LDOS Contribution from Radiative Modes

4.2 Spontaneous Emission in a One-Dimensional Dielectric Stack

4.2.1 LDOS Contribution from Radiative Modes

Although theintensityof the electric field is a scalar function ofz only, we must treat the fields of the radiative modes as vectors because of the different mode polarizations.

First, we define thez-component of the wavevector in mediumj with refractive index

n_j as

k_jz = k²n_j²−k_x²−k_y²1/2

= k²n_j² −β²1/2

. (4.19)

Note that kjz can be imaginary for modes originating in a medium with index nµ if n_µ > n_j. The transverse polarization vectors for the TE and TM radiative modes are

ˆ

n_k,TE = sinϕˆx−cosϕˆy , (4.20a) ˆ

n^±_k,TM= 1

β²+|k_jz|²1/2 (k_jzcosϕˆx+k_jzsinϕˆy∓βˆz) , (4.20b) where the electric field is in the xy-plane for TE modes, but TM modes have an out-of-plane component that can be positive or negative depending on the direction of propagation.

For the general five-layer stack in Fig. 4.1, the electric field for radiative TE modes can be expressed as

E_k,TE,µ(z) =































U_TE,µ⁽¹⁾ nˆ_k,TEe^ik1zz +V_TE,µ⁽¹⁾ nˆ_k,TEe^−ik1zz , z >0 U_TE,µ⁽²⁾ nˆ_k,TEe^ik2zz +V_TE,µ⁽²⁾ nˆ_k,TEe^−ik2zz , 0> z >−a U_TE,µ⁽³⁾ nˆ_k,TEe^ik3zz +V_TE,µ⁽³⁾ nˆ_k,TEe^−ik3zz , −a > z > −b U_TE,µ⁽⁴⁾ nˆ_k,TEe^ik4zz +V_TE,µ⁽⁴⁾ nˆ_k,TEe^−ik4zz , −b > z >−c U_TE,µ⁽⁵⁾ nˆ_k,TEe^ik5zz +V_TE,µ⁽⁵⁾ nˆ_k,TEe^−ik5zz , z <−c .

(4.21)

For TM modes, the magnetic field is continuous across the index interfaces and ori- ented along ˆn_k,TE. Solving for the electric field by the relationE(r) = _ω(r)ⁱ ∇ ×H(r) yields

Ek,TM,µ(z) =































(^β2+|k1z|²)^1/2nµ

(0/µ0)^1/2kn²₁

h

U_TM,µ⁽¹⁾ nˆ⁺_k,TMe^ik1zz+V_TM,µ⁽¹⁾ nˆ⁻_k,TMe^−ik1zzi

, z > 0 (^β2+|k2z|2)^1/2nµ

(0/µ0)^1/2kn²₂

h

U_TM,µ⁽²⁾ nˆ⁺_k,TMe^ik2zz+V_TM,µ⁽²⁾ nˆ⁻_k,TMe^−ik2zzi

, 0> z >−a (^β2+|k3z|2)^1/2nµ

(0/µ0)^1/2kn²₃

h

U_TM,µ⁽³⁾ nˆ⁺_k,TMe^ik3zz+V_TM,µ⁽³⁾ nˆ⁻_k,TMe^−ik3zz i

, −a > z >−b (^β2+|k4z|²)^1/2nµ

(0/µ0)^1/2kn²₄

h

U_TM,µ⁽⁴⁾ nˆ⁺_k,TMe^ik4zz+V_TM,µ⁽⁴⁾ nˆ⁻_k,TMe^−ik4zzi

, −b > z >−c (^β2+|k5z|2)^1/2nµ

(0/µ0)^1/2kn²₅

h

U_TM,µ⁽⁵⁾ nˆ⁺_k,TMe^ik5zz+V_TM,µ⁽⁵⁾ nˆ⁻_k,TMe^−ik5zzi

, z < −c . (4.22) When written in this general way, Eqs. 4.21 and 4.22 can be expressed in the more compact form:

E_k,TE,µ(z) = U_TE,µ^(j) nˆ_k,TEe^ikjzz+V_TE,µ^(j) nˆ_k,TEe^−ikjzz (4.23) for TE modes, and

E_k,TM,µ(z) = (β²+|k_jz|²)^1/2n_µ (₀/µ₀)^1/2kn²_j

h

U_TM,µ^(j) nˆ⁺_k,TMe^ikjzz+V_TM,µ^(j) nˆ⁻_k,TMe^−ikjzzi

(4.24)

for TM modes, where it should be understood that the field is a piecewise function with subdomains j. The constants U_ζ,µ^(j) and V_ζ,µ^(j) are completely determined using the appropriate boundary conditions at the dielectric interfaces and by imposing the normalization condition. For modes incident from region 1, the only non-zero component in medium 5 is the one traveling in the negative z direction, so, for both polarizations, U_ζ,µ=1⁽⁵⁾ = 0. For the incident plane wave, the normalization condition in Eq. 4.17 yields

V_TE,µ=1⁽¹⁾ = 1 (2π)^3/2

k k_1z

1/2

, (4.25a)

V_TM,µ=1⁽¹⁾ =(₀/µ₀)^1/2 (2π)^3/2

k k_1z

1/2

. (4.25b)

Similarly, for modes incident from region 5, we have V_ζ,µ=5⁽¹⁾ = 0 and

U_TE,µ=5⁽⁵⁾ = 1 (2π)^3/2

k k_5z

1/2

, (4.26a)

U_TM,µ=5⁽⁵⁾ =(₀/µ₀)^1/2 (2π)^3/2

k k_5z

1/2

. (4.26b)

The other U_ζ,µ^(j) and V_ζ,µ^(j) are found, for TE modes, by imposing continuity of the electric field and its first derivative at each dielectric interface and, for TM modes, by imposing continuity of the magnetic field and the in-plane component of the electric field, analogous to the analysis in Chapter 2 for guided modes. Although it is possible to write down these coefficients explicitly, the expressions are very unwieldy for a full five-layer stack. The calculations presented in the following sections have made use of the symbolic solver inMathematica to determine each coefficient in terms ofβ and n_j.

At this point, we can insert Eqs. 4.23 and 4.24 into the expression forρ(z) to cal- culate the LDOS contribution from radiative modes. Since we are primarily interested in the LDOS enhancement, it is useful to normalize ρ(z) by the LDOS of an emitter in a homogeneous medium. The well-know result for the spontaneous emission rate in vacuum is [85]

Γ_vac = ω₀³|D_ab|² 3π₀~c³ = π

₀~

|D_ab|²ρ_vac , (4.27) where we have defined the vacuum density of optical states, ρvac = k₀³/(3π²). Note that in a homogeneous dielectric with index n_ref,

Γ_ref = ω₀³|D_ab|²

3π(₀n²_ref)~(c/n_ref)³ = π

₀~|D_ab|²ρ_ref , (4.28) where

ρ_r =ρ_vacn_ref = k₀³

3π²n_ref . (4.29)

In a structure supporting no guided modes, the radiative spontaneous emission rate enhancement relative to an emitter embedded in a homogeneous reference materials is then

Γ(z)

Γ_ref = ρ(z)

ρ_ref = 2π³ k²₀n_ref

X

µ=1,5

Z k0nµ

0

U_TE,µ^(j) e^ikjzz+V_TE,µ^(j) e^−ikjzz

2

+(β²+|k_jz|²)n_µ (₀/µ₀)kn²_j

U_TM,µ^(j) e^ikjzz+V_TM,µ^(j) e^−ikjzz

2

βdβ . (4.30) We note that the prefactor for the TM-mode part of the integrand in Eq. 4.30 has two terms: the term proportional toβ²is due to the in-plane electric field component while the term proportional to|k_jz|²is due to the out-of-plane component. Thus, the sum of the TE-mode term and the in-plane TM-mode term represent the LDOS contribution from dipoles oriented in the x- and y-directions, ρ^k(z), while the contribution from dipoles along the z-direction,ρ^⊥(z), is given by the out-of-plane TM-mode term. We can then explicitly express the LDOS enhancement in terms of contributions from dipoles parallel and perpendicular to the plane of the dielectric interface:

ρ(z)

ρ_ref = ρ^k(z)

ρ_ref + ρ^⊥(z)

ρ_ref . (4.31)

As an experimentally relevant test structure supporting only radiative modes, a single Si-SiO₂ interface is analyzed using the equations outlined above, where the waveguiding layers (regionsj = 2, 3, and 4) have zero thickness, as shown in Fig. 4.2.

A wavelength of λ₀ = 2π/k₀ = 1537 nm is considered, corresponding to the ⁴I_13/2 to ⁴I_15/2 transition for Er³⁺ ions embedded in glass (the material system considered experimentally in Section 4.3). The relevant indices are n₁ = 1.444 for SiO₂ and n₅ = 3.476 for silicon.

The integrals in Eq. 4.30 were integrated numerically in Mathematica for values of z in 1-nm increments. For µ = 5, there is a singularity in the integrand at β = k₀n₁, which is a standard problem when considering modes incident from the higher- index region of geometries with asymmetric cover and cladding layers. By simply

Figure 4.2. Calculated radiative spontaneous emission rate enhancement near a Si-SiO₂interface as a function of dipole emitter position assuming an isotropic distribution of dipole orientations and a free-space emission wavelength of λ0 = 1537 nm. In both panels, the total LDOS enhancement is plotted in black. In the upper panel, the fractional contributions from TE- and TM-polarized radiative modes are shown, while the lower panel shows the fractions contributed by dipoles parallel and perpendicular to the interface.

excluding the singularity point in the integration domain, the numerical integration will converge reliably.

Figure 4.2 shows the total LDOS relative to vacuum across the Si-SiO₂ interface.

Also shown are the fractional contributions from dipoles parallel and perpendicular to the plane of the interface and the contributions from radiative TE and TM modes.

Note that the TE contribution is continuous and slope continuous across the interface, while the contribution from dipoles parallel to the interface is continuous, but with a slope discontinuity due to the TM-mode contribution. Perhaps the most striking feature is the discontinuity in the LDOS enhancement due to dipoles perpendicular to the interface, which, very close to the interface, is even higher in the low-index medium than the bulk rate enhancement in the high-index medium. Finally, as expected, the total LDOS tends toward ρ_vacn_j for dipole positions far from the interface.