# Modelling Miniature Incandescent Light Bulbs for Thermal Infrared *‘THz Torch’* Applications

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## Abstract

The ‘*THz Torch*’ concept is an emerging technology that was recently introduced by the authors for implementing secure wireless communications over short distances within the thermal infrared (20-100 THz, 15 μm to 3 μm). In order to predict the band-limited output radiated power from ‘*THz Torch*’ transmitters, for the first time, this paper reports on a detailed investigation into the radiation mechanisms associated with the basic thermal transducer. We demonstrate how both primary and secondary sources of radiation emitted from miniature incandescent light bulbs contribute to the total band-limited output power. The former is generated by the heated tungsten filament within the bulb, while the latter is due to the increased temperature of its glass envelope. Using analytical thermodynamic modelling, the band-limited output radiated power is calculated, showing good agreement with experimental results. Finally, the output radiated power to input DC power conversion efficiency for this transducer is determined, as a function of bias current and operation within different spectral ranges. This modelling approach can serve as an invaluable tool for engineering solutions that can achieve optimal performances with both single and multi-channel ‘*THz Torch*’ systems.

## Keywords

*THz Torch*thermal infrared thermodynamics band-limited blackbody source

## 1 Introduction

The thermal infrared frequency bands, from 20 to 40 THz (15 μm to 7.5 μm) and 60 to 100 THz (5 μm to 3 μm), are best known for applications in thermography. There has been little in the way of enabling technologies within this part of the electromagnetic spectrum to support wireless communications. However, this largely unused spectral range offers opportunities for the development of secure communications. To this end, the ‘*THz Torch*’ concept was recently introduced by the authors [1-6]. The ‘*THz Torch*’ technology fundamentally exploits engineered blackbody radiation, by partitioning thermally-generated noise power into pre-defined frequency channels. The band-limited power in each channel is then independently pulsed-modulated, transmitted and detected, creating a robust form of short-range secure communications in the thermal infrared. In this paper, the radiation mechanisms associated with the basic transducer within the ‘*THz Torch*’ transmitter will be investigated, leading to the calculation of output radiated power and then output radiated power to input DC power conversion efficiency for this transducer.

The first incandescent light bulb to employ a tungsten filament was patented in 1904 by Just and Hanaman [7], offering greater luminosity in the visible spectrum, when compared to carbon filaments. Since then, a great deal of research has been undertaken to investigate the optical, electrical, chemical and thermal properties of tungsten materials; as well as the characteristics of tungsten light bulbs [8-14]. More recently, due to advances in materials and nanotechnology, higher luminous efficiency has been achieved by improving the emissivity of the filaments or reducing the infrared radiation contribution to the blackbody spectrum without reducing the radiation at visible wavelengths [15-17].

With traditional incandescent light bulb applications, only output radiated power within the visible spectrum is considered useful, while the remaining energy is considered to be lost. This explains why they are highly inefficient when compared to white light-emitting diodes (LEDs). Fortunately, these bulbs represent a low cost far/mid-infrared thermal source, which has been exploited by the authors to implement secure short range wireless communications. However, for the thermal infrared, the modelling approaches used for the visible spectral range is insufficient for predicting the band-limited output power from incandescent light bulbs. Therefore, an analytical thermodynamic modelling approach needs to be developed, so that the performance of both single and multi-channel *‘THz Torch’* thermal infrared systems can be optimised.

In the thermal infrared, technologies have also been developed to improve the emission efficiency of thermal sources at specific wavelengths by modifying the blackbody radiation using periodic microstructures [18-22]. However, most of these studies are based on sophisticated and time-consuming technologies, such as electron beam lithography or a repetitive etching-and-deposition process [17]. Therefore, these devices cannot be easily mass produced. Most of the thermal infrared sources on the market still rely on *untrimmed* blackbody radiation using materials with high emissivity. It should be noted that although such thermal-based sources offer many benefits (e.g., simplicity, ease of tuning and affordability), the main drawback is that there is no signal coherency, as with all unmodulated noise sources. Thus, only the intensity of band-limited output power can be controlled.

*THz Torch*’ transmitter consists of five Eiko 8666-40984 miniature incandescent light bulbs, having a length of 6.3 mm and diameter 2.6 mm [1-6]. These bulbs are connected in series and assembled into a compact cylindrical package having an outer diameter of 8.2 mm, as shown in Fig. 1. In the thermal infrared, such commercial-off-the-shelf (COTS) bulbs are not perfect blackbody radiators (since their tungsten filaments have low emissivity), their windows have relatively poor transmission characteristics (due to the high absorption and reflectance of the glass envelopes) and without an additional collimating lens and back reflector the spreading loss is high. Nevertheless, these COTS transducers offer a very low cost means of converting input electrical power into output thermal infrared radiation.

The thermal analysis of incandescent light bulbs involves all three fundamental methods of heat transfer: radiation, conduction and convection. This analysis is inherently complex, as it requires the study of a number of interacting mechanisms: (a) primary radiation from the filament (which appears mostly in the frequency spectrum between millimeter-wave and beyond visible); (b) absorption of primary radiation by the glass envelope, causing it to heat up; (c) thermal convection inside the glass envelope, causing the filament to heat up the glass envelope; (d) thermal conduction within the glass envelope and also within the two electrical leads to the outside world, which act as poor heat sinks; (e) secondary radiation from the glass envelope (which appears mostly in the thermal infrared spectral region, due to a much lower outside surface temperature); (f) thermal conduction from the glass envelope to the contacting environments on both sides; and (g) thermal convection outside the glass envelope. Due to this inherent complexity, it is not possible to individually quantify the effects of all the mechanisms.

The output radiated power contributed by the primary source of radiation lies in the spectral region dominated by high transmittance through the glass envelope; while that from secondary radiation lies in the spectral region dominated by high absorptance within the glass envelope. The power transmittance for typical (soda lime silica) window glass will first be calculated, and the band-limited output radiant intensity due to primary radiation will be determined. Then, the outside surface temperature of the glass envelope will be measured directly and the band-limited output radiant intensity due to secondary radiation will be determined. The combined band-limited output radiated power is then measured, using a calibrated thermal detector, which verifies our modelling of the two radiation sources for this basic transducer.

## 2 Primary Radiation Modelling

Primary radiation is defined as the output power that is generated directly from the tungsten filament and passes through the glass envelope. When a bias electrical current is applied, the temperature of the filament increases, due to Joule heating. In theory, the filament will radiate electromagnetic energy across the whole frequency spectrum. An inert gas (e.g., argon) is normally used to fill the inside of the bulb; to prevent the tungsten filament from oxidizing, which would otherwise result in the catastrophic failure of the transducer.

### 2.1 Filament Working Temperature Estimation

*CSA*[cm

^{2}] and length

*l*[cm]. The electrical resistivity of tungsten

*ρ(T)*, as a function of absolute temperature

*T*, can be expressed in terms of the filament resistance

*R(T)*:

*ρ(T)*against temperature, shown in Fig. 2, can be accurately modelled by the following empirical quadratic expression

At room temperature, *ρ*(288 K) = 5.14 × 10^{-6} Ω.cm [23]. The combined resistance of five Eiko 8666-40984 bulbs connected in series is directly measured to be 23.6 Ω at room temperature. Therefore, if the parasitic resistances of the short electrical leads are considered negligible (which is a good assumption), this gives an average value of *R*(288 K) = 4.72 Ω for each bulb filament. As a result, the effective ratio of cross-sectional area to length can be extracted using (1) and (2), to give a value of (*CSA/l*)_{ eff } = 10.9 nm, which is assumed to be temperature independent. By indirectly measuring the resistance of a bulb, at a specific bias current, the working temperature can now be estimated to an acceptable degree of accuracy.

### 2.2 Ideal Spectral Radiance Estimation

*I*

_{ BB }(

*λ*,

*T*) can be obtained by applying Planck’s law, to give

*I*

_{ BB }(

*λ*,

*T*) is the power radiated per unit area of emitting surface in the normal direction per unit solid angle per unit wavelength at the absolute temperature

*T*for the blackbody radiator;

*λ*is the free space wavelength;

*h*is the Planck constant;

*c i*s the speed of light in a vacuum;

*k*

_{ B }is the Boltzmann constant; and

*b*is Wien’s displacement constant. Fig. 3(a) shows the ideal spectral radiance against wavelength at different temperatures.

*ε*(

*λ*,

*T*) of the emitter must be included. If the ambient temperature is assumed to be

*T*

_{ 0 }[K], the net spectral radiance emanating from the radiator is given by

With the earliest experiments, the Eiko 8666-40984 bulbs had a quiescent DC biasing current of 44 mA, which gives an estimated filament working temperature of 772 K and a corresponding spectral radiance peak at 80 THz (3.75 μm), as shown in Fig. 3(b). This peak frequency can be easily adjusted by changing the bias current. With a larger bias current, one can obtain higher spectral radiance levels, yielding an increased integrated output power for transmission. However, the penalty for this is a decrease in the band-limited output radiated power to input DC power conversion efficiency for this transducer; which may be an issue where available DC supply power is at a premium (e.g., coin battery powered security key fob applications).

*A*

_{ eff }[m

^{2}] is the total effective radiating area of the radiator; and

*λ*

_{ 1 }and

*λ*

_{ 2 }are the free space wavelengths associated with the lower and upper frequencies of interest, respectively.

### 2.3 Bulb Filament Emissivity Estimation

*ε*(

*λ*,

*T*), defined as the ratio of energy radiated by a material to that radiated by an ideal blackbody at the same wavelength and temperature, is a parameter that characterises this efficiency. In practice, if the spectral range is not too large, the emissivity at the surface of a material is only a function of temperature (to a good approximation). Measured data [23] for the spectrum-average emissivity of tungsten \( \overline{\varepsilon_{filament}\left({T}_{filament}\right)} \), shown in Fig. 4, can be accurately modelled by the following linear expression

### 2.4 Band-limited Radiant Intensity from Primary Radiation

*T*

_{ filament }represents the working temperature for all identical filaments, extracted after determining the filament’s resistance;

*A*

_{ eff _ filament }=

*A*

_{ filament }/2 [m

^{2}] is the estimated total effective radiating area (which assumes that only radiation in the outward direction is considered) for all five filaments, if the total radiant intensity for the 5-bulb array is required;

*T*

_{ G }(

*λ*) is the power transmittance of the glass envelope; and

*I*

_{ NET }(

*λ*,

*T*

_{ filament }) is the net spectral radiance of the filament at

*T*

_{ filament }.

*T*

_{ G }(

*λ*) and

*A*

_{ eff _ filament }first have to be obtained. With the former, the inert gas within the glass envelope is assumed to be air (which is a good assumption); resulting in air-glass and glass-air boundaries. This problem can be represented by an analogous 2-port network (adapted from [24]), as illustrated in Fig. 5. Here, \( {\widehat{P}}_i \), \( {\widehat{P}}_r \) \( {\widehat{P}}_a \) and \( {\widehat{P}}_t \) represent the incident, reflected, absorbed and transmitted power of the associated electromagnetic waves, respectively. Within the glass, the propagation constant

*γ*=

*α*+

*jβ*, where

*α*is the attenuation constant and

*β*is the phase constant; and

*T*

_{ h }is the thickness of the glass.

*μ*is the intrinsic permeability;

*σ*is the intrinsic conductivity;

*ε*is the intrinsic permittivity; and the termination impedance of the equivalent 2-port network

*Z*

_{ T }→

*η*

_{0}is the intrinsic impedance of free space (i.e., the air dielectric).

_{21}and overall input voltage-wave reflection coefficient S

_{11}can be represented as [24]

_{21}and S

_{11}, measured values for the index of refraction

*ñ*(

*λ*) =

*n*(

*λ*) −

*jk*(

*λ*), where

*n*(

*λ*) is the refractive index and

*k*(

*λ*) is the extinction coefficient, for typical (soda lime silica) window glass was first needed [25]. The propagation constant through the glass

*γ*is expressed as

By applying (16) to (9)-(12), *ρ* _{1}, *ρ* _{2}, *τ* _{1} and *τ* _{2} can be obtained. As a result, *S* _{21} and *S* _{11} can then be calculated using (13) and (14). The overall power transmittance and reflectance are determined from |*S* _{21}|^{2} and |*S* _{11}|^{2}, respectively, while power absorptance is given by 1 − |*S* _{11}|^{2} − |*S* _{21}|^{2}.

*T*

_{ G }(

*λ*), reflectance and absorptance from 1 to 100 THz (assuming typical window glass at room temperature) is shown in Fig. 6. Here, \( {\left|\tau \right|}^2={\left|{e}^{-\alpha {\mathrm{T}}_h}\right|}^2 \) and |

*ρ*|

^{2}= |

*ρ*

_{1}|

^{2}and 1 − |

*τ*|

^{2}− |

*ρ*|

^{2}correspond to the power transmittance, reflectance and absorptance, respectively, without considering multiple reflections; these parameters are often plotted in the open literature as first-order approximations [25]. It can be seen in Fig. 6 that these first-order approximations are very accurate below ~65 THz (4.6 μm), where absorptance dominates.

It can also be seen from Fig. 6 that typical window glass can be considered opaque below ~60 THz (5 μm). For most conventional applications, this would only allow its use in its transparent region above ~70 THz (4.3 μm). However, for our ‘*THz Torch*’ applications, the high absorptance will contribute to the secondary source of radiation (due to the increase in outer surface temperature).

*A*

_{ eff _ filament }, the filament is assumed to be a uniform cylinder. Therefore, if

*CSA*and

*l*can be physically measured then the other can be extracted using the previously estimated value of (

*CSA/l*)

_{ eff }. Using a scanning-electron microscope (SEM), the average value for the diameter of the filaments was measured to be 22.40 μm, as shown in Fig. 7. Therefore, for the 5-bulb array configuration, the total effective radiating area is

*A*

_{ filament =}12.73 mm

^{2}and

*A*

_{ eff _ filament }=

*A*

_{ filament }/2 = 6.37 mm

^{2}.

### 2.5 Filament Thermal Time Constants

Filament thermal time constants are also important parameters for ‘*THz Torch*’ applications having transient behaviour in the electrical stimulus of the transducer (e.g., direct modulation); this will set fundamental limits on signalling rates. When the bulb is in the ON state, having a step response function, at its initial temperature *T(0)*, there is a large injection of current and the temperature of the filament increases. Since the resistivity of tungsten has a positive temperature coefficient, the instantaneous bulb resistance also increases; from its initial value of *R(T(0))* until a steady-state value is reached, at thermal equilibrium, where the input power is exactly balanced out by all of the dissipative (i.e., heat transfer) loss mechanisms.

*t*is the instantaneous heating time starting from the initial working temperature

*T*(0);

*ΔT*

_{ MAX }= [

*T*(∞) −

*T*(0)] is the maximum change in filament temperature; and

*T*(∞) is the final steady-state working temperature in thermal equilibrium. With this example,

*T*(0) = 300 K and

*T*(∞) = 772 K. The turn-ON thermal time constant was found experimentally to be

*τ*

_{ H }= 645 ms.

*τ*

_{ C }is defined in a similar way to

*τ*

_{ H }; being the time taken to decrease from 90% to 10% of the temperature difference between the initial and final steady-state temperatures. Cooling of the hot filament is a relaxation process [14]. Therefore,

*τ*

_{ C }is expected to be larger than

*τ*

_{ H }. Using the measuring technique proposed in [4], turn-OFF responses for instantaneous current and extracted filament temperature can be obtained, as shown in Fig. 9. A simple empirical curve fit can be applied to the instantaneous turn-OFF filament temperature, shown in Fig. 10(b), as given by the following expression with less than 3% error.

*t*is the instantaneous cooling time from the initial working temperature

*T*(0);

*ΔT*

_{ MAX }= [

*T*(0) −

*T*(∞)] is the maximum change in filament temperature; and

*T*(∞) is the final working temperature; with this example,

*T*(0) = 772 K and

*T*(∞) = 300 K. The turn-OFF thermal time constant was found to be

*τ*

_{ C }= 2,415 ms. As expected, this value is much larger than

*τ*

_{ H }. This severely limits the switching speed for the bulb and sets a practical limit on the data rate for on-off keying (OOK) digital modulation – although this is not an issue with applications that do not required high data rates.

## 3 Secondary Radiation Modelling

### 3.1 Band-limited Radiant Intensity from Secondary Radiation

*A*

_{ eff _ glass }[m

^{2}] is the total effective radiating area for all the glass envelopes and

*ε*

_{ glass }(

*λ*,

*T*

_{ glass }) is the emissivity of the glass envelope with an outer surface temperature

*T*

_{ glass }. Here, we assume that

*ε*

_{ glass }(

*λ*,

*T*

_{ glass }) does not change significantly as temperature increases from the ambient room temperature of 300 K to the highest elevated temperature of 366 K; this is a reasonable assumption, as stated in [23]. Furthermore, according to Kirchhoff's law of thermal radiation, emissivity is equal to the power absorptance in thermal equilibrium. Therefore, the values for power absorptance shown in Fig. 6 can be used directly to represent the emissivity of the glass envelope

*ε*

_{ glass }(

*λ*), which is now only wavelength-dependent.

With our particular 5-bulb array configuration, the radiant intensity from secondary radiation can be further separated out into two parts: the central higher temperature region and its surrounding lower temperature region, as shown in Fig. 10.

*T*

_{ high }and

*T*

_{ low }are the average temperatures for the high and low temperature regions, respectively;

*A*

_{ eff _ glass _ high }≈

*D*

^{2}(1 −

*π*/4) = 1.45 mm

^{2}is the total effective radiating area of the higher temperature region for the five-bulb array;

*A*

_{ eff _ glass _ low }≈ 8

*π*(

*D*/2)

^{2}= 42.47 mm

^{2}is the total effective radiating area of the lower temperature region; and

*D*= 2.6 mm is the diameter of the bulb’s glass envelope.

The outer surface temperature of the glass envelope depends on the filament’s emissivity, emitting area, temperature, position and shape. Instead of using complex thermodynamic modelling to simulate its outer surface temperature distribution, a more direct approach is to measure its temperature using a thermal camera. An experiment using a FLIR E60 thermal camera was performed. This camera uses an uncooled microbolometer focal plane array with 320×240 pixels. Note that the THz band-pass filter and associated aperture were removed, in order obtain the actual temperature distribution for the 5-bulb array.

Measured outer surface temperatures for the 5-bulb array at different bias currents

Bias Current (mA) | Measured Temperatures (K) | ||
---|---|---|---|

Array Maximum | Array Average | Outer Bulb Centre | |

0 | 294.5 | 294.5 | 294.5 |

40 | 308.9 | 304.0 | 303.9 |

44 | 312.3 | 306.3 | 306.0 |

50 | 318.9 | 310.6 | 310.4 |

55 | 325.3 | 315.3 | 315.0 |

60 | 331.8 | 319.9 | 319.4 |

65 | 338.9 | 324.9 | 324.4 |

70 | 346.9 | 330.9 | 330.4 |

75 | 355.7 | 337.3 | 336.7 |

80 | 366.1 | 344.9 | 344.2 |

### 3.2 Glass Envelope Thermal Time Constants

*‘THz Torch’*applications if external modulators (e.g., programmable mechanical shutters or spatial light modulators) are employed.

## 4 Calculated and Measured Band-limited Output Radiated Power

*T*

_{ G }(

*λ*) and emissivity

*ε*

_{ glass }(

*λ*) of the glass envelope. Fig. 13 shows the calculated net spectral radiance from these two radiation mechanisms at a bias current of 44 mA. It is interesting to note that below ~50 THz (6 μm), secondary radiation dominates the output power; while primary radiation dominates above ~50 THz.

*A*

_{ eff _ filament }= 6.37 mm

^{2}and

*A*

_{ eff _ glass }= 43.92 mm

^{2}are taken into account for the filament and glass envelope, respectively, the resulting radiant intensities and band-limited output radiated powers are now of the same order of magnitude, for the bandwidth between 1 to 100 THz (300 μm to 3 μm), as shown in Fig. 14. As the bias current increases, the peak in spectral radiance increases in frequency, as dictated by (3). However, when compared to the secondary radiation mechanism, the output power from the primary radiation mechanism is more strongly coupled to changes in bias current.

## 5 Band-limited Output Radiated Power to Input DC Power Conversion Efficiency Calculations

In this section, the band-limited output radiated power and conversion efficiency for both single and multi-channel *‘THz Torch’* transmitters, employing the same 5-bulb array configuration described previously, can be calculated. The spectral range for the first proof-of-concept single-channel *‘THz Torch’* system was defined over the 25 to 50 THz (12 μm to 6 μm) octave bandwidth [1,3,4], while four non-overlapping spectral ranges for the 4-channel multiplexing systems are: 15 to 34 THz (20 μm to 8.8 μm) for Channel A; 42 to 57 THz (7.1 μm to 5.3 μm) for Channel B; 60 to 72 THz (5 μm to 4.2 μm) for Channel C; and 75 to 89 THz (4 μm to 3.4 μm) for Channel D [2,4-6].

*R*(772 K) = 17.02 Ω. Therefore, the input DC power is

*P*

_{ DC }=

*I*

^{2}

*R*= 33 mW. As a result, the conversion efficiency can be estimated to be ~1% for the single channel and Channel A thermal sources and approximately half this value for the other channel sources at this bias point. It should be noted that the insertion loss of the THz band-pass filter and also spreading losses are not taken into account. As the bias current increases, the efficiency for the single channel and Channel A and B sources will decrease, as the spectral peak for the primary radiation moves higher in frequency, and more power is radiated above the spectral ranges of these channels. For this reason, Channel C and D sources show an increase in power conversion efficiency at higher bias currents.

## 6 Conclusions

The ‘*THz Torch*’ concept was recently proposed as a low cost means of establishing secure communications over short distances. In order to accurately characterize the thermal infrared transmitter and, in turn, predict the band-limited output radiated power for each channel, a detailed investigation of the associated radiation mechanisms has been given here for the first time. It is found that, with the use of incandescent light bulbs, the output radiated power has contributions from both the primary and secondary radiation sources. At a fixed bias current of 44 mA, these two radiation mechanism can generate similar band-limited (1-100 THz) output radiation power levels. In addition, the thermal time constants for both the tungsten filaments and glass envelopes have been investigated. For channels above ~70 THz (4.3 μm), where primary radiation dominates, the cooling thermal time constant of the filaments dictates switching speed. For channels below ~60 THz (5 μm), where secondary radiation dominates, the cooling thermal time constant of the glass envelope dictates switching speed; this is two to three orders of magnitude slower than those associated with the filaments.

Low cost near-infrared LEDs can provide even higher efficiency and switching speeds, but their output spectral frequency cannot be tuned. In contrast, the spectral peak of thermal sources can be continuously tuned over a vast spectral range, simply by changing the quiescent DC bias current.

The thermodynamic modelling approach reported here can accurately estimate the band-limited output radiated power of the thermal sources, and this has been verified by experimental results. Our modelling approach can serve as an invaluable tool for engineering solutions that can achieve optimal performances with both single and multi-channel *‘THz Torch’* systems. Moreover, the modelling methodology presented in this paper can be further extended to other incandescent light bulbs or more bespoke thermal sources, having different material systems, to predict the band-limited output power in the spectrum of interest.

## Notes

### Acknowledgements

This work was partially supported by the China Scholarship Council (CSC).

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