Theory of light emission from a dipole source embedded in a chiral sculptured thin film

Tom G. Mackay; Akhlesh Lakhtakia

doi:10.1364/OE.15.014689

1. Introduction

A chiral sculptured thin film (CSTF) is an assembly of parallel helical nanowires grown on a substrate by physical vapor deposition techniques [1, 2]. At optical wavelengths, a CSTF may be viewed as a unidirectionally nonhomogeneous continuum that locally possesses or-thorhombic symmetry and globally is structurally chiral [3, Chap. 9]. Just like cholesteric liquid crystals [4], CSTFs also exhibit the circular Bragg phenomenon; i.e., a structurally right/left- handed CSTF of sufficient thickness almost completely reflects right/left-circularly polarized (RCP/LCP) light which is normally incident, but LCP/RCP light is reflected very little, within a wavelength regime called the Bragg regime. This phenomenon has been exploited to use CSTFs as circular polarization [5] and spectral-hole [6] filters, among other applications [7, 8].

The multiscale porosity [9] of CSTFs makes them highly promising platforms for optical sensing applications [10, 11]. Unlike porous-silicon platforms [12, 13], CSTFs can be used to generate and detect signals with a high degree of circular polarization and a narrow bandwidth [14, 15, 16]. This attribute is going to be significant for optical biosensors that exploit luminescence [17, 18], as well as for light-emitting devices with precise control over the circular polarization state and the emission wavelengths [19].

Photoluminescence and electroluminescence are common phenomenons wherein a material emits light in response to optical illumination and electrical excitation, respectively. These phenomenons underlie not only the commercial use of fluorescent tagging and identification techniques [20], phosphorescent dyes in displays, and dye-doped lasers [21], but are also responsible for the extraordinary interest of researchers in quantum dots these days [22]. Layers of quantum dots have indeed been inserted in CSTFs, and preliminary experiments have upheld the promise of quantum-dot-impregnated CSTFs as sources of circularly polarized light [16, 19].

These experimental results prompted us to continue the development of a theory of light emission from sources embedded in CSTFs. Earlier research had been confined to a transversely uniform distribution of sources that could emit light only normally with respect to the two pupils of a CSTF [10, 23]. In contrast, in this paper, we are concerned with emission from a single point-dipole source embedded in a CSTF. Light would then be emitted in all directions, which necessitates the use of a full–fledged spectral–Green–function formalism [24, 26].

The plan of this paper is as follows. Section 2 contains a detailed derivation of the theory of light emission by a point electric dipole embedded in a CSTF. Section 3 provides key numerical results that illustrate the chief features of light emitted by a dipole–containing CSTF, and discussions thereof. An exp(-iωt) time-dependence is implicit, with ω as the angular frequency; vectors are denoted in boldface, with the Cartesian unit vectors denoted by u _x,y,z; dyadics are double–underlined; and column vectors as well as matrixes are denoted in boldface and enclosed within square brackets. The permittivity and the permeability of free space are denoted by ε₀ and μ₀, respectively; $k_{0} = ω \sqrt{ε_{0} μ_{0}}$ is the free-space wavenumber; λ₀ = 2π/k ₀ is the free–space wavelength; and $η_{0} = \sqrt{μ_{0} / ε_{0}}$ is the intrinsic impedance of free space.

2. Theory

The theory of planewave reflection and transmission involving a CSTF is comprehensively described elsewhere [3, Chap. 9]. It has been applied to develop a spectral-Green-function formalism for radiation inside a CSTF [26] as well as to investigate the shielding of a point–dipole source by a CSTF [27]. Here we extend these earlier treatments in order to accommodate radiation by a point–dipole source embedded inside a CSTF.

Suppose the region -L ≤ z ≤ L is occupied by a CSTF whose frequency–domain constitutive relations are

D (r, ω) = ε_{0} {\underset{͇}{ε}}_{r}^{CSTF} (z, ω) ∙ E (r, ω), B (r, ω) = μ_{0} H (r, ω) .

The relative permittivity dyadic of the CSTF is written in dyadic notation as [3, Chap. 9]

{\underset{͇}{ε}}_{r}^{CSTF} (z, ω) = {\underset{͇}{S}}_{z} (z, h) ∙ {\underset{͇}{S}}_{y} (χ) ∙ {\underset{͇}{ε}}_{ref} (ω) {\underset{͇}{∙ S}}_{y}^{- 1} (χ) ∙ {\underset{͇}{S}}_{z}^{- 1} (z, h),

where

{\underset{͇}{ε}}_{ref} (ω) = ε_{b} (ω) u_{x} u_{x} + ε_{c} (ω) u_{y} u_{y} + ε_{a} (ω) u_{z} u_{z},

{\underset{͇}{S}}_{y} (χ) = (u_{x} u_{x} + u_{z} u_{z}) \cos χ + (u_{z} u_{x} - u_{x} u_{z}) \sin χ + u_{y} u_{y},

{\underset{͇}{S}}_{z} (z, h) = (u_{x} u_{x} + u_{y} u_{y}) \cos [h \frac{π (z + L)}{Ω}] + (u_{y} u_{x} - u_{x} u_{y}) \sin [h \frac{π (z + L)}{Ω}] + u_{z} u_{z},

with 2Ω being the pitch of the helical nanowires, χ the nanowire inclination angle with respect to the xy plane, and h = ±1 the structural handedness parameter. The regions z < -L andz > L are vacuous.

An electric dipole of moment p/2 and oriented in the direction of the unit vector u _J is embedded at r = d u _z, -L < d < L; hence, the source current density phasor

J (r, ω) =- \frac{iωp}{2} u_{J} δ (z - d) δ (x) δ (y),

where δ (∙) is the Dirac delta function. In consonance with the spectral–Green–function formalism [26], we express J(r, ω)) as the spatial Fourier transform

J (r, ω) = \frac{1}{4 π^{2}} \int_{0}^{\infty} d κ κ \int_{0}^{2 π} d ψ j (z, κ, ψ, ω) \exp [i κ (x \cos ψ + y \sin ψ)),

wherein j(z, κ, ψ, ω) = -(iωp/2)u _Jδ(z - d) follows from (6). In order to describe the dipole orientation consistently with (2)–(5), we choose u _J ∈ {S͇ _z (d,h)∙u _τ,S͇(d,h)∙u _n,S͇ _z(d,h)∙u _b} , where the unit vectors

u_{τ} = u_{x} \cos χ + u_{z} \sin χ, u_{n} = {- u}_{x} \sin χ + u_{z} \cos χ, u_{b} = - u_{y} .

2.1. Spectral Green function for fields inside the CSTF

Let us now consider only a single spatial–Fourier component of J(r, ω)), i.e.,

J (r, κ, ψ, ω) = j (z, κ, ψ, ω) \exp [iκ (x \cos ψ + y \sin ψ)],

In consonance with (9), the electromagnetic field phasors for -L < z < L may be expressed as

E (r, κ, ψ, ω) = e (z, κ, ψ, ω) \exp [iκ (x \cos ψ + y \sin ψ)],

H (r, κ, ψ, ω) = h (z, κ, ψ, ω) \exp [iκ (x \cos ψ + y \sin ψ)] .

Substitution of the phasors (9)–(11) and constitutive relations (1) into the frequency–domain Maxwell curl postulates leads to:

two algebraic equations which may be solved to find
$e_{z} (z, κ, ψ, ω) = \frac{1}{ε_{a} \cos^{2} χ + ε_{b} \sin^{2} χ} ((ε_{a} - ε_{b}) \sin χ \cos χ \times {e_{x} (z, κ, ψ, ω) \cos [h \frac{π (z + L)}{Ω}] + e_{y} (z, κ, ψ, ω) \sin [h \frac{π (z + L)}{Ω}]} + \frac{1}{ω ε_{0}} {κ [h_{x} (z, κ, ψ, ω) \sin ψ - h_{y} (z, κ, ψ, ω) \cos ψ] - {ij}_{z} (z, κ, ψ, ω)})$
and
$h_{z} (z, κ, ψ, ω) = - \frac{κ}{ω μ_{0}} [e_{x} (z, κ, ψ, ω) \sin ψ - e_{y} (z, κ, ψ, ω) \cos ψ];$
and
four ordinary differential equations which may be compactly expressed as
$\frac{d}{dz} [f (z, κ, ψ, ω)] = i [P (z, κ, ψ, ω)] [f (z, κ, ψ, ω)] - \frac{iωp}{2} δ (z - d) [g (z, κ, ψ, ω)], z \in (- L, L),$
with the column vectors
$[f (z, κ, ψ, ω)] = [\begin{matrix} e_{x} (z, κ, ψ, ω) \\ e_{y} (z, κ, ψ, ω) \\ h_{x} (z, κ, ψ, ω) \\ h_{y} (z, κ, ψ, ω) \end{matrix}]$
and
$[g (z, κ, ψ, ω)] = \frac{u_{J} {∙ u}_{z}}{ε_{a} \cos^{2} χ + ε_{b} \sin^{2} χ} [\begin{matrix} \frac{κ \cos ψ}{ω ε_{0}} \\ \frac{κ \sin ψ}{ω ε_{0}} \\ (ε_{a} - ε_{b}) \sin χ \cos χ \sin [h \frac{π (z + L)}{Ω}] \\ (ε_{b} - ε_{a}) \sin χ \cos χ \cos [h \frac{π (z + L)}{Ω}] \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ u_{J} {∙ u}_{y} \\ {- u}_{J} {∙ u}_{x} \end{matrix}] .$
The 4×4 matrix [P(z,κ,ψ,ω] is straightforward to derive but too cumbersome to reproduce here; see [3, eqn. (9.116)]. Without the source term, (14) for unidirectionally nonhomogeneous mediums is often attributed to Berreman [28].
The solution of the matrix ordinary differential equation (14) is given as [24, 26]
$[f (z, κ, ψ, ω)] = [G (z, - L, κ, ψ, ω)] [f (- L, κ, ψ, ω)] + \int_{- L}^{z} [G (z, z_{s}, κ, ψ, ω)] [g (z_{s}, κ, ψ, ω)] d z_{s}, z \in [- L, L] .$
Herein the spectral Green function
$[G (z, z_{s},κ, ψ, ω)] = [M (z, κ, ψ, ω)] {[M (z_{s}, κ, ψ, ω)]}^{- 1}$
is given in terms of the 4×4 matrizant [M(z,κ,ψ,ω)] which satisfies the matrix ordinary differential equation
$\frac{d}{dz} [M (z, κ, ψ, ω)] = i [P (z, κ, ψ, ω)] [M (z, κ, ψ, ω)]$
and the boundary condition [M(-L,κ,ψ,ω)] = [I], with [I] being the 4×4 identity matrix. The general solution (17) yields the particular solution
$[f (L, κ, ψ, ω)] = [M (2 L,κ, ψ, ω)] [f (- L,κ, ψ, ω)] + [M (2 L,κ, ψ, ω)] {[M (d + L,κ, ψ, ω)]}^{- 1} [g (d, κ, ψ, ω)]$
of (14). For computing the numerical results presented in Section 3, we adopted the piecewise uniform approximation technique [3, Chap. 9] to evaluate [M(2L,κ,ψ,ω)] and [M(d + L,κ,ψ,ω)].

2.2. Boundary-value problem

Light is emitted into the vacuous regions z < -L and z > L due to the source term (6). The electromagnetic field phasors for z < -L may be expressed as

E (r, κ, ψ, ω) = \frac{1}{\sqrt{2}} [{- b}_{L} (κ, ψ, ω) (i s - p_{-}) + b_{R} (κ, ψ, ω) (i s + p_{-})] \times \exp {i [κ (x \cos ψ + y \sin ψ) - α_{0} (z + L)]},

H (r, κ, ψ, ω) = \frac{1}{η_{0} \sqrt{2}} [b_{L} (κ, ψ, ω) (i s - p_{-}) + b_{R} (κ, ψ, ω) (i s + p_{-})] \times \exp {i [κ (x \cos ψ + y \sin ψ) - α_{0} (z + L)]},

while those for z > L may be expressed in similar terms as

E (r, κ, ψ, ω) = \frac{1}{\sqrt{2}} [c_{L} (κ, ψ, ω) (i s - p_{+}) - c_{R} (κ, ψ, ω) (i s + p_{+})] \times \exp {i [κ (x \cos ψ + y \sin ψ) + α_{0} (z - L)]},

H (r, κ, ψ, ω) = \frac{1}{η_{0} \sqrt{2}} [c_{L} (κ, ψ, ω) (i s - p_{+}) - c_{R} (κ, ψ, ω) (i s + p_{+})] \times \exp {i [κ (x \cos ψ + y \sin ψ) + α_{0} (z - L)]},

where $α_{0} = + \sqrt{k_{0}^{2} - κ^{2}}$ . The scalars b_L (κ, ψ, ω) and c_L (κ, ψ, ω) are the amplitudes of the LCP components; b_R(κ, ψ, ω) and c_R (κ, ψ, ω) are the amplitudes of the RCP components; and the unit vectors

s = - u_{x} \sin ψ + u_{y} \cos ψ, p_{\pm} = \mp (\frac{α_{0}}{k_{0}}) (u_{x} \cos ψ + u_{y} \sin ψ) + (\frac{κ}{k_{0}}) u_{z}

delineate the s- and p-polarization states. The ω-dependences of p _± and α_o have not been explicitly identified here.

The boundary values

[f (- L, κ, ψ, ω)] = \frac{1}{\sqrt{2}} [K (κ, ψ, ω)] [\begin{matrix} 0 \\ 0 \\ - i [b_{L} (κ, ψ, ω) - b_{R} (κ, ψ, ω)] \\ [b_{L} (κ, ψ, ω) + b_{R} (κ, ψ, ω)] \end{matrix}]

and

[f (L, κ, ψ, ω)] = \frac{1}{\sqrt{2}} [K (κ, ψ, ω)] [\begin{matrix} i [c_{L} (κ, ψ, ω) - c_{R} (κ, ψ, ω)] \\ - [c_{L} (κ, ψ, ω) + c_{R} (κ, ψ, ω)] \\ 0 \\ 0 \end{matrix}]

emerge from the continuity of the tangential components of E(r, κ, ψ, ω) and H(r, κ, ψ, ω) across the pupils z = -L and z = L, respectively, of the CSTF, with the 4×4 matrix

[K (κ, ψ, ω)] = [\begin{matrix} - \sin ψ & - (\frac{α_{0}}{k_{0}}) \cos ψ & - \sin ψ & (\frac{α_{0}}{k_{0}}) \cos ψ \\ \cos ψ & - (\frac{α_{0}}{k_{0}}) \sin ψ & \cos ψ & (\frac{α_{0}}{k_{0}}) \sin ψ \\ - η_{0}^{- 1} (\frac{α_{0}}{k_{0}}) \cos ψ & η_{0}^{- 1} \sin ψ & η_{0}^{- 1} (\frac{α_{0}}{k_{0}}) \sin ψ & η_{0}^{- 1} \sin ψ \\ - η_{0}^{- 1} (\frac{α_{0}}{k_{0}}) \sin ψ & - η_{0}^{- 1} \cos ψ & η_{0}^{- 1} (\frac{α_{0}}{k_{0}}) \cos ψ & - η_{0}^{- 1} \cos ψ \end{matrix}]

After substituting (26) and (27) into (20), the four amplitudes b_L,R (κ, ψ, ω) and c_L,R (κ, ψ, ω) can be determined by standard algebraic techniques.

2.3. Emitted electromagnetic field phasors

The spectral electromagnetic phasors E(r, κ, ψ, ω) and H(r, κ, ψ, ω) in both vacuous regions z > L and z < -L are delivered from (21)–(24). Inverting the spatial Fourier transform, we can find the emitted electromagnetic phasors

E (r, ω) = \frac{1}{4 π^{2}} \int_{0}^{\infty} dκ \int_{0}^{2 π} dψ κ E (z, κ, ψ, ω) \exp [iκ (x \cos ψ + y \sin ψ)],

H (r, ω) = \frac{1}{4 π^{2}} \int_{0}^{\infty} dκ \int_{0}^{2 π} dψ κH (z, κ, ψ, ω) \exp [iκ (x \cos ψ + y \sin ψ)]

for z ∉ [-L,L].

Let us choose the observation point r ^obs ≡ (r^obs , θ^obs, ψ^obs) (in a spherical coordinate system) to evaluate the emitted field phasors in the far zone in the two vacuous regions; i.e., in the limit k ₀|r| → ∞. We find the asymptotic approximations [29, 30]

E (r^{obs}, ω) \approx \frac{i \cos θ^{obs}}{2 \sqrt{2 π}} [{- b}_{L}^{obs} ({i s}^{obs} - p_{-}^{obs}) - b_{R}^{obs} ({i s}^{obs} + p_{-}^{obs})] \frac{\exp (i k_{0} {\tilde{r}}_{-}^{obs})}{k_{0} {\tilde{r}}_{-}^{obs}}

H (r^{obs}, ω) \approx \frac{\cos θ^{obs}}{η_{0} 2 \sqrt{2 π}} [{- b}_{L}^{obs} ({i s}^{obs} - p_{-}^{obs}) - b_{R}^{obs} ({i s}^{obs} + p_{-}^{obs})] \frac{\exp (i k_{0} {\tilde{r}}_{-}^{obs})}{k_{0} {\tilde{r}}_{-}^{obs}}

for z^obs < -L, and

E (r^{obs}, ω) \approx \frac{i \cos θ^{obs}}{2 \sqrt{2 π}} [c_{L}^{obs} ({i s}^{obs} - p_{+}^{obs}) - c_{R}^{obs} ({i s}^{obs} + p_{+}^{obs})] \frac{\exp (i k_{0} {\tilde{r}}_{+}^{obs})}{k_{0} {\tilde{r}}_{+}^{obs}},

H (r^{obs}, ω) \approx \frac{\cos θ^{obs}}{η_{0} 2 \sqrt{2 π}} [c_{L}^{obs} ({i s}^{obs} - p_{+}^{obs}) - c_{R}^{obs} ({i s}^{obs} + p_{+}^{obs})] \frac{\exp (i k_{0} {\tilde{r}}_{+}^{obs})}{k_{0} {\tilde{r}}_{+}^{obs}}

for z^obs > L, where b_L,R^obs = b_L,R(κ^obs,ψ^obs,ω), c_L,R^obs = c_L,R(κ^obs,ψ^obs,ω), κ^obs = k ₀sinθ^obs, and r̃^obs = |r ^obs ±L u _z|. The approximations (31)–(34) are appropriate at distances far from the CSTF pupils but do not possess physical significance in the vicinity of θ^obs = π/2 [29, 30].

The time–averaged Poynting vector, P(r, ω) = (1/2)Re[E(r, ω) × H ^*(r, ω)], at the observation point r ^obs = r^obs r̂ ^obs in the far zone is given approximately as

P (r^{obs}, ω) \approx {\begin{matrix} \frac{1}{2 η_{0}} ({∣ b_{L}^{obs} ∣}^{2} + {∣ b_{R}^{obs} ∣}^{2}) {(\frac{\cos θ^{obs}}{2 π k_{0} {\tilde{r}}_{-}^{obs}})}^{2} {\hat{r}}^{obs}, z^{obs} < - L \\ \frac{1}{2 η_{0}} ({∣ c_{L}^{obs} ∣}^{2} + {∣ c_{R}^{obs} ∣}^{2}) {(\frac{\cos θ^{obs}}{2 π k_{0} {\tilde{r}}_{+}^{obs}})}^{2} {{\hat{r}}^{obs}, z}^{obs} > L \end{matrix} .

3. Numerical results

In characterizing the light emitted from the dipole–containing CSTF, it is helpful to distinguish between LCP and RCP contributions to the time–averaged Poynting vector, because of the circular–polarization–discriminatory properties of CSTFs. Therefore, we introduce

P_{LCP} (r^{obs}, ω) \approx {\begin{matrix} \frac{1}{2 η_{0}} ({∣ b_{L}^{obs} ∣}^{2}) {(\frac{\cos θ^{obs}}{2 π k_{0} {\tilde{r}}_{‒}^{obs}})}^{2} {\hat{r}}^{obs}, z^{obs} < - L \\ \frac{1}{2 η_{0}} ({∣ c_{L}^{obs} ∣}^{2}) {(\frac{\cos θ^{obs}}{2 π k_{0} {\tilde{r}}_{+}^{obs}})}^{2} {{\hat{r}}^{obs}, z}^{obs} > L \end{matrix}

and

P_{RCP} (r^{obs}, ω) \approx {\begin{matrix} \frac{1}{2 η_{0}} ({∣ b_{R}^{obs} ∣}^{2}) {(\frac{\cos θ^{obs}}{2 π k_{0} {\tilde{r}}_{+}^{obs}})}^{2} {\hat{r}}^{obs}, z^{obs} < - L \\ \frac{1}{2 η_{0}} ({∣ c_{R}^{obs} ∣}^{2}) {(\frac{\cos θ^{obs}}{2 π k_{0} {\tilde{r}}_{-}^{obs}})}^{2} {{\hat{r}}^{obs}, z}^{obs} > L \end{matrix},

and characterize the light emitted from the CSTF in terms of the relative magnitudes

P_{j} (r^{obs}, ω) = \frac{∣ P_{j} (r^{obs}, ω) ∣}{ω^{2} {∣ p ∣}^{2}} \times 10^{13}, j \in {LCP, RCP} .

In addition, we introduce the real–valued parameters Γ_j, j ∈ {LCP,RCP}, computed as

Γ_{j} = 10^{3} \int_{ψ^{obs} = ρ_{1}}^{ρ_{2}} {dθ}^{obs} \int_{ψ^{obs} = 0}^{2 π} {dψ}^{obs} {(r^{obs})}^{2} \sin θ^{obs} P_{j} (r^{obs}, ω), j \in {LCP, RCP},

where ρ₁ = 0, ρ₂ = 0.95π/2 for z^obs > L, and ρ₁ = π - (0.95π/2), ρ₂ = π for z < -L. Thus, Γ_LCP and Γ_RCP provide a measure of the rates of energy flow into the vacuous regions z < L and z < -L.

Fig. 1. Projections of P_LCP onto the z = 0 plane for the regions z^obs > L (top row) and z < - L (bottom row), when S͇ _z ^-1 (d,h)∙u _J ∈ {u _n,u _τ,u _b}, λ₀ = 727nm, d = 0, and ε_a = ε_b = ε_c = 1.

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A single–resonance Lorentzian model was selected for the relative permittivity scalars of the CSTF:

ε_{a, b, c} = 1 + \frac{q_{a, b, c}}{1 + {(\frac{1}{N_{a, b, c}} - i \frac{λ_{a, b, c}}{λ_{0}})}^{2}} .

Here, q_a,b,c are the oscillator strengths, while the parameters λ_a,b,c and N_a,b,c describe the resonance and absorption bands, respectively. We selected q_a = 2.0, q_b, = 2.6, q_c = 2.1, λ_a = λ_c = 180nm, λ_b = 150nm and N_a,b,c = 500. The structural parameters χ = 30° and Ω = 200 nm were chosen to ensure that the Bragg regime for normally incident plane waves is given by 702 ≤ λ₀ ≤ 752 nm [31], while the thickness ratio L/Ω. = 30 was taken so that the circular Bragg phenomenon is fully developed. Also, we chose the CSTF to be structurally righ-handed (h = +1). For all calculations reported here, we chose the free-space wavelength λ₀ ∈ {652,727,802} nm and the observation radius r^obs = 10⁵λ₀.

Let us begin by considering a point–dipole source which lies midway through the CSTF (d = 0); accordingly, u _J ∈ {u _n,u _τ,u _b}. In order to provide a reference for our numerical results, we initially converted the CSTF into free space by setting ε_a = ε_b = ε_c = 1. In Fig. 1, the projection of P_LCP is mapped onto the disc of radius r^obs (normalized to unity) in the z = 0 plane. The associated values of Γ_LCP are also indicated in Fig. 1. The data presented in Fig. 1 were obtained using Γ₀ = 727 nm. The symmetry about the y axis but not about the x axis of the maps in the left and center columns of Fig. 1 indicates the tilted orientations of u _n and u _τ relative to the x and z axes, whereas the fact that u _b lies along the y axis is clearly reflected in the symmetries of the maps about the x and the y axes in the right column of the same figure. The corresponding maps computed using λ₀ = 652 nm and 802 nm are indistinguishable from those in Fig. 1, because free space is nondispersive. Furthermore, the corresponding maps of P_RCP are indistinguishable from those in Fig. 1, because free space is isotropic and achiral.

Fig. 2. Projections of P_LCP and P_RCP onto the z = 0 plane for the regions z^obs > L (top two rows) and z < - L (bottom two rows), when u _J =S͇ _z(d,h)∙u _n,λ₀ ∈ {652,727,802} nm and h = 1. The relative permittivity parameters are prescribed by eqn. (40). The dipole source is located at d = 0.

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Let us now turn to the dipole embedded centrally in the CSTF. In Figs. 2, 3 and 4, the projections of P_LCP and P_RCP onto the z = 0 plane are mapped for u _J = u _n, u _τ, and u _b, respectively. The values of Γ_LCP and Γ_RCP are provided for each map, as appropriate. The following symmetries were analytically deduced for these maps and were supported by visual examination of specific features therein:

b^obs_L,R(κ^obs, ψ^obs)=-b^obs_L,R(-κ^obs,ψ^obs+π) and c^obs_L,R(κ^obs,ψ^obs)=−c^obs_L,R(-κ^obs,ψ^obs+π) for u _J=u _n,u _τ,and u _b;
b^obs_L,R(κ^obs, ψ^obs)=c^obs_L,R(-κ^obs,-ψ^obs) for u _J=u _n and u _τ
b^obs_L,R(κ^obs, ψ^obs)=− c^obs_L,R(-κ^obs,-ψ^obs) for u _J=u _b.

Fig. 3. As Fig. 2 but with u _J = S͇ _z(d,h)∙u _τ.

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On comparing the values of Γ_LCP and Γ_RCP in Figs. 2–4 with those in Fig. 1, it is clear that the CSTF is a load on the dipole source; i.e., the dipole embedded in the CSTF is a less efficient emitter than the dipole in free space.

There are significant differences between the LCP and the RCP emission patterns in Figs. 2–4: Firstly, when |cosθ^obs| ≃ 1 (i.e., either normal or almost normal to the two pupils of the CSTF), these differences are more obvious for λ₀ = 727 nm than for λ₀ = 652 nm and 802 nm This can be explained by the circular Bragg phenomenon for normally incident plane waves. Being structurally right–handed, the CSTF preferentially reflects normally incident RCP light for λ₀ ∈ [702,752] nm. Experimental data reported in Figs. 3a and 3b of Zhang et al. [19] for a CSTF with a central layer of quantum dots indicate that our dipole-containing CSTF must then be a preferential emitter of LCP light in the same wavelength regime. Secondly, when 0.4 < |cosθ_obs| < 0.8 the difference between the LCP and the RCP emission patterns is more obvious for λ₀ = 652 nm than for λ₀ = 727 nm and 802 nm. This is indicative of the blueshift of the circular Bragg phenomenon for obliquely incident plane waves [3]. Thirdly, when |cosθ_obs| < 0.4, the distinction between the LCP and RCP patterns is very little, if any, for all three wavelengths. This reflects the huge diminishment of the circular Bragg phenomenon for highly oblique planewave incidence [32].

Fig. 4. As Fig. 2 but with u _J = S͇ _z(d,h)∙u _b.

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A notable feature of Figs. 2–4 is the presence of emission rings. In order to take account of the resonant effects of the CSTF pupils z = -L and z = L, we repeated the computations for these figures, but with the CSTF replaced by an isotropic film of relative permittivity that is the arithmetic mean of ε_a, ε_b, and ε_c of (40). As is the case for Figs. 2–4, the dipole source was still located at d = 0. Plots of the projections of P_LCP onto the z = 0 plane, and the corresponding values of Γ_LCP, are provided in Fig. 5 for u _J = u _n, u _τ, and u _b at the free-space wavelength λ₀ = 727 nm. The corresponding plots of P_RCP are identical to those presented in Fig. 5 because the isotropic film is structurally achiral, and are therefore not displayed here. The annular emission features apparent in Fig. 5 have the same spatial frequency as those observed in Figs. 2–4. We infer that these are Fabry–Perot rings arising due to thickness resonances of the CSTF. The modulation of the Fabry–Perot ring pattern by the structural handedness of the CSTF can be deduced from Figs. 2–4.

Fig. 5. Projections of P_LCP onto the z = 0 plane for the reqions z^obs > L (top row) and z < - L (bottom row), when S͇ _z(d,h)∙u _J ∈ {u _n,u _τ,u _b}, λ₀ = 727nm and h = 1. The CSTF has been replaced by an isotropic material with relative permittivity that is the arithmetic mean of ε_a, ε_b, and ε_c of eqn. (40). The dipole source is located at d = 0.

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Finally, we considered a dipole source located very close to the z = L pupil of the CSTF; we set d = L - 40 nm. The corresponding projections of P_LCP and P_RCP onto the z = 0 plane are plotted in Figs. 6, 7 and 8, for u _J = S͇ _z(d,h)∙u _n,S͇ _z(d,h)∙u _τ, and S͇ _z(d,h)∙u _b, respectively. Asymmetries between the emission intensities through the closer and the distant pupils are highly conspicuous. In particular, we observe that in the Bragg regime, RCP light is preferentially emitted through the pupil closer to the source (z = L), whereas LCP light is preferentially emitted through the pupil farther from the source (z= -L). Also, light is generally emitted with greater intensity than for the symmetrically placed dipole source.

The effects of the location and orientation of a point source will become crucial when numerous point sources are embedded in a CSTF, e.g., by dispersing quantum dots [16, 19] or DNA fragments [33] in a CSTF, which are later excited optically, electrically, or chemically to emit light. The presented formalism is robust enough for theoretical examinations of such problems, when the interactions between the individual sources are sufficiently small to be ignored, and can also be used to develop a statistical understanding [34] after studying emission due to different ensembles of sources.

4. Closing remarks

Light emitted from a CSTF containing a point–dipole source bears the signatures of the structural handedness of the CSTF as well as the location and orientation of the source. Specifically, the emitted radiation can be circularly polarized to a high degree within the wavelength regime of the circular Bragg phenomenon relevant to directions offset from the direction normal to the two pupils; the nature of circular polarization depends on the orientation and location of the dipole source. These emission characteristics augur well for the future development of CSTFs as platforms for optical sensing devices such as genomic sensors [8, 10, 33] as well as for light emitters with controlled circular polarization and bandwidth [15, 16, 19]. Accordingly, we will apply the developed theory to study emission characteristics of a CSTF containing mono-layer and volumetric distributions of point-dipole sources, and eventually develop an integral–equation formalism for light emission from an arbitrary source embedded in a CSTF.

Fig. 6. Projections of P_LCP and P_RCP onto the z = 0 plane for the regions z^obs > L (top two rows) and z^obs < -L (bottom two rows), when u _J =S͇ _z(d,h)∙u _n,λ₀ ∈ {652,727,802} nm and h = 1. The relative permittivity parameters are prescribed by eqn. (40). The dipole source is located at d = L - 40 nm.

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Fig. 7. As Fig. 6 but with u _J = S͇ _z(d,h)∙u _τ.

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Fig. 8. As Fig. 6 but with u _J = S͇ _z(d,h)∙u _b.

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Acknowledgment

TGM is supported by a Royal Society of Edinburgh/Scottish Executive Support Research Fellowship. AL is supported in part by the Charles Godfrey Binder Endowment.

References and links

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Theory of light emission from a dipole source embedded in a chiral sculptured thin film

Abstract

Corrections

1. Introduction

2. Theory

2.1. Spectral Green function for fields inside the CSTF

2.2. Boundary-value problem

2.3. Emitted electromagnetic field phasors

3. Numerical results

4. Closing remarks

Acknowledgment

References and links

Cited By

Figures (8)

Equations (40)

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