Single mode thermal emission

Lena Simone Fohrmann; Alexander Yu. Petrov; Slawa Lang; Dirk Jalas; Thomas F. Krauss; Manfred Eich

doi:10.1364/OE.23.027672

Optics Express
Vol. 23,
Issue 21,
pp. 27672-27682
(2015)
•https://doi.org/10.1364/OE.23.027672

Single mode thermal emission

Lena Simone Fohrmann, Alexander Yu. Petrov, Slawa Lang, Dirk Jalas, Thomas F. Krauss, and Manfred Eich

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Lena Simone Fohrmann, Alexander Yu. Petrov, Slawa Lang, Dirk Jalas, Thomas F. Krauss, and Manfred Eich, "Single mode thermal emission," Opt. Express 23, 27672-27682 (2015)

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Abstract

We report on the properties of a thermal emitter which radiates into a single mode waveguide. We show that the maximal power of thermal radiation into a propagating single mode is limited only by the temperature of the thermal emitter and does not depend on other parameters of the waveguide. Furthermore, we show that the power of the thermal emitter cannot be increased by resonant coupling. For a given temperature, the enhancement of the total emitted power is only possible if the number of excited modes is increased. Either a narrowband or a broadband thermal excitation of the mode is possible, depending on the properties of the emitter. We finally discuss an example system, namely a thermal source for silicon photonics.

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Fig. 1 (a) Possible modes in a one dimensional resonator defined by a waveguide section between two mirrors at a distance L. Only resonator modes are allowed whose effective wavelength λ_eff fulfil the condition of n⋅λ_eff = 2L. (b) Spectral power density for a propagating mode in one direction with one polarization at 1000 K, 800 K and 600 K.

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Fig. 2 Power division and combination in a system with n ports. (a) A signal enters at port 1 and is divided into multiple signals that exit the system via the ports 1 to n. As a consequence of power conservation, the sum of the output powers is limited by the input power, so ∑_i|b_i|² = |a₁|². (b) The signals of several input ports are combined into port 1. Due to the unitarity of the system, for an incoherent signal, the output power cannot exceed the powers at the single input port.

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Fig. 3 (a) Concept of a broadband thermal emitter based on a SOI waveguide. The waveguide has a rectangular cross section over a length of 2L + L_a and is underetched. The thermal emission is produced by heating an emission section in the center of the waveguide which is designed to have a high emissivity. The emitted power is directly coupled into the silicon waveguide. (b) Resonant thermal emitter design where the emission section is embedded into a photonic bandgap cavity to resonantly select one frequency. The resonance frequency and 3dB bandwidth can be tuned by choosing the sizes and distances of the photonic crystal holes.

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Fig. 4 Comparison of the spectral power density emitted by the broadband thermal emitter P_λ,b(black curve) and the narrowband resonant thermal emitter P_λ,n (red curve) at T = 1000 K. The emission range of the resonant thermal emitter is adapted to the CO₂ specific wavelength range between 4.2 µm and 4.3 µm.

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Equations (18)

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e (ν, T) = \frac{h ν}{\exp (\frac{h ν}{k_{B} T}) - 1}

L = n \cdot \frac{λ_{e f f}}{2} = n \cdot \frac{π}{k_{e f f}}, n = 1, 2, 3, \dots

d N = \frac{d k}{Δ k} = 2 L \cdot \frac{d ν}{v_{g}},

g (ν) = \frac{1}{L} \cdot \frac{d N}{d ν} = \frac{2}{v_{g}}

P_{ν} (T) = \frac{1}{2} v_{g} \cdot g (ν) \cdot e (ν, T) = \frac{h ν}{\exp (\frac{h ν}{k T}) - 1}

P_{λ} (T) = \frac{h c^{2}}{λ^{3} (\exp (\frac{h c}{λ k T}) - 1)}

P (T) = \int_{0}^{\infty} P_{ν} d ν = \frac{{(k T)}^{2}}{h} \cdot \frac{π^{2}}{6}

\frac{1}{Q} = \frac{1}{Q_{0}} + \frac{1}{Q_{e}}

b_{i} = \sum_{j} S_{i j} a_{j}

P_{o u t} = {| b_{1} |}^{2} + {| b_{2} |}^{2} + \dots + {| b_{n} |}^{2} = \sum_{i} {| S_{i 1} |}^{2} \cdot {| a_{1} |}^{2} = {| a_{1} |}^{2}

\sum_{i} {| S_{i j} |}^{2} = 1

\sum_{j} {| S_{i j} |}^{2} = 1

P_{o u t} = {| b_{1} |}^{2} = {(S_{11} a_{1} + S_{12} a_{2} + \dots + S_{1 n} a_{n})}^{2} = \sum_{j} {| S_{1 j} |}^{2} P_{0} + 2 \cdot \sum_{j} \sum_{k < j} S_{1 j} S_{1 k} a_{j} a_{k}

〈 d 〉 = Δ t Δ ν \cdot n_{0},

n_{0} = \frac{1}{\exp (\frac{h ν}{k T}) - 1}

σ^{2} = Δ t Δ ν \cdot n_{0} (1 + n_{0})

S N R = \frac{P_{s i g n a l}}{P_{n o i s e}} = \frac{h ν \cdot 〈 d 〉}{h ν \cdot σ} = \sqrt{Δ t Δ ν \cdot \frac{n_{0}}{1 + n_{0}}}

S N R = \sqrt{N_{m} Δ t Δ ν \cdot \frac{n_{0}}{1 + n_{0}}}

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