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

1. Introduction

Thermal radiation emerges if matter is heated to a temperature greater than zero. In 1860, Kirchhoff [1] introduced the concept of a black body that completely absorbs the incident light for all frequencies. He stated that the thermal emission of a body cannot exceed the thermal radiation originating from a black body of the same temperature. Later on, Planck [2] developed a formulation for the maximal emitted radiation of a black body at a given temperature which is well known as Planck’s law of black-body radiation. Hence, it was accepted that the spectral power density emitted by a black body into vacuum only depends on the radiation frequency and its temperature.

Here, we consider a 1D system. We calculate the thermal radiation generated by this system and the power that can be coupled into a single propagating mode, before discussing possible applications. Some aspects of thermal radiation emitted into a single mode have already been discussed by a number of authors, e.g [3–7].

We consolidate the available information and provide a simple derivation and explanation for some important properties of single mode emission. Unlike the case for out-of-plane thermal emission, which has been studied in detail previously [8,9], we now look at the in-plane case. Analogous to the derivation of black body emission in 3 dimensions [10], we develop an expression for the spectral power density emitted into a single mode. In both cases, the density of states depends on the refractive index of the environment. Nevertheless, in contrast to the 3D case where the resulting spectral power density is still dependent on the refractive index, the influence of the refractive index on the 1D spectral power density cancels out, resulting in an expression for the single mode spectral power density that is not dependent on the geometry and refractive index of the waveguide. We also discuss the impossibility of power combining and cover the topics of photon noise and resonant excitation of thermal radiation in propagating modes.

Considering guided modes in integrated optics, we discuss the application of a single mode thermal emitter as an integrated MIR source. The proposed thermal emitter is based on a single mode silicon on insulator (SOI) waveguide where one part of the device is heated which therefore emits broadband thermal radiation directly into the waveguide. A critically coupled photonic crystal resonator can be used to generate frequency selective emission.

The paper is structured as follows: In section 2., we derive the power limit for thermal radiation into a single mode waveguide. The resulting total power over all wavelengths turns out to be proportional to the square of the emitter’s temperature. In section 3., the coupling from a thermal cavity into an external waveguide is discussed. The process is optimized for critical coupling. We show that a resonant cavity emitter cannot increase the power carried by a propagating single mode above the power maximum calculated in the prior section. In Section 4., we consider the power combination of multiple incoherent sources. As an example, the combination of signals from n waveguides into one waveguide is discussed. The resulting power is still limited by the maximum of single mode thermal radiation even for non-reciprocal structures. Section 5. gives an estimation for the noise of single mode thermal light detection. Besides the photon statistics of the signal, the signal-to-noise ratio is defined by the integration time and bandwidth of the detector. In section 6., a broadband thermal emitter design is presented that consists of a single mode SOI-strip waveguide. The thermal emitter can also be converted into a narrowband emitter if the thermal emission section of the waveguide is integrated into a photonic crystal cavity. Therefore, one wavelength is resonantly enhanced.

2. Maximal power emitted into a single mode waveguide

In our derivation for the maximal spectral power density, only propagating modes are taken into account. We limit our consideration to a medium without dissipation, developing an approach that is similar to the derivation of the black body radiation in 3D [10]. This derivation does not consider the mechanism for the thermal excitation. To excite the thermal radiation, dissipation will be required which will be considered in the next section.

For the estimation of the maximal thermal radiation into a single mode waveguide, we consider a one-dimensional resonator in thermal equilibrium. The spectral energy in each mode of the resonator is given by the Bose-Einstein statistics for photons [4]:

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

with the Planck constant h, the Boltzmann constant k_B, frequency ν and temperature T. Zero-point energy is neglected here.

For the calculation of the density of states, we assume a one dimensional resonator defined by a section of a straight waveguide terminated by two perfectly reflecting mirrors separated by distance L (see Fig. 1(a)). As the electromagnetic field must be zero at the resonator walls, only resonator modes are allowed for which

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

and where λ_eff is the effective wavelength and k_eff defines the effective wavenumber of the propagating mode in the waveguide. Therefore, the wavenumber spacing between two neighboring modes in the resonator is Δk = π/L and we can calculate the gradient of the number of modes dN within a frequency range dν = 1/(2π)⋅v_g dk:

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

where v_g denotes the group velocity of the mode which is the derivative of the angular frequency ω with respect to the wavenumber k. The density of states g(ν) for one polarization is given by the number of modes per frequency range normalized to the cavity length L:

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

The one dimensional cavity contains radiation in forward and backward direction. To calculate the spectral power density for only one direction, a factor of 1/2 has to be taken into account. Knowing the mode occupation of photons (given by Eq. (1)) and the density of states in the cavity (Eq. (4), we can thus determine the spectral power density of a propagating single mode for one polarization and one direction [11]:

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)}

λ is the vacuum wavelength of radiation and c is the speed of light in vacuum. The spectral power density over frequency for one polarization and one direction has no maximum. However the representation over wavelength exhibits a maximum at a temperature specific wavelength λ_max ≈5.1 mm⋅K/T (see Fig. 1(b)). The maximum of the spectral power density is shifted to smaller wavelengths with increasing temperature. During the derivation, we have considered a waveguide of arbitrary cross section. The final Eq. (5) and (6) show that there is no dependence of the maximal power on the properties of the waveguide. Thus the excitation of a single mode via thermal emission has a fundamental limit. This statement is supported by the fact that radiation in one single mode waveguide can be adiabatically coupled into a second single mode waveguide of different geometry. As a consequence, the limit for the spectral power density must be identical in all possible single mode waveguides. Note that at some frequency, every real waveguide will start to support more than one mode per polarization. However, the cutoffs can be at such large frequencies that the higher order modes are almost thermally unoccupied, making the considered case indeed a realistic one.

The total power in a propagating mode existing at all frequencies increases proportional to T², in contrast to T⁴ for the three dimensional case:

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

Thus a temperature bath of 1000 K can deliver a thermal radiation power of 0.47 µW into a non-radiating single mode waveguide.

3. Excitation of single mode thermal radiation

The thermal radiation into a single mode waveguide can be coupled in from a blackbody source, for example, by positioning a single mode fiber in front of the black body source. More efficient is the use of an absorptive section of the waveguide that itself works as a single mode emitter. Here we apply Kirchoff’s law to such a one dimensional system. The section of the waveguide has a single mode emissivity equal to its absorptivity. Thus a section that absorbs 100% of the input power at a certain wavelength will provide maximal spectral power density according to Eq. (6) at this wavelength when heated to temperature T. Kirchoff’s law for a one dimensional system can be proven similar to a 3D system by imposing the validity of second law of thermodynamics [12], namely that there cannot be any net exchange of power between two bodies at the same temperature. An absorptivity α(λ) leads to an absorption equal to α(λ)P_λ(T) and an emissivity ϵ(λ) leads to an emission that is equal to ϵ(λ)P_λ(T). If we consider only radiative heat transfer at thermal equilibrium, the two processes will compensate each other so that α(λ) = ϵ(λ). Thus the emissivity can never exceed 1.

Spectrally selective emission can be obtained by having spectrally selective absorbers. This can be achieved by the coupling of a lossy resonator to a radiation waveguide. Thermal energy can leave a thermally excited resonator through such radiation waveguides. The coupled mode theory in [3] rigorously describes this constellation. Here we give a simplified explanation. We start with the absorption properties of a single mode waveguide terminated by a lossy cavity. Light that is guided in the waveguide can be coupled into the cavity. The excited mode inside the cavity is determined by the decay rate due to internal losses 1/τ₀ and the decay rate caused by the backcoupling to the waveguide 1/τ_e. The overall quality factor of the cavity Q which describes the total energy decay inside the system is composed of the internal quality factor 1/Q₀ ~1/τ₀ and the external quality factor 1/Q_e ~1/τ_e according to

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

In general, three cases can be distinguished for the coupling between a waveguide and a cavity. If Q₀ < Q_e, internal loss is the dominant decay process and the system is referred to as being undercoupled. In this case, the resonance has a very low amplitude and most of the power is reflected back into the waveguide. On the contrary, if Q₀ > Q_e, the amount of energy that is lost inside the cavity is low compared to the amount coupled back into the waveguide. The system is overcoupled. If both quality factors are matched (Q₀ = Q_e) the coupling is optimized [11]. This case is typically referred to as critical coupling, which occurs when the coupling between the waveguide and the cavity matches the internal loss in the waveguide, and on resonance, complete absorption is achieved. Again, an absorption of 100% at the resonance frequency will lead to a maximal spectral power density P_ν,max = e(ν,T) of thermal emission at this frequency.

4. Impossibility of thermal power combining

The maximal thermal power coupled into a single mode waveguide is determined by the temperature of the emitter as shown in Eq. (7). Using the unitarity of scattering matrices, it can be shown that the power in a single mode waveguide cannot be increased by adding the power from other thermal sources into the same waveguide.

For our derivation, we will consider waves that enter the system in port j and exit at port i. The variables a_j and b_i describe the amplitudes of the incoming and outgoing waves respectively. They are related by a scattering matrix S according to:

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

S_ij can be considered as the complex probability amplitude for a photon entering the system in port j to leave in port i. The amplitudes are normalized such that their squared magnitude is equal to the signal powers.

Figure 2(a) shows an exemplary system where a signal entering at port 1 is divided into several output signals at the ports 1… n. Due to power conservation, the overall output power is given by

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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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}

This means that if a power of 1 unit is coupled into one input port, the sum of the output powers will also be 1 unit in a lossless system. Therefore, the sum of each matrix entry within one column squared is 1. For the column j it follows:

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

Thus the scattering matrix of a lossless system is unitary, i.e. S⁺S = 1. Due to the fact that S = S(S⁺S) = (SS⁺)S, it follows SS⁺ = 1 and the sum of each matrix entry within one row squared is also 1. For the row i it is:

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

Considering a system with n ports as shown in Fig. 2(b) and input powers of |a₁|² = |a₂|² = … = |a_n|² = P₀, the output power in port 1 is given by:

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}

Using Eq. (12) the first summand of Eq. (13) gives the power at one of the input ports P₀. The second summand depends on the phase relation between the complex input amplitudes. In case of coherent light the input signals have a fixed phase relation. However, for incoherent light the phase difference between waves are fluctuating. Therefore, the second term vanishes and the output power is given by the input power P_out = P₀.

A combination of multiple thermal sources thus will not help to increase the power in a single mode waveguide. Irrespective of the number of combined thermal sources with equal individual powers the output signal will never exceed the signal power in one of the individual input ports. The statement is true even for non-reciprocal systems when S_ij ≠ S_ji.

5. The signal-to-noise ratio of single mode thermal radiation

Thermal radiation is subjected to intrinsic fluctuations of the photon flux, which is referred to as photon noise. It is important to evaluate this noise in single mode systems as it can post a fundamental limitation for their application. We consider the thermal noise in a single mode detector with bandwidth Δν. Accordingly, the average number of photons that is detected in a single mode waveguide during a time interval Δt is given by

〈 d 〉 = Δ t Δ ν \cdot n_{0},

where n₀ is the average number of photons in a single mode waveguide per time and frequency interval obtained from Eq. (5):

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

Referring to an integration time of Δt, the variance of the detected photons σ² can be derived from Bose-Einstein statistics [4]:

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

The first summand results from the Poisson-distribution of the photons. For a small number of photons per mode (n₀ < 1), the variance is directly proportional to the average number of detected photons. The second term is caused by photon bunching and cannot be neglected if n₀ > 1.

Correspondingly, the signal-to-noise ratio at the detector output is determined as follows,

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}}}

The last factor in the square root of Eq. (17) corresponds to the signal-to-noise ratio of the mode occupation in a thermal resonator that follows from the Bose-Einstein statistics and is always less than 1. Considering the radiation statistics in a single mode waveguide, the averaging time and frequency bandwidth thus need to be taken into account in addition to the occupation statistics. For high values of ΔtΔν this fact leads to a signal-to-noise ratio of a single mode detector that can be much larger than one. For a resonant thermal source, the bandwidth Δν is proportional to 1/τ, where τ is the total decay time of the resonator. Thus ΔtΔν = Δt/τ defines the effective number of emission events the detector integrates over and the square root of this number increases the signal-to-noise ratio.

The SNR of a thermal source can be further increased if several waveguides or several modes of the same waveguide are added up at the detector. This does not contradict the previous section on the limits of power combining into one waveguide. However, a detector can add up the power from several waveguides without combining them in a propagating single mode. In this case, the power increases proportional to the number of modes N_m, whereas the noise and the signal-to-noise ratio increase proportional to N_m^0.5:

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

6. Application: Silicon waveguide based single mode thermal emitter

We propose a thermal emitter that is based on a single mode silicon on insulator waveguide. One part of the waveguide is heated and therefore releases broadband thermal radiation in the mid-infrared range also directly into the waveguide. Such a MIR source would be useful for optical spectroscopy in integrated photonics. One of the major application can be gas detection, ranging from toxic gas sensing [13], CO₂ control in automobiles [14], measurement of atmospheric composition changes [15], breath analysis for clinical diagnosis [16] or process control. In terms of mass production of optical components, the need for heterogeneous integration is often a bottleneck, and monolithic integration and CMOS compatibility are essential requirements for large scale, low cost mass-manufacture. Therefore, it is very desirable to find a route towards integrated light sources for MIR spectroscopy.

We have recently suggested that a silicon on insulator (SOI) platform can be used for building a broadband single mode thermal emitter [17]. The proposed light source is integrated into a silicon waveguide and it consists of an emission section that is heated up in order to emit thermal radiation into the waveguide as shown in Fig. 3.

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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The oxide below the silicon waveguide layer is etched away to minimize optical leakage into the substrate and to avoid MIR absorption in the silica. Intrinsic silicon is transparent between 1 and 8 µm [18]. The absorption section can be created by using an absorbing coating on top of the silicon waveguide or by doping the silicon. The width w and height h of the waveguide are designed to allow single mode propagation over the frequency range of interest. As we consider thermal emission into a single mode, the discussions and calculations of the previous sections are valid for our light source. In the following, we assume a waveguide absorption section with an absorptivity of 1 for the guided mode and thus an emissivity of 1 for the light source. The heating of the emission section can be realized electrically, similar to the approach recently presented by O’Regan et al. [19]. The efficiency of the light source is then directly connected to the heating power provided by the electric circuit. Air convection, conduction and outward radiation will produce losses and, require an increase of the heating power in order to keep the emission section at a temperature T. Therefore the losses of the light source reduce the efficiency, but will not reduce the power emitted into the waveguide, which only depends on the temperature and is given by Eq. (7). Thus, for a temperature of T = 1000 K, the proposed light source provides a total power of P = 0.47 µW into the waveguide.

One possible application of the proposed thermal emitter is CO₂ optical sensing. CO₂ exhibits absorption lines between 4.2 and 4.3 µm [20]. According to Eq. (6) and assuming a temperature of T = 1000 K, the power emitted by the proposed broadband light source in this wavelength range is 3 nW. If an optical wave of this power is travelling with a part of its field distribution interacting with the surrounding CO₂-gas, then the power alteration at the detector compared to the power detected in the absence of CO₂ allows the determination of the gas concentration according to the Beer-Lambert law [21]. As an example and in relation to section 5., we calculate the SNR produced in a single mode detector with a spectral bandwidth of Δν = 2 THz (100 nm), assuming an integration time of Δt = 1 s. If we consider an emission temperature of T = 1000 K and a signal frequency of ν = 70 THz, the signal-to-noise ratio will be around 2.9∙10⁵, which is high enough to give accurate detection results.

Thus, the proposed thermal emitter can be used for optical sensing applications. A heating temperature of T = 1000 K offers a power of a few nW within a bandwidth of some hundred nm. For the detection of gases, an additional filter is required to limit the emission to the gas specific wavelength range and therefore avoid an overlap to the specific regions of other gases.

The sensitivity, selectivity as well as the power efficiency of a light source within the gas sensor could be significantly improved if the emitted bandwidth could be matched to the gas specific wavelength range [22–24]. In such a narrowband emitter the thermal emission is limited to a defined wavelength range and no power is lost into wavelengths outside of the specified bandwidth.

A critically coupled photonic crystal resonator can be used to generate such frequency selective emission. Therefore, a modification of our broadband silicon thermal emitter can be considered to resonantly select one wavelength using a photonic bandgap cavity. Such a narrowband emitter with a tailored bandwidth can be used for gas sensing without the requirement of additional filters. Figure 3(b) shows a redesign of the thermal emitter using a photonic crystal cavity similar to [25]. The emission section is positioned inside of this cavity. By adjusting the hole sizes and distances, the cavity can be designed to exhibit low vertical scattering into the surrounding air compared to the coupling in the horizontal direction. Coupling into the non-desired direction can be suppressed by increasing the reflectivity of the photonic crystal mirrors on this side. As discussed in section 3., the cavity should be optimized for critical coupling to offer maximal efficiency. Therefore, the internal loss that is caused by absorption needs to be adjusted to the coupling loss between the cavity and the waveguide. By varying the hole sizes and distances, the resonance frequency and coupling quality factor of the cavity can be tailored. The internal quality factor can be adjusted by the doping level or absorptive coating of the emission section which describes the absorption of the emitter. Given the bandwidth required for carbon dioxide detection, a cavity should be designed with a resonance frequency at 4.25 µm and a bandwidth of Δν = 100 nm to enable thermal emission throughout the carbon dioxide specific range.

Based on a silicon waveguide with a cross section of A = 500 nm⋅1000 nm, we used CST Microwave Studio to simulate the absorptivity α_n inside a cavity similar to that shown in Fig. 3(b) with 3 holes on the right side and radius r = 0.21 μm, a lattice constant of a = 1.2 µm, cavity length l = 1.89 µm and a doping concentration of n = 4⋅10¹⁹/cm³ inside of the emission section. According to chapter (3), the emission of the cavity is equivalent to its absorption and can be determined by multiplying the absorptivity with the spectral power density, P_λ,n = α_n(λ)⋅P_λ(T). The red curve in Fig. 4 shows the result for the spectral power density of the cavity heated to T = 1000 K. The spectral power density has a maximum at λ = 4.25 µm and exhibits a bandwidth of Δν = 100 nm and therefore the cavity is suitable for CO₂ sensing.

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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It must be noted again that the maximal spectral power density of the resonant thermal emitter is still the same as for the broadband emitter (black curve in Fig. 4) and it is only defined by the temperature and wavelength. Nevertheless a resonant enhancement will reduce the power consumption as no power is lost by emission into wavelengths outside of the resonant emission range. For the proposed thermal emitter design, the losses by thermal conduction are significantly higher than the thermal radiation losses. Therefore, the resonant concept will only slightly increase the efficiency compared to that of the broadband approach. However, it is possible to decrease the thermal conductance, e.g. as it is done in the thermal emitter design of de Zoysa et al. [26] where the thermal conductance is orders of magnitude smaller than the thermal radiation. In this way the heating efficiency of a resonant thermal emitter can be improved by orders of magnitude compared to the broadband design.

7. Conclusion

In summary, we have investigated the thermal emission in a single mode waveguide. We have shown that the maximal radiation power into such a waveguide is limited by the temperature of the thermal emitter and does not depend on the geometry of the waveguide, its refractive index or group velocity. Combining several waveguides will not improve the power that is coupled into a propagating single mode. The single mode waveguide can be efficiently excited by a thermal resonator if the critical coupling condition is fulfilled. In this case, the coupling constant between the resonator and the waveguide is equal to the loss factor of the resonator. Nevertheless, resonant coupling cannot increase the maximal spectral power density carried by a single mode waveguide.

Based on single mode emission, we have presented the design of broadband and narrowband single mode thermal emitters which can be used as integrated MIR sources. The sources are based on the thermal emission of heated absorptive waveguide sections. The broadband emitter at 1000 K can provide a net emission power up to 0.47 µW. The resonant emitter can be used to generate selective emission with a power limited by the bandwidth. The photon noise in such a single mode thermal source is still small enough to allow a reasonable signal-to-noise ratio of approximately 3⋅10⁵ for detection to be achieved.

Acknowledgments

This publication was supported by the German Research Foundation (DFG) and the Hamburg University of Technology (TUHH) in the funding programme “Open Access Publishing”. The authors gratefully acknowledge financial support from the German Research Foundation (DFG) via SFB 986 “Tailor-Made Multi-Scale Materials Systems: M3,” Project C1. The authors also acknowledge the support from CST, Darmstadt, Germany, with their Microwave Studio software. In addition, we would like to thank Roel Baets and Gunther Roelkens of Gent University, Belgium, for numerous inspiring discussions.

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Single mode thermal emission

Abstract

1. Introduction

2. Maximal power emitted into a single mode waveguide

3. Excitation of single mode thermal radiation

4. Impossibility of thermal power combining

5. The signal-to-noise ratio of single mode thermal radiation

6. Application: Silicon waveguide based single mode thermal emitter

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (18)

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