attention to the fact that q is a two-dimensional vector within the plane of the boundary, and it is the same for all four plane waves, whereas w_j_,_j₊₁ has the same value on the same side of the boundary, but is different on different sides.

(5)

z

w_j

w_j _ z

z

+j

k −

kj

q q

q

As we have seen in the preceding section, the lengths of the wave vectors are proportional to the index of refraction, n_j_,_j₊₁. Setting for convenience the length of the wave vector in vacuum to unity, we thus have the general relation

2 1 , 2 1 , 1

,j+ = j j+ = + jj+

j k q w

n .

Furthermore, the angle of incidence of the plane waves is given by

1 , 1

sinθ_j,_j₊ =q n_j _j₊ ,

which yields directly Snell’s law of refraction because q is the same in all media.

Next, we take into account the boundary conditions for the electric field: The electric field components parallel to the boundary, as well as the products of the dielectric constant (ε_j =n²_j) with the components perpendicular to the boundary, have to be the same on both sides of the boundary.

Let us first consider the p-wave case. Simple geometry gives the components of the electric field parallel and perpendicular to the interface as ±w_j n_jE^±_p_,_j and q n_jE^±_p_,_j, respectively (the same holds for j+1). Thus, the boundary conditions can be written in a compact matrix form as:















 −

=















 −

− + + + +

+

+ + +

+

− +

1 ,

1 1

1 1 1

1

, ,

j p

j j

j j j

j

j p

j j

j j j j

E E n

n

n w n

w E

E n

n

n w n w

where q was cancelled out. The inverse of the matrix on the l.h.s. is given by:

w n w n

n n w

n w n

k k k k

k k k

 −

 

 = −

 



−1 1

2 .

Thus, the connection between the electric field components is explicitly given by:

(6)













 



+ +

−

+

−

= +















 



 −



 





= −













− + + +

− + + + +

+

+ + +

+

− +

1 ,

1 1

1 1 1

1

, ,

2 1

1 1 2

1

j p

j j

j j j

j

j j j

j j

j

j p

E E n n w n n w

n n w n n w

E E n

n

n w n

w n w n

n w

n E

E

where the short-hand notation w≡w_j₊₁ w_j and n≡n_j₊₁ n_j is introduced.

Consider the special case of a plane wave incident from the j-side. Then, there will be no E component, but a reflected ⁻_j₊₁ E and a transmitted ⁻_j E component. We thus ⁺_j₊₁ find









+ +

−

+

−

= +











 ⁺ ₊

− +

2 0

1 _, ₁

,

, p j

j p

j

p E

n n w n n w

n n w n n w E

E ,

and for the electric field amplitude in j+1:

1 1 1

, 1 1

, ,

1 , 1 ,

ˆ 2 ˆ 2

w n w n

E w n n n

n w

E

j j j

j p j j j j p j p j

p j p

+ +

+ + + +

+ + +

+ + = +

=ê + ê

E

with

1 1 1 ,

ˆ ˆ ˆ

+ + +

+

= −

j j j

p n

q

w q z

ê a unit vector along E⁺_p_,_j₊₁, where q ˆˆ,z denote unit vectors along q and perpendicular to the boundary, respectively. Finally, the spatial field dependence is given by the exponential term ^exp

(

ⁱq⋅ñ+^iw^{j 1}+ ^z

)

where z is the coordi- nate perpendicular to the boundary, and ρρ the two-dimensional coordinate vector within the boundary plane. An important special case is when w becomes imaginary, _j₊₁ leading toe an evanescent wave on the j+1-side. This happens if

(

² ²

)

⁰

2 1 2 2

1 2

1= ₊ − = ₊ − − <

+ j j j j

j n q n n w

w ,

which shows that a condition for the emergence of an evanescent wave is n_j₊1 <n_j. The angle of incidence for which w_j = n²_j₊₁−n²_j is called the critical angle, where total reflection starts and evanescent waves emerge.

The reflection and transmission coefficients are defined as

(7)

+

−

=

j p

E R E

,

, and ₊

+ +

=

j p

E T E

, 1 ,

so that we obtain:

w n

w R n

+

= ₂²− and

w n T n

= ₂2+ .

An important peculiarity of total internal reflection is the occurrence of a phase jump between incident and reflected wave, which can be seen by writing the electric field components E_p^±_,_j as (see matrix equation for the boundary conditions above)

+ +

−

+ + +



 



− +

=



 

 +

=

1 , ,

2 1 2 1

j p j

p

j p j

p

E n n E w

If w is purely imaginary, the factor in the brackets adds an additional phase _j₊₁ φ^± to the E^±_p_,_j components, with

( )

ⁱ^φ^± ^∝^±^w_n ⁺ⁿ

exp or

2

Im Re

Im

tan n

w n n

w n n w

= ±



 



± +



 



± +

=

φ^± ,

so that



 



− 

=



 



− 



 

−

= φ

− φ

= φ

∆ ⁻ ⁺ ₂ ₂ Im₂

arctan Im 2

arctan arctan Im

n w n

w n

w .

Very similar considerations hold for s-waves. Because all electric field vectors are now parallel to the interfaces, one uses a second boundary condition involving the continu- ity of the parallel component of the magnetic field across boundaries. For a plane wave

(8)

with electric field vector E parallel to the boundary, the parallel component of the _s,_j magnetic field is given by ^q^ˆ^⋅

(

^k^ˆ ^j^×^E^s^,^j

)

⁼^w^j^E^s^,^j. Thus, the boundary conditions read:













 



= −













 



− ⁻ ₊

+ + +

− + +

1 ,

1 , 1

, 1 1

1 1

j s

j j j

s j s

j

j E

E w E w

E w

w ,

so that, with





−

− −

 =

 



−

1 1 2

1 1

1 ¹

j j

j w

w w w

w ,

one finds

1 . 1

1 1 2 1

1 1

1 1 2 1

1 ,

1 , 1

,













 



+

−

= +















 





 −



 





= −













− + + +

− + + + +

− + +

j s

j j

j s

E E w w

w w

E E w w

w w E

E

The reflection and transmission coefficients at a single interface

(

^j^,^j⁺¹

)

^{now are}

w R w

+

= − 1

1 and T

= w +

2

1 ,

and the phase jump under total reflection is ∆φ = −2 arctan Im( w , and we again em-) ployed the short-hand notation w≡w_j₊₁ w_j and n≡n_j₊₁ n_j .

(9)

3. Oscillating electric dipole in a homogeneous environment

After our exceeding review of plane electromagnetic waves and their interaction with planar interfaces, let us now return to the electromagnetic field of a free electric dipole within a homogeneous environment.

Starting point of our considerations is the fourth of Maxwell’s equations for a oscillating electric dipole at the coordinate system’s origin, ^r⁼

(

⁰^,⁰^,⁰

)

^{, thus}

( )

^r

p E

j E B

rot =− ωε + π =− ωε − πω δ c

i c

c

i 4 4

.

Together with the third of Maxwell’s equations, ^rot^E⁼

(

^iω ^c

)

^B, this can be rewritten to

( )

^r

p E

E rot

rot −εω₂² = 4πω₂ ² δ c

c .

Passing to Fourier space yields

(

^k ^E

)

^E

( )

^E ^k

⁽

^k ^E

⁾

^p

k× × −εk₀² = k² −εk₀² − ⋅ =4πk₀²

− ,

where the definition k0 =ω c was used. Multiplying both sides by k gives a expres- sion for k⋅E:

p k E

k ⋅

ε

− π

=

⋅ 4

so that

( ) [

^p ^k

^{( )}

^k ^p

]

E ε − ⋅

ε

− ε

= π ₂ ₀²

0 2

4 k

k k

when switching back from Fourier to direct space, E is given by

( ) ( )

^exp

( ) [ ( ) ]

^.

2

4 2

2 0 0 2 3 3

p k k r p

k r k

E ε − ⋅

ε

−

⋅ ε π

= π

∫

^d _k ⁱ _k ^k

Representing k in appropriately oriented spherical coordinates

(

^k^,^η,^ψ

)

^{so that}

ψ

=

⋅r krcos

k , this result can be rewritten as

(10)

( ) ( ) ( ) [ ( ) ]

( )

^sin ^exp

⁽

^cos

⁾

^,

1

cos sin exp

2 8

2 0 2 0

2

0 2

0

2 2 0

0 2 0

2

0 3 2

k k d ikr

k dk k

k k k d ikr

k dk

ε

− ψ ψ

ψ +

επ ε

=

⋅

− ε ε

− ψ ψ

π ψ ε

= π

∫

π

∞ π

∞

p div grad

p k k p r

E

where the integration over η yielded a factor 2π. Let us next consider the integrals only. The integral over ψ can be taken directly, giving

( ) ( ) ( ) ( )

2 0 2 2

0 2 0

2 0 2 0

2

0

exp 1

exp exp

1 cos sin exp

−

,

or

( )

^r ⁼_ε¹

(

^ε^k⁰² ⁺^grad^div

)

_^^p^exp

⁽

^ink_r ⁰^r

⁾

_^

E .

Without restriction of generality, let us calculate the action of the differential operators in a spherical co-ordinate system

(

^r^,^θ,^φ

)

with its polar axis θ=0 oriented along p.

The we have (using in the following the abbreviation k=nk₀)

(11)

( ) ( ) ( )

^ikr

r r ik r

r r

ik r k

r ikr r ik r

ikr

1 exp ˆ

1ˆ ˆ 2 ˆ

2

ˆ exp 1

exp

2 3

2 2

2



 



 



 

 −

×

−

 ⋅

 

− − +

=



 



  ⋅



 

 −

=

 



p r r r r p

r p grad

p div grad

or finally

( ) ( ) ( ) ( ) ( )

kr ikr kr kr

i kr kr

k i

k 1 exp

ˆ 1 3 ˆ

1 3 ₂ ₂

2

0 









 

 + − +

 ⋅

 

− − +

= rr p p

r

E .

This result describes the field of electromagnetic waves propagating away from the dipole’s position (always remember that there is the factor ^exp

(

⁻^iω^t

)

present which we have all the time omitted). The next figure shows the contours of constant field amplitude ^E

( )

^r for a vertical dipole orientation.

^ikr

E − ⋅

and

( ) ( ) ( ) ^{( )}

( ) ( ) ( )

^ikr

r k k r

ikr i

k

r k ikr

ik ik

ˆ exp exp ~

exp 1

1 1

0 0

2

0 0

p r p

rot

p div grad rot

r E rot r

B

 ×

 

= 



 ε +

=

,

so that for the far-field Poynting vector we obtain

[ ( ) ] ( )

2

[

²

( )

²

]

4 0 2

4

0 ˆ ˆ

ˆ 8 ˆ

8 ˆ

~ p r r p r p r r p

S − ⋅

= π

×

⋅

π − p

r cnk r

cnk .

Thus, the radiation points away from the dipole’s position, and has the angular distribution

π θ φ =

θ θ

2 2 4 0 2

8 sin sin

p cnk d

d S

d .

The total power of emission is obtained by integrating over all angular directions,

2 4 0 2

2 2 0 2

0

0 3

sin 1

sin 8 cnk p

n p d ck d

S_tot θ=

φ π θ θ

=

∫

^π

∫

^π ^.

What do we learn from these results? The angular distribution of radiation of a dipole follows a simple sin²θ-dependence, where θ is the angle between dipole orientation and radiation direction. This distribution is shown in the net figure.

(13)

Thus, there is no far-field energy transport along the direction of the dipole axis. The total emitted power is proportional to k₀⁴ ~λ⁻⁴, thus inversely proportional to the fourth power of the wavelength. Because the electrodynamics of light scattering on dipoles follows a similar mathematics as developed here, we have recovered the well- known result that scattering intensity (scattering cross section) increases inversely proportional to the fourth power of the wavelength. Finally, we have the interesting result that the total power of emission depends also on the refractive index n of the embedding medium: This is the first encounter of the phenomenon that the dielectric properties of the environment can have a direct impact on the emission properties of a dipole (and thus on those of a fluorescing molecule).

Until now, we were only interested in the purely mathematical derivation of the electromagnetic field of an oscillating electric dipole. Now, it is time to stop and to consider the connection of the found results with the emission of a fluorescing molecule.

Nearly all fluorescing molecules of practical interest are electric dipole emitters. Of course they are also quantum-mechanical entities in the sense that they are not emitting a continuous train of energy but emit their energy in quantized units, photons. These photons are characterized by their energy (which is directly connected with the oscillation frequency ω by E=hω) and polarization (corresponding to the electric field vector orientation in our consideration). However, what does, for example, the angular distribution and total power of emission as derived above tell us about the single molecule’s fluorescence? The angular distribution of emission, as derived above within the framework of Maxwell’s electrodynamics for an oscillating dipole, corresponds to an angular probability distribution of photon emission: By repeatedly exciting a molecule and measuring the direction of subsequent photon emission, one obtains the same angular distribution as that calculated above. The total power of emission has a similar quantum-mechanical interpretation: The larger that value, the faster the energy is emitted away from the molecule, which means the shorter is the (average) lifetime of the excited state (fluorescence decay time). This correspondence between classical quantities such as angular distribution of radiation or total emission power, and quantum- mechanical quantities such as angular distribution of photon emission or lifetime of the

(14)

4. Förster Resonance Energy Transfer

In the conclusion of the preceding section, we considered the far-field emission of the dipole, neglecting the so called near-field components of the electric (and magnetic) field that fall off faster than r . Here, we will concentrate on these near-field compo-⁻¹ nents, and we will study the action of an oscillating dipole onto another electric dipole causing the hopping of the energy from an optically excited molecule to an excitable molecule in its ground state (Förster Resonance Energy Transfer, or FRET).

Starting point is the r -near-field part of electric field of the oscillating electric dipole ⁻³ (cf. with results of previous section):

( ) [ ( ) ]

2

^{( )}

3

ˆ exp 3ˆ

r n

r _d _d ikr

d r r p p

E ≈ ⋅ − .

Here, the subscript d indicates that the quantities refer to a so-called donor molecule, which will act by its electromagnetic near-field on a suitable acceptor molecule, if there is so-called spectral overlap between the donor’s frequency of emission and the acceptor’s frequency of optical absorption.

The excitation rate of a potential acceptor molecule is given by the product of its absorption cross section and the number of (virtual) photons per area per second coming from the donor. This photon flux is calculated as the Poynting energy flux, proportional to

(

nc ⁸π

)

⋅Ed² divided by the energy per photon λ hc. Taking furthermore into account that the acceptor dipole excitation is proportional to the scalar product of acceptor dipole (p ) orientation and exciting electric field (_a E ) _d orientation, and that the macroscopically measured absorption cross section σ is averaged over all possible dipole orientations (and thus only 31 of its maximum value), one finds for the acceptor excitation rate k : _a

( )

a d d

a hc

k nc λ ⋅ φ

σ π

= ˆ ²

3 8 p E .

where φ_d is the donor’s fluorescence quantum yield (the probability that the donor’s energy is given away electromagnetically and not by e.g. molecular collisions). Insert- ing the explicit expression for E yields _d

(15)

( )( ) ( )

² ² ₃ ₆ ²

6 3 2

8 ˆ 3

ˆ ˆ ˆ ˆ 3ˆ 8

3 φ κ

π

= σλ

⋅

−

⋅ φ ⋅

π

= σλ

r n h p r

n h

k_a p^d ^d r p_a r p_d p_a p_d ^d ^d .

which defines the so called orientation factor κ as

( )( ) ( )

[ ]

²

2 2 2

sin sin cos cos

cos 2

cos cos

cos 3

ˆ ˆ ˆ ˆ ˆ 3 ˆ

d a d

a d a

d a d a

θ θ φ

− θ θ

=

φ

− θ θ

=

⋅

−

⋅

=

κ r p r p p p

Here, θ_a and θ_d are the angles between the donor and acceptor dipoles and the connecting vector r , φ is the angle between the two dipoles, and φ the angle between the planes formed by the donor dipole and r and the acceptor dipole and r , see figure.

θd

φ

r

p

_d

p

a

θa

If the donor and acceptor can freely rotate, one obtains an averaged value for κ² as

[ ]

3 2 9 4 2 1 9 4

sin sin

cos sin

sin cos cos cos 4 cos

cos 4

sin sin cos cos

cos 2

2 2

2 2 2

= +

=

θ θ

φ + θ θ φ θ θ

− θ θ

=

θ θ φ

− θ θ

= κ

d a

d a d

a d

a

d a d

a

In the above equation for k , one has still the a priori unknown quantity _a p . It can be _d found by looking at the total energy emission of the free donor. From a quantum mechanical point of view, its energy is that of one photon, hc λ. From an classical electromagnetic point of view, it is given by the emission rate of a dipole with amplitude

p times the average lifetime of the donor’s excited state, d τ_d. Equaling both values yields:

d dk cn

hc = p τ

λ

4 0 2

3 1

(16)

d d

d k n

h n

k p h

τ

= π τ

= π ₃

0 3

0 2

2 3 2

3

and

d d

a k n r k n r

h

k h κ φ

τ π

= σ φ τ κ

π π

= σλ ₄ ₄ ₆ ²

0 2

6 4 3

0 8

9 2

3 8

3 .

Until now, donor emission and acceptor excitation was assumed to happen at exactly one wavelength. The complete acceptor excitation rate has to be integrated over all wavelengths, leading to

( ) ( ) ( )

∫ ( )

∫

λ λ

λ λ σ λ φ λ

τ κ π

= π

d d d

d tot

a d F

F d r

k n

4 2

6 4 4

, 8 2

9

where ^F^d

( )

^λ

∫

^d^λ^F^d

( )

^λ is the normalized emission spectrum of the donor. The so called Förster radius is given by

( ) ( ) ( )

∫ ( )

∫

λ λ

λ λ σ λ φ λ

π π

= κ

d d

d d F

F d R n

4

4 4 2 6

0 8 2

9 ,

so that

6 0 ,

1 



 





=τ r k R

d tot

a .

Taking into account the relation between absorption cross section (in cm²) and molar extinction ε (in l/cm/mol),

ε ε≈

= σ

A

A N

N

2303 10

ln 10³

,

one finds the usual textbook expression for k_a_,_tot

( ) ( )

∫ ( )

∫

λ λ

λ λ ε λ λ τ

φ κ π

= ⋅

d a d

d d A

tot

a d F

F d r n k N

4

6 2

4 , 5

128 10 ln

9000 .

In practice, FRET can be measured by either intensity or lifetime measurements. The fluorescence intensity of the donor is given by the ratio

(17)

a nr f

f

da k k k

I k

+

~ + ,

where k and _f k are the radiative (fluorescent) and non-radiative (e.g. molecular col-_nr lision induced) transition rates of the free donor molecule from its excited to its ground state. The subscript da refers to the donor intensity in presence of the acceptor. The intensity of the acceptor is given directly by I_a ~φ_ak_a, where φ_a is the acceptor’s fluorescence quantum yield. Thus, we find the relations

6 6 0

6

r R

r k

k k

k I

I I

a nr f

a d

da d

= + +

= +

−

where I is the donor intensity at equal excitation and detection conditions, but in ab-_d sence of the acceptor. If R is known a priori, this relation can be used to infer the ₀ value of r from intensity measurements.

An alternative is to use lifetime measurements. As already mentioned, the fluorescence lifetime or lifetime of the exited state is the inverse of the total transition rate from the excited to the ground state. Thus, for the lifetimes τ_da and τ_d of the donor in presence and in absence of the acceptor we find the relation

6 6 0

6

r R

r k

k k

k

a nr f

a

d da d

= + +

= + τ

τ

−

τ ,

which can again be used for determining r.

(18)

5. Fluorescing molecules on dielectric planar substrates

Until now, we have considered only an emitting molecule within a homogeneous environment or close to another absorbing molecule. Nonetheless, we have already seen that the properties of the molecule’s environment, ether represented by the refractive index of the embedding medium either by the optical-absorption capability of a nearby molecule, change the fundamental emission properties such as the excited state lifetime. Here, we will look closer onto this phenomenon, and more specifically, we will study the impact of the presence of a planar substrate on the molecule’s emission properties.

E ε − ⋅

ε

−

⋅ ε π

= π

∫

² ⁰²

0 2 3

3 exp

2

4 k

k k

i d

m m m

,

where R=r−r₀, and the index m refers always to characteristics of the medium where the dipole is located. We will employ cylindrical coordinates with their z-axis perpendicular to the interface and pointing from the dipole towards the interface. The general geometry and several vectors used are shown in the following scheme.

emp

e_s

e_x e_y e_z

km

e_s

ep

k η

ψ

η_m ^e^mp e_s

− +

+

km⁻

dipole position

Performing the integration along the wave vector component that is vertical to the interface, along a closed contour in the complex plane similar to that employed in Sec- tion 3, Cauchy’s theorem yields

(19)

( )

[ ] ( ⁽

0

⁾

0

)

2 2

2 k exp i i z z

w d i

m m

m m m m

− +

−

⋅

⋅ πε −

=

∫

^q ^p ^k^± ^k^± ^p ^q ^ñ ^ñ ^w

E ,

where we have introduced cylindrical co-ordinates R=

(

ñ−ñ0,z−z0

)

and

(

m

)

m = ±w

± q,

k , and the abbreviations w_m = k_m² −q² and k_m = ε_mk₀ are used (w _m being value of the vertical k-component at the pole of the integrand’s denominator).

The ±-superscript in k in accordance whether the target point r is below or above ^±_m the dipole’s position, and the value of w has always non-negative imaginary part, _m corresponding to the fact that we admit only waves propagating away from the dipole, see also figure above. The found representation is the so called Weyl representation of the electric field of an oscillating dipole. This representation is perfectly suited to study the interaction of the dipole with a planar substrate at z=0 . Upon interaction, every plane wave in the Weyl representation is refracted and diffracted at the interface ac- cording to the formulas derived in Section 2.

Let us consider the case of a simple interface dividing the upper medium (z < 0) with dielectric constant ε_m from a lower medium (z > 0) with dielectric constant ε. Firstly, we separate the p- and s-waves within the Weyl representation above. Let us introduce two unit vectors e and ⁺_mp e that are perpendicular to each other and to the wave vec-_s tor k , and with _m e being parallel to the interface (cf. figure above). These unit vec-_s tors correspond to the polarizations of the p- and s-components of the plane wave propagating along k and allow the recasting of the Weyl representation into the form _m

( ) ⁽ ⁾

[ ] [ ⁽

⁰

⁾

⁰

]

2

0 exp

2 i iw z z

w d ik

m s

s mp

mp m

− +

−

⋅

⋅ + π ⋅

=

∫∫

^q ê^± ê^± ^p ê ê ^p ^q ^ñ ^ñ

E .

The unit vectors themselves have the explicit forms (in Cartesian co-ordinates, as shown in above figure)

(

m m

)

m

(

m m m

)

mp = w ψ w ψ −q k = ψ η ψ η − η

+ cos , sin , cos cos ,sin cos , sin

e ,

(

⁻^sin^ψ^,^cos^ψ^,⁰

)

=

×

=

×

= ⁺_m ⁺_mp _m ⁻_m ⁻_mp _m

s k e k k e k

e .

(20)

Secondly, every plane wave in the above representation is reflected and transmitted at the interface, with reflection and transmission coefficients R_p,_s and T_p,_s for plane p- and s-waves that are given by (see Section 2)

,

, w w

w R w

w w

w R w

m s m m m

m

p m +

= − ε

+ ε

ε

−

= ε

and

w w T w w w

nw T n

m s m m m

m

p m = +

ε +

= ε 2

2 ,

.

Now, it is straightforward to write down expressions for the electric fields transmitted through and reflected at the surface:

( ) ⁽ ⁾

[

^T ^T

] [

ⁱ

⁽ ⁾

^iw ^z ^iwz

]

w d ik

m s

s s mp

p p m

T ⋅ + ⋅ ⋅ − + +

= ₂π⁰²

∫∫

^q ê ê⁺ ^p ê ê ^p êxp ^q ^ñ ^ñ⁰ ⁰

E

( ) ( )

[

^R ^R

] [

ⁱ

( )

^iw ^z ^iwz

]

w d ik

m s

s s mp

p mp m

R ⋅ + ⋅ ⋅ − + −

= ₂π⁰²

∫∫

^q ê⁻ ê⁺ ^p ê ê ^p êxp ^q ^ñ ^ñ⁰ ⁰

E