1. Introduction

The Mid Wavelength InfraRed (MWIR) absorption spectrum is a material fingerprint so that MWIR spectroscopy plays a key role in many applications such as chemical analysis, astrophysics, gas sensing or security. MWIR also provides new opportunities for the development of short range free space communications robust to scattering and for energy conversion using thermophotovoltaics. For all these applications, compact, robust and inexpensive MWIR sources are needed. Light emitting diodes (LEDs) are a natural candidate but suffer from serious limitations in this spectral range. Their efficiency drops in the MWIR for a fundamental reason: the spontaneous emission rate decays as $\lambda ^{-3}$ where $\lambda$ is the wavelength so that non-radiative processes such as Auger recombination occur faster than light emission by electron-hole recombination. As a result, available MWIR LEDs have a significantly reduced efficiency at ambient temperature as compared to visible LEDs. Among other available MWIR sources, quantum cascade lasers are remarkably bright infrared sources and can reach large modulation frequencies but are expensive. Optical parametric oscillators (OPO) are highly tunable and very fast sources but are bulky and expensive. The only available compact and cheap MWIR sources are incandescent emitters such as hot membranes and globars [1]. These sources suffer from many limitations. Their emission is quasi isotropic and broadband. The brightness is low as the number of photons per mode is limited by Bose-Einstein statistics. Finally, available incandescent sources cannot be modulated faster than a few tens of Hz due to their thermal inertia [2,3].

At first glance, all these features seem highly detrimental to the potential of incandescent sources. However, it is important to note that these properties are not imposed by fundamental limitations stemming from physics laws, except for the low brightness. Hence, it is possible to significantly modify light emission properties of incandescent sources. To this aim, it is possible to use Kirchhoff law which states that emissivity is equal to absorptivity [4,5]. It turns out that many resonant nanostructures can be designed to control the absorption dependence on angle and spectrum. Accordingly, directional sources [6,7] and quasi-monochromatic emitters in the near field [8–10] and in the far field [11–13] have been reported. It is also possible to design incandescent sources with high efficiency [14–16] and combine directivity and optimized spectrum [17]. Recent reviews summarize the state of the art [18–20].

Finally, the possibility of modulating incandescent sources at frequencies much larger than 100 Hz by modulating the emissivity has been suggested [21,22]. Emissivity modulation up to 30 kHz has been demonstrated with mechanical electrosystems [23]. Electrical modulation of the emissivity of a stack of quantum wells up to 600 kHz has been reported [20]: the emitter is narrowband and can achieve a contrast in emissivity of 0.5 but the operating temperature of GaAs/AlGaAs was 200$^{\circ }$C. Emission with temperatures up to 500$^{\circ }$C has been achieved with GaN/AlGaN quantum wells at the expense of a lower modulation rate of 50 kHz and a lower emissivity contrast [24]. Other mechanisms for electrical modulation of optical properties can also be used [25–27]. Nanosecond modulation has been demonstrated using an external laser to modulate the doping and therefore the emissivity [28]. For all these devices, the temperature is assumed to be homogeneous in the incandescent source so that Kirchhoff’s law can be used to design the source.

An alternative to emissivity modulation is temperature modulation. Hot membranes can be modulated up to 100 Hz due their thermal inertia [1,2]. Achieving faster modulation rates has been reported with two class of non-equilibrium systems. First, systems with an electronic temperature much higher than the lattice temperature benefit from a fast cooling rate on the picosecond range due to the electron-phonon interaction. A second class of sources includes thin hot emitters deposited on a cold substrate. Here, the electrons are thermalized with the emitter phonons. The fast dynamics is ruled by the conduction time through the emitter thickness. This time scales as $h^2/\kappa$ where $h$ is a typical thickness and $\kappa$ is the thermal diffusivity so that a thin emitter can have a fast cooling rate.

Let us first review experiments belonging to the first class. Thermal emission due to hot electrons in metallic nanostructures has been reported in the visible. Light emission by nanoscale constrictions has been studied by many groups [29–31]. Emission in the near infrared (NIR) by graphene has been reported recently and pulses on the order of 100 ps have been obtained [32–34]. Despite these performances that surpass light emitting diodes and can be compared with laser sources, there are a number of limitations. The emitting sources have a small emitting area with a characteristic length on the order of a few micrometers, the emissivity is low, on the order of 2% for a layer of graphene and the emitted radiation is broadband and unpolarized. Finally, the power emitted in the MWIR is too low to be detected. Emission in the MWIR requires larger samples with well-controlled fabrication procedures. An alternative to graphene has been reported using hot electrons in quantum wells to study emission in the MWIR [35] and modulation up to 500 kHz has been observed. For quantum wells, the electrons are in equilibrium with optical phonons and the relaxation time is controlled by the coupling to acoustic phonons.

We now turn to the second class of emitters where the electrons are in equilibrium with phonons so that the time response is governed by heat diffusion and no longer by electron-photon interaction. Using a small hot volume deposited with a good thermal contact on a cold substrate which behaves as a thermal sink, it is possible to ensure a fast thermal relaxation. A NIR source based on carbon nanotubes [36] with an area of 60 $\mathrm {\mu }\textrm {m}^2$ has been reported with a modulation frequency up to 1 GHz and an estimated power on the order of $10^{-11}$ W/sr. Moving to the MWIR range, modulation frequencies up to 100 kHz have been demonstrated [37,38] by means of electrically heated graphene multilayers on SiO$_2$/Si. The control of the emission spectrum by combining graphene emitters with photonic crystals [34] or plasmonic metasurfaces [39] has been reported. A recent paper [40] reports a detailed study of an electrically-heated platinum metasurface with an area of $10^4$ $\mathrm {\mu }\textrm {m}^2$ emitting in the MWIR transparency window with a linearly polarized emissivity reaching 0.8 at 5.1 $\mathrm {\mu }$m with a 1.5 $\mathrm {\mu }$m FWHM and sustaining an emission modulation beyond 10 MHz. This class of emitters appears to be robust and can be fabricated over large surfaces using standard techniques. However, it has been reported that there is a trade-off between frequency and modulation. For instance, the estimated wall-plug efficiency in [40] was $6.9 \, 10^{-6}$ at 20 kHz, and $2.2 \, 10^{-7}$ at 20 MHz.

The analysis of the trade-off between efficiency and modulation frequency involves exploring the interplay between many factors including the emitter size, the operating temperature, the input voltage, the thermal management of the device and the modulation regime, pulsed or harmonic. To address these issues, we need to compute the spatial and temporal dependence of the temperature field. We also need a model of thermal emission for nonequilibrium bodies with a temperature field depending on position and time. The usual Kirchhoff law cannot be used as we deal with emitting bodies with a temperature which is neither uniform nor stationary. Here we analyze the temperature field and we use the local Kirchhoff law [5] to compute the emission by a body with a temperature field $T(\mathbf {r},t)$ which depends on space and time. The paper has two goals: firstly, to introduce a framework to analyze these issues, secondly, to clarify the role of the different parameters and provide some guidelines to increase the efficiency of fast incandescent metasurfaces.

2. Description of the emitter

2.1 Qualitative analysis of the trade-off between fast modulation and efficiency

Let us first discuss with simple arguments the origin of the trade-off between modulation frequency and efficiency. The basic idea for fast temperature modulation is to have a hot emitter on a cold substrate with a small volume so as to minimize its thermal inertia [41]. Let us call $A$ its area, $V=h A$ its volume, $\rho$ and $C_p$ the mass per unit volume and specific heat respectively. We will refer to the reference temperature as $T_0$ and the temperature of the sample as $T$. The spectrally averaged emissivity of the sample will be denoted $\bar {\varepsilon }$.

For a hot emitter, three main types of energy transfer to its environment can occur: conduction to the substrate, convection and radiative processes. A simple power balance gives the time scales on which convection takes place $\tau _{\mathrm {conv}} \sim \frac {\rho C_p V}{h_{\mathrm {cv}} A}$ where $h_{\mathrm {cv}}$ is a convection heat transfer coefficient between the hot emitter and the surrounding air which is typically on the order of 10 $\mathrm {W m^{-2} K^{-1}}$. Similarly, radiative losses occur on time scales on the order of $\tau _{\mathrm {rad}}\sim \frac {\rho C_p V \Delta T}{\bar {\varepsilon } \sigma T^4 A}$ where $\Delta T$ is the body temperature increase, $T$ its temperature and $\sigma$ is the Stefan-Boltzmann constant. Finally, conduction time constant is given by $\tau _{\mathrm {cond}} \sim h^2/\kappa$ where $\kappa$ is the body thermal diffusivity. This time provides an estimate of the diffusion time across the body. Taking $\rho C_p = 10^6 \, \mathrm {J m^{-3} K^{-1}}$ which is a typical order of magnitude, $\Delta T = 100^{\circ }$C and assuming the emitter to be of cubic shape with characteristic length $L$, $\tau _{\mathrm {conv}}$ and $\tau _{\mathrm {rad}}$ are on the same order of magnitude and depend linearly on $L$ whereas $\tau _{\mathrm {cond}}$ depends quadratically on $L$. As a consequence, for small objects with characteristic length $L \ll \frac {\kappa \rho C_p \Delta T}{\bar {\varepsilon } \sigma T^4} \sim 1$ m, conduction is the main physical phenomenon governing the dynamics of the source cooling. Fast temperature modulation can then be achieved. For example, taking $h=50$ nm and $\kappa =10^{-5} \mathrm {m^2 s^{-1}}$, the cooling time is on the order of 0.25 ns. This crude estimate assumes that the substrate is cold so that it only provides a lower bound of the cooling time.

In summary, thermal conduction from a hot tiny emitter to a cold substrate ensures fast cooling. However, for an incandescent source to be optimal, the power supplied should be radiated and not lost through conduction. Therefore, such an incandescent source cannot be both fast and efficient at the same time. The purpose of the paper is to discuss how to optimize the trade-off between the two phenomena.

2.2 Description of the geometry of the emitter

The following study will be based on the device sketched in Fig. 1. It consists in an array of quasi-1D platinum metallic wires (MWs) emitters deposited on a semi-infinite cold substrate. While the numerical values are taken from Ref. [40], the analysis is general and can also be applied to similar devices (see e.g. [38]). A voltage applied to each end of the MWs drives the device, which heats up via Joule effect. The device emits polarized MWIR radiation and can be modulated beyond 10 MHz.

 

Fig. 1. Electrically modulated device emitting polarized, spectrally selective MWIR radiation. Geometrical parameters are listed in Table 1.

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Table 1. Geometrical parameters of device sketched in Fig. 1 and thermal parameters used in this paper.

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The driving idea leading to such design is fast modulation of emission through temperature modulation. As mentioned earlier, conduction to a colder underlying material is the dominant loss channel for the energy stored in the MWs. A good thermal contact between the wires and the substrate is therefore needed and should be unaltered at high temperature. Platinum on silicon nitride ($\mathrm {SiN_x}$) is stable and can go at least up to $650^\circ$C [1]. The device is large enough (100 $\mathrm {\mu }$m$\times$ 100 $\mathrm {\mu }$m) to emit detectable radiation despite a small height (25 nm) needed for a short time response. In order to increase the emissivity and control the emission spectrum, the $\mathrm {SiN_x}$ layer can be deposited on a mirror. This is ignored in the present study for the sake of simplicity.

3. Modeling the wall-plug efficiency

3.1 Modeling the emitted power

In order to estimate the device efficiency, a theoretical model of the emitted power is needed. The device is not isothermal because only the MWs are heated by Joule effect. Hence, Kirchhoff law cannot be used to compute the emission. However, it is possible to use a local form of Kirchhoff law [5]. The local absorption cross-section density is equal to the local emission density of a volume element $\mathrm {d}^3\mathbf {r}$ at point $\mathbf {r}$. The local emission density can be used to compute the body emission if the temperature field is known. Denoting the local emission density $\eta ^{(\ell )}(\mathbf {u}, \mathbf {r}, \lambda )$ where $\ell = s, p$ stands for polarization, the spectral power radiated by the device in direction $\mathbf {u}$ per unit solid angle $\mathrm {d} \Omega$ at time $t$ reads

Comparing Eq. (2) to the radiometric definition of the emitted spectral power:

Note that if the local polarized emissivity is peaked around $\lambda _0$ with spectral width $\Delta \lambda$ as in ref. [40], we can introduce a spectrally averaged emissivity $\overline {\varepsilon ^{(\ell )}}$. We can express the spectrally integrated flux $\phi _{\mathrm {rad}}(\mathbf {u},t)$ as follows:

This form can be extended using a piecewise model of the emissivity with a central wavelength $\lambda _{0,i}$, a spectral width $\Delta \lambda _i$ and an emissivity $\overline {\varepsilon _i^{(p)}}$. Eq. (2) finally reads

Without loss of generality, we will consider an emissivity spectrum with a single peak in the following discussion.

Note that if the substrate is not transparent and if the time scale is such that the temperature increase in the substrate is not negligible, the substrate contribution will need to be accounted for: it is expected to have a different spectrum, polarization and time dependence. In the following, we will take $\lambda _0 = 5.1$ $\mathrm {\mu }$m and $\Delta \lambda = 1.5$ $\mathrm {\mu }$m as in Ref. [40] to carry out numerical estimates, without loss of generality. Table 1 lists numerical values of other relevant physical quantities.

3.2 Definition of the wall-plug efficiency

The wall-plug efficiency is defined as the ratio between the fraction of radiated power which is modulated and the electric power supplied to the emitter. We define the modulated radiated power as the contrast in the emitted signal compared to a reference level of the emitted signal. Here, we consider either sinusoidal modulation of the applied voltage or a series of pulses, the repetition rate being the modulation frequency. As both regimes can be used for detection or communication applications, it is of interest to investigate which one performs better in terms of efficiency. To proceed, we define more specifically the efficiency for a periodic series of pulses and for harmonic modulation. The temperature in the MWs reads $T(t) = T_0 + T_{\mathrm {DC}} + \Delta T(t)$ where $T_0$ is the temperature in the absence of heating, $T_{\mathrm {DC}}$ the stationary temperature increase and $\Delta T(t)$ the time-dependent temperature increase.

Pulse modulation We consider a constant voltage $U_0$ applied to the device from $t=0$ until $t_{\mathrm {stop}}$. The temperature in the MWs rises to a maximum at the end of the heating pulse and decreases once the voltage is switched off. In this context, it is reasonable to define the useful signal as the total energy that is emitted by the MWs until a threshold time $t_{\mathrm {lim}}$ at which the contrast in the instantaneous radiated power becomes less than 1% of the maximum contrast, i.e. $I_{\mathrm {BB}}(\lambda _0, T_0 + T_{\mathrm {DC}} +\Delta T(t_{\mathrm {lim}})) - I_{\mathrm {BB}}(\lambda _0, T_0 + T_{\mathrm {DC}} ) = 0.01 \left [ I_{\mathrm {BB}}(\lambda _0, T_0 + T_{\mathrm {DC}} +\Delta T_{\mathrm {Max}}) - I_{\mathrm {BB}}(\lambda _0, T_0 + T_{\mathrm {DC}} ) \right ]$. We will refer to $t_{\mathrm {lim}}$ as the signal pulse width. Note that a natural modulation rate of the source is given by $1/t_{\mathrm {lim}}$ in order to ensure a large contrast. Note also that by increasing $t_{\mathrm {stop}}$, the maximum temperature increases so that $t_{\mathrm {lim}}$ also increases.

Denoting $\bar {\varepsilon }$ the emissivity averaged over polarization $(\overline {\varepsilon ^{(s)}}+\overline {\varepsilon ^{(p)}})/2$ at $\lambda _0$ and $\Delta \lambda$ the spectral emissivity width and assuming the emitter to be isotropic, we can integrate the emitted power over the solid angle. The efficiency $\rho _{\mathrm {impulse}}$ in the pulse regime is then given by:

Harmonic modulation The sample is driven by a generator with an oscillating voltage $U_0 \cos (\omega _{\mathrm {elec}} t)$ delivering a RMS power $P_{\textrm {elec,tot}} = U_0^2/(2 R_{\mathrm {sample}})$. The MWs temperature $T(t) = T_0+T_{\mathrm {DC}}+\Delta T(t)$ will consequently oscillate twice as fast with $\Delta T(t) = \Delta T \, \cos (\omega _{\mathrm {th}}t)$, where $\omega _{\mathrm {th}} = 2\omega _{\mathrm {elec}}$. In contrast to pulse feeding, the information conveyed by the source is given by the oscillation amplitude in the emitted signal at $\omega _{\mathrm {th}}$. The maximum contrast of the instantaneous emitted power in the fundamental mode $\omega _{\mathrm {th}}$ of the emitted power is the relevant quantity. The efficiency $\rho _{\mathrm {mod}}$ is thus given by:

The input power would be $P_{\mathrm {elec}} = U_0^2/\left (2 R_{0}\right )$, assuming the MWs electrical resistance remains constant despite temperature variations.

From these definitions, it is seen that several parameters can be used to leverage the efficiency of the source: the applied voltage (or the input power), the reference signal level, the emitter area and the pulse rate or modulation frequency.

4. Temperature modulation

In order to estimate the efficiency of the device, we have to derive the temperature reached in the emitting part of the device. When computing the temperature modulation of the MWs, it is critical to compute the heating of the substrate as a function of time and position. Let us consider the pulsed regime to illustrate this statement. The cooling by heat diffusion through the platinum wire thickness occurs on a timescale on the order of $\tau _{\mathrm {diff,Pt}} = h_{\mathrm {MW}}^2/\kappa _{\mathrm {MW}} \sim 10^{-11}$s. The power dissipated in the MWs during this time is given by $\tau _{\mathrm {diff,Pt}} \, U_0^2/R_{0}$. The maximum temperature increase of the MWs can be estimated by assuming that this power has not diffused towards the substrate yet, resulting in an increase of temperature given by

The temperature increase in the MWs is then expected to be on the order of

4.1 Pulse modulation

The non-stationary temperature field in the substrate can be written as $T(x,y,z,t)=T_0 + T_{\mathrm {DC}} +\Delta T(x,y,z,t)$. The temperature increase $\Delta T(x,y,z,t)$ is obtained by solving the diffusion equation in the substrate with the relevant boundary conditions. The initial condition is $\Delta T(x,y,z,0)=0$. The boundary conditions are given for the field temperature and for the flux: $\Delta T(x,y,z,t)$ takes finite value for $x,y\to \pm \infty$ and $z\to +\infty$; the conduction flux $-K_{\mathrm {sub}}(\partial \Delta T/\partial z)$ at $z=0$ is equal to the electric power density below the wire $P_{\mathrm {elec}}\Lambda /(wL_{\mathrm {sample}}^2)$ and is zero between the wires accounting for negligible losses in air. Neglecting radiation and natural convection is an excellent approximation as a typical flux density exchanged with an environment at temperature $T_{0}$ is given by $h(T(x,y,0)-T_{0})$ with typically ${h=10 \, \mathrm {W m^{-2} K^{-1}}}$. Using a typical temperature difference of 100 K yields a flux density of $10^3 \, \mathrm {W m^{-2}}$ which is five orders of magnitude smaller than what is used with an electrical power of 1 W dissipated in an area $L_{\mathrm {sample}}^2=10^{-8} \, \mathrm {m}^{2}$.

The solution is given as a convolution product of the imposed flux over space and time and the Green function of the problem. The flux is non zero in the area of the $2N+1$ wires and for $t \in [0,t_{\mathrm {stop}}]$. Details are provided in section A1 of the appendix. The solution for $t<t_{\mathrm {stop}}$ is

4.2 Harmonic modulation

Harmonic modulation of the electrical voltage at frequency $\omega _{\mathrm {elec}}$ results in a power modulation at $2\omega _{\mathrm {elec}}$. The temperature is thus driven at $\omega _{\mathrm {th}}=2\omega _{\mathrm {elec}}$. While the system is two-dimensional, a one-dimensional solution can be used. For modulation times longer than $\Lambda ^2/\kappa _{\mathrm {sub}}$, the heat can diffuse between two MWs before the temperature field changes in time. As a consequence, the temperature field in the substrate does not vary between MWs and is homogeneous below the source. This is the regime of interest for modulation frequencies below $\kappa _{\mathrm {sub}}/\Lambda ^2$ which is on the order of $40$ MHz for the values given in Table 1.

Depending on the total size of the source $L_{\mathrm {sample}}$, two different trends are observed. If the modulation time is small compared to $L_{\mathrm {sample}}^2/\kappa _{\mathrm {sub}}$ while remaining large compared to $\Lambda ^2/\kappa _{\mathrm {sub}}$, the one-dimensional solution is valid. The conduction flux is normal to the interface. This is the regime of interest for our application (see Fig. 2). We will see that it leads to a temperature varying as $1/\sqrt {\omega _{\mathrm {th}}}$. If the modulation is slow with a modulation time larger than $L_{\mathrm {sample}}^2/\kappa _{\mathrm {sub}}$, then the whole source appears as a point-like source so that the conduction flux is radial.

 

Fig. 2. Sketch of the MWs. For $w^2/\kappa _{\mathrm {sub}}\ll 2\pi /\omega _{\mathrm {th}} \lesssim \Lambda ^2/\kappa _{\mathrm {sub}}$, the concentric half-circles represent temperature wavefronts and for $\Lambda ^2/\kappa _{\mathrm {sub}}\ll 2\pi /\omega _{\mathrm {th}} \ll L_{\mathrm {sample}}^2/\kappa _{\mathrm {sub}}$, the lines stand for the temperature wavefronts in the 1D regime.

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In what follows, we consider the one dimensional case. We solve the harmonic diffusion equation, assuming that the electrical power supplied to the MWs is fully dissipated through heat conduction so that the flux is imposed at $z=0$. In addition, we assume that the temperature increase at infinity is zero $\Delta T(+\infty,t)=0$. Details are given in section A2 of the appendix. The temperature field in the substrate is given by a damped thermal wave [42]:

In addition to this temperature modulation, a DC heating is expected due to the RMS input power flowing through the substrate. The induced temperature difference across the substrate is given by $T_{\textrm {DC}} = R_{\mathrm {th}} U_0^2/(2R_0)$, where $R_{\mathrm {th}}$ is the thermal resistance of the substrate.

We can now use the dependence of the temperature modulation to explore the trade-off between modulation and efficiency. An important issue is the temperature amplitude. Given that Planck’s law is not linear with temperature, operating at high temperature may be beneficial to enhance radiation. We thus discuss separately the regimes where the radiated power is linear and nonlinear with the temperature increase.

5. Linear regime

The Planck function has a nonlinear dependence on the temperature due to the Bose-Einstein distribution $[\exp (hc/\lambda _0 k_BT)-1]^{-1}$. The validity of the linearization of Planck function is dependent both on the temperature and the frequency. Inserting a first order Taylor expansion in the definition of the efficiency Eq. (6) and Eq. (7) leads respectively to

5.1 Influence of the emitting area $L_{\mathrm {sample}}^2$

Although somewhat counter-intuitive at first sight, the sample area does not influence the efficiency of the source. Indeed, in the linear regime and for a fixed input power, a larger emitting area also corresponds to a smaller temperature increase as we will see.

Pulse modulation The sample surface appears as a prefactor in Eq. (14). However, this term is compensated by its inverse in the temperature dependence Eq. (11). A more intricate dependence on the sample size in $\Xi _n$ can be neglected after time integration in Eq. (11).

Harmonic modulation In the linear regime, the efficiency of the source is directly proportional to the MWs temperature increase $\Delta T$ which in turn depends on geometrical and thermal parameters as described in Eq. (13). Here again, the efficiency of the harmonically modulated source in the linear regime does not depend on the size of the source.

Increasing the area can be beneficial to increase the emitted power only if the maximum temperature increase needs to be kept at low values due to other reasons such as material degradation for example.

5.2 Influence of the thermal resistance of the substrate

The two forms of the efficiency are proportional to the derivative of the Planck function taken at $T_0 + T_{\mathrm {DC}}$. As Planck function increases non-linearly with temperature, it is beneficial to increase this temperature. As mentioned at the end of section 4.2, the DC temperature increase in the sample due to the RMS input power $T_{\textrm {DC}} = R_{\mathrm {th}} U_0^2/(2R_0)$ which depends linearly on the electrical power. It is thus seen that increasing the thermal resistance $R_{\mathrm {th}}$ by insulating the system will result in an increase of the operating temperature which is beneficial for the efficiency of the source. This effect is clearly seen in Fig. 3 where the efficiency is plotted for two different operating temperatures $T_0 + T_{\mathrm {DC}}$ equal to $500$ K and $950$ K.

 

Fig. 3. Device efficiency. (a): Efficiency for harmonic feeding as defined in Eq. (15) against modulation frequency $\omega _{\mathrm {th}}$ for $U_0 = 20$V and for reference temperatures $T_0 + T_{\mathrm {DC}} = 500, 950$ K. Note that the curve at 500 K is truncated for low frequencies to limit the curve to the domain of validity of the infinite substrate approximation. (b): Efficiency for pulsed feeding as defined in Eq. (14) against $1/t_{\mathrm {lim}}$ for $U_0 = 20$ V and for reference temperatures $T_0 + T_{\mathrm {DC}} = 500, 950$K.

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5.3 Influence of the modulation frequency

Harmonic modulation In the linear regime, the efficiency given by Eq. (14) depends on the temperature increase normalized by the electric power. From Eq. (13), a key point is the decrease of the temperature $\Delta T(\omega _{\mathrm {th}})$ with increasing modulation frequency with a $\left (\omega _{\mathrm {th}}\right )^{-1/2}$ behavior. This directly affects the efficiency of the source as a function of frequency as seen in Fig. 3(a).

Pulse modulation The impact of time dynamics on the efficiency is less straightforward for a pulse modulation and numerical estimations are needed. We plot in Fig. 3(b) the evolution of the efficiency for a pulse voltage of 20 V as a function of the inverse pulse duration $1/t_{\mathrm {lim}}$, where $t_{\mathrm {lim}}$ is defined in section 3.2. Briefly, the parameter $t_{\mathrm {lim}}$ measures the signal pulse width. In the same way as for the harmonic case, a drop of the efficiency is observed as the pulse width decreases.

In summary, we found that the efficiency does not depend on the emitter area. It can be improved significantly by increasing the emitter DC temperature using a large thermal resistance of the sample. This procedure improves the efficiency without extra energy cost. The modulation type does not affect significantly the efficiency.

6. Nonlinear regime

In this section, we briefly explore numerically the efficiency without using the linearized formulas. We aim at exploring the validity of the linearized formulas and the potential benefits of the nonlinearity of Planck’s law.

Harmonic modulation In Fig. 4(a), we plot the efficiency in the harmonic regime when operating at $U_0 = 20$V resulting in a reference temperature of $724$ K and considering emission at 2 $\mathrm {\mu }$m and 5.1 $\mathrm {\mu }$m for different modulation frequencies. It is seen that the linearized formula is in agreement with the exact form at 5.1 $\mathrm {\mu }$m but a nonlinear increase shows up for shorter wavelengths. The temperatures corresponding to the different modulation frequencies are plotted in Fig. 4(c). In agreement with the analytical formula, they decay as $1/\sqrt {\omega _{\mathrm {th}}}$. These temperatures are obtained with an electrical power of 8 W used to heat a $100\times 100$ $\mathrm {\mu }\textrm {m}^2$ square emitter. In summary, when operating at temperatures dictated by the materials below 1000 K and in the mid infrared, the emitter runs in the linear regime.

 

Fig. 4. (a): Device efficiency against signal modulation frequency, according to definitions in Eq. (7) (exact) and Eq. (15) (linearized) under harmonic feeding for ${U_0 = 20 \textrm { V}}$ resulting in $T_0+T_{\mathrm {DC}} = 724$ K. (b): Device efficiency against $1/t_{\mathrm {lim}}$ according to definitions in Eq. (6) (exact) and Eq. (14) (linearized) under pulsed feeding for ${U_0 = 50 \textrm { V}}$ and for reference temperatures ${T_0 + T_{\mathrm {DC}} = 500\textrm { K}}$. (c) and (d): Corresponding temperature increase in the pulsed and harmonic regimes respectively.

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Pulse modulation We now consider the pulsed regime. It is seen in Fig. 4(d) that the maximum temperature can be much larger when using short pulses with large voltages. The temperature variation when driving the system with voltages of $50$ V may exceed $1000$ K. The maximum temperature depends on the duration of the heating pulse. It can be evaluated using a simple energy conservation argument. Let us assume that the heat flux is 1D. The heat delivered by the heater with area $L_{\mathrm {sample}}^2$ is confined in a sub-volume $L^2_{\mathrm {sample}} \delta z$ where $\delta z$ is the typical length over which heat diffuses in the substrate during the heating time. We can write the energy balance yielding $P_{\mathrm {elec}} t_{\mathrm {heating}} = \rho C_p \, \delta z \, f L_{\mathrm {sample}}^2 \Delta T_{\mathrm {Max}}$. An order of magnitude estimation gives $\delta z = \sqrt {\kappa _{\mathrm {sub}} t_{\mathrm {heating}}}$ therefore

As expected, the maximum temperature increases as the heating time increases.

We now plot the efficiency as a function of the pulse repetition period chosen to be $t_{\mathrm {lim}}$ defined in 3.2. It is seen in Fig. 4(b) that the linearized form of the efficiency underestimates the efficiency for large temperatures, particularly for short wavelengths. In this nonlinear regime, the emitted signal is increased when reducing the emitter area.

Finally, we emphasize a key difference between the harmonic and the pulsed regime. The data in Fig. 4(c) were obtained with a RMS electric power of 8 W whereas the power used in Fig. 4(d) varies with the repetition rate but is always below 1 W: the same peak temperature and therefore the same peak signal can be obtained with a much lower RMS power when operating in the pulsed regime.

In summary, for a fixed electrical power, smaller surfaces leads to larger signals and better efficiency when entering the nonlinear regime.

7. Conclusion

In summary, we have introduced a framework to model the efficiency of a modulated incandescent source. The model is based on the local Kirchhoff law which allows us to compute the flux emitted by a body with a temperature field depending on space and time. The second ingredient is the derivation of the spatial and temporal dependence of the temperature field. Using this model, we have analyzed the factors influencing the efficiency of the source. Although a drop in the efficiency of such sources at high frequencies is unavoidable, it is possible to adjust parameters to boost it.

We have shown that, for a fixed electrical power, the emitted power does not depend on the sample area in the linear regime. An important parameter is the DC operating temperature. In all regimes, be it harmonic or pulsed, the emitted signal and the efficiency increase when boosting the operating temperature so that only material limitations at high temperatures should be taken into account. Increasing the operating DC temperature can be achieved with no further energetic cost by increasing the conduction resistance of the substrate.

We have found that the efficiency is similar for harmonic and pulsed modulation in the linear regime. However, a given signal peak amplitude can be obtained in the pulsed regime with an average input electrical power smaller by an order of magnitude. Finally, we have found that an increased efficiency can be obtained due to the non-linearity of Planck function when considering large temperature modulations, particularly at short wavelengths and in the pulsed regime. Larger temperature modulations can be obtained by reducing the area of the emitter at constant input power.