1. Introduction

Infrared (IR) detectors are widely used nowadays with many possible applications in astronomy, medicine, military, automotive industry, and many more. They can be divided in two large groups - thermal and quantum [1] - based on the working principle. Quantum-detection sensors produce electrical output due to the direct conversion of IR signal into electric energy. They have high detection efficiency and very fast response, but cryogenic cooling is required, making them sensitive and expensive. Thermal detectors are heated as a result of absorption, altering some physical property of a material with subsequent conversion to electrical signal [2]. They are slower, but work at room temperature. New ideas abound, such as IR detection based on graphene [3], carbon nanotubes [4], quantum dots [5], superconducting nano-wires [6] or up-conversion [7]. Common to all of them is a complicated construction, complex manufacturing technologies and intricate underlying physics.

Generally, arrays of IR thermal detectors were used for imaging camera construction [8]. Based on their mode of operation and cooling requirements, thermal cameras are classified as cooled (requiring cryogenic system such as Stirling machine) and uncooled, which are much simpler and more versatile.

A large group of uncooled cameras are based on bi-material, micromechanical cantilevers, whose radiation-induced mechanical deformation is detected electrically or optically. To amplify the thermo-mechanical response, cantilevers are made of two materials with different thermal expansion coefficient (usually metals or semiconductors, but materials of biologic origin, such as chitin or proteins, present a viable option [9], too).

Natural biological structures can be turned into IR detectors by using thermally-induced spectral change of wing-scales iridescence [10]. It was proposed that thermal dilatation of wing scales increases the distance between the Brag-grating-like lamellas and changes the refractive index, thus inducing a slight shift of the iridescence spectrum. Due to low thermal expansion coefficient (50 × 10⁻⁶ K⁻¹) the final effect is very small and differential spectrometry is used for detection. The thermal response was further improved by coating wing scales with carbon nanotubes. In another line of research [11], wing scales were covered with a thin layer of gold in order to produce bi-material structure which bends upon heating. In both cases, the detection principle renders spatially resolved imaging very difficult, and strongly limited by the thermal expansion coefficient, leaving little room for further improvements.

Here we present an infrared imaging system with Morpho menelaus butterfly wing as a sensing medium. Digital holographic interferometry was used to detect thermally-induced displacements. The spatial resolution is of the order of the butterfly wing-scale (approximately 50 × 100 µm [12]), the temperature sensitivity is 2.8 mK and the response frequency around 200 Hz, which is quite comparable to high level commercial cameras. Sensitivity is much higher than could have been expected from the thermal dilatation coefficient alone and we propose that photophoretic forces are responsible for the effect. We have calculated the photophoretic pressure and the resulting displacements of butterfly scales, finding them in good agreement with the experimental results.

2. Infrared imaging on a butterfly’s wing

The detection of infrared radiation was done using a Morpho menelaus butterfly wing, whose circular section, 1 cm diameter, was extracted from a wing, putting some effort to choose a zone devoid of wing-veins as much as possible. Section was attached to the holder with a double adhesive tape, without any special preparations, see Fig. 1(a). As can be seen in Fig. 1(b), the wing is densely covered with iridescent particles shaped as micro-cantilevers possessing nanoscale, layered structures, see inset of a Fig. 1(b). The wing-section was irradiated with near IR laser beam for localized heating of the wing. A thermal process was monitored holographically, using a setup and method described in a Section A.1 of the Appendix. We were able to reconstruct a single diffraction order of a hologram at 2048 × 2048 resolution due to properties of the shifted Fresnel transform [13]. All the details of the wing were clearly revealed, as shown in Fig. 1(c). By producing an interferogram between the reference hologram (recorded before switching-on the 980 nm laser) and the one recorded during the thermal process, we were able to faithfully observe the elongated laser beam shape, as shown in Fig. 1(d).

 

Fig. 1 (a) A photograph of a circular section of a Morpho menelaus butterfly wing attached to a holder. (b) SEM image showing the cantilever-like structure of the Morpho menelaus wing-scale and (see inset) ultrastructure of its surface, with a number of lamellae. (c) A hologram reconstruction of a section of the butterfly wing. (d) A holographic image obtained after irradiation of a butterfly wing with a 980 nm pulsed laser beam (having elliptical beam profile 2.7 mm × 6.8 mm, 8.7 mW power and 128 ms pulse length, corresponding to ~2 mJ/cm² energy density).

Download Full Size | PDF

The laser beam power was kept constant and the pulse duration was decreased until no beam trace was discernible by the holographic interferometry. In this way the detection threshold was established at several µJ/cm², depending on the wavelength. This is because we used a butterfly wing as it is, without further functionalization or alteration and the temperature increase is determined by its natural absorptivity. It is well known that the melanin is a constituent of wing scales, with absorptivity being much higher at UV than in IR [14], thus explaining the lower detection threshold in UV.

We used commercial thermal camera (FLIR SC620, 640 × 480 pixel, 40 mK thermal resolution, and maximum speed of 120 fps at reduced resolution) to establish the relation between the temperature of the wing and holographically measured phase difference (see section A.1 of the Appendix for details).

Thus, the same thermal process was monitored by the thermal camera [depicted in Figs. 2(a) and 2(b)] and our method [Fig. 2(c)]. We have found that the temperature rise of ΔT = 0.4 K, recorded by the IR camera corresponds to holographically detected phase difference ΔΦ = 5.1 rad, i.e. to the displacement of 0.432 µm. The related thermo-mechanical response is, therefore, 1.08 µm/K. Based on that and experimentally recorded phase-noise level (Φ_n = 0.037 rad, corresponding to resolvable displacement of the order of 3.1 nm), we found the temperature detection sensitivity of 2.9 mK.

 

Fig. 2 (a) Thermal camera image of the butterfly wing section irradiated with the laser beam (at 980 nm wavelength, 8.7 mW power and 128 ms pulse length). (b) Temperature variation (black line) recorded by the thermal camera during laser irradiation. Fast oscillations are camera noise. A laser pulse shape (red line) is shown, too. (c) A corresponding phase difference recorded holographically under the same conditions.

Download Full Size | PDF

According to the theory of linear time-invariant systems [15], impulse response can be determined by differentiating its Heaviside step response, and a transfer function is further found by Fourier transforming the impulse response. Thermal systems can be treated as linear, because conductive, convective and radiation thermal energy exchange can be modeled by the linear differential equations. That is why the thermal response of the wing was determined by irradiation with a long laser pulse (at 980 nm and 8.7 mW). The response was Fourier transformed to obtain the transfer function, displayed in Fig. 3. For a 10 dB signal-to-noise ratio (SNR) the detection frequency is 30 Hz, while for 20 dB SNR detection frequency is larger than 200 Hz, see Fig. 3.

 

Fig. 3 Attenuation of the butterfly’s wing response as a function of signal frequency.

Download Full Size | PDF

3. Thermo-mechanics of the wing-scales

Radiation-induced temperature rise produces a thermal dilatation and bending of wing-scales, which were calculated at 0.005 µm/K and 0.029 µm/K, respectively (see Section A.2 of the Appendix for a detailed analysis). This is more than an order of magnitude lower than experimentally observed 1.08 µm/K. To resolve the discrepancy, we were motivated to take a closer look at the morphological, absorptive, thermal and mechanical properties of the Morpho menelaus wing-scales.

Each scale is almost a flat plate (approximately 100 µm long, 50 µm wide and 2 µm thick [12]) embedded in the wing membrane by its pedicle [Fig. 1(b)] and acting just like a mechanical cantilever. The scale’s main, plate-like, body has two laminae: the lower one being almost flat, while the upper is structured with densely packed ridges, made of a number of lamellas [10], see inset in Fig. 1(b). As of the material properties (mostly chitin), a thermal expansion coefficient is k = 50 × 10⁻⁶ K⁻¹ [10, 16], modulus of elasticity E is estimated between 2 GPa and 20 GPa [17] and we measured the absorption coefficient of the wing α = 4.5 × 10⁻⁴ nm⁻¹ at 980 nm.

Here we claim that the additional mechanical force comes from molecular interaction between the surrounding gas and the heated nanostructures of the wing-scales The mean free path of molecules in air is generally accepted to be around 66 nm at the ambient pressure (~10⁵ Pa) [18], according to kinetic theory of gasses. This is quite comparable to the lamellar spacing of the wing-scale (150 nm). Under such circumstances, the inter-lamellar space is so small that only a few hundred atoms fit in-between, see Fig. 4. Numerically, the situation is described by the Knudsen number K_n, defined as a ratio between the mean free path of the surrounding fluid and the characteristic dimension of the system (in our case, the distance between lamellas). If a thermal gradient is established in a thermodynamic system with the Knudsen number between 0.1 and 10, an additional, so-called radiometric (Knudsen) forces appear [19]. This is exactly the situation of a radiatively-heated wing-scale, where the temperature difference is established and the Knudsen number is 0.45.

 

Fig. 4 Scheme of the cross-section of a Morpho butterfly scale. Black dots represent molecules of air at atmospheric conditions. A radiation-induced temperature gradient (black dashed line) is established across the wing scale (T_s<T_c). Dimensions are in nanometers but the drawing is not to the scale. L – lamellae, UL – upper lamina, LL – lower lamina.

Download Full Size | PDF

The radiometric forces have been investigated in many different circumstances connected with thermal transpiration [20], AFM cantilevers in partial vacuum [21] or pumps without moving parts [22]. The same effect has been used to mechanically manipulate large particles using vortex or bottle beams [23].

We have studied the effects of radiometric forces using a theory developed by Passian [21], which was slightly amended and experimentally tested in [24]. For ambient conditions corresponding to our experiments (normal atmospheric pressure 10⁵ Pa and room temperature 295 K) and the temperature gradient of 0.31 K (corresponding to temperature increase of 1 K) we have calculated that the resulting photophoretic pressure is 26.3 Pa (see Section A.3.1 of the Appendix for details).

In order to estimate the associated mechanical deformation, we use a simplified mechanical model of a butterfly wing-scale, shown schematically in Fig. 5(a). This is a combination of a thin, corrugated, hollow box (imitating the main body - flattened part - of the scale) attached to a short beam, rigidly constrained at its end (approximating the scale pedicle). An approximate analytic solution for the maximum displacement of the mechanical model was found, showing the influence of mechanical parameters of the system on thermo-mechanical response (see Section A.3.2 of the Appendix). More accurate solution was found using finite element method (FEM) based on the model shown in Fig. 5(b). The corresponding displacement field was calculated and shown in Fig. 5(c), based on mechanical parameters given in Table 1. of the A.3.2 Appendix section.

 

Fig. 5 (a) Scheme of the butterfly wing-scale with dimensional parameters. (b) Perspective view of the FEM wing-scale model. (c) Perspective view of the butterfly wing-scale deflection field.

Download Full Size | PDF

Table 1. Environmental, dimensional and mechanical parameters of the butterfly wing-scale model.

View Table

By taking dimensional parameters of the wing-scales from [10,12] [which closely correspond to our SEM images – see inset in Fig. 1(b)], photophporetic pressure calculated above and modulus of elasticity of 18 GPa, taken from the available literature [17], we found that scale-tip displacement is 1.16 µm/K, in agreement with experimentally obtained value.

4. Discussion

We used holography for detection, because imaging is very sensitive, straightforward and easy to implement. We found that the classical problems of mechanical stability, ambient conditions like air currents, vibration, and acoustic disturbances, were completely unimportant in our single-beam setup, described earlier and in the Appendix. Heating of the butterfly wing by the holographic reference beam itself is unimportant because it is constant, of low intensity (110 µW/cm2 at 532 nm) and makes uniform background which cancels in interferometric measurements. However, intensity stability of the beam must be such that the induced temperature variations are below the detection threshold, which was satisfied in our experiments (laser stability 1.28%, RMS noise 0.17%, according to manufacturer). The angle between the reference beam and observation direction is small and the holographic sensitivity (influenced by the sensitivity vector [25]) is close to 1. This means that the one fringe displacement is very close to the wavelength (532 nm in our case). For larger photophoretic displacements, we have observed phase wrapping, which was easily corrected (by phase un-wrapping [25]).

Butterfly wings are not an ideal imaging detector. Wing-scales are not distributed evenly, their shape, position, anatomy and mechanical properties are not uniform, some zones are depleted [such as wing veins – as can be seen in Fig. 1(a)] and the wing surface is wavy. As a result, different wing zones and wing scales produce variable response. This is a well known problem in thermal camera construction, corrected by, so called, non-uniformity compensation. Here we made no attempt to improve the situation because the proper equipment was not available to us. For the same reason, we were also unable to measure the noise equivalent thermal difference (NETD), because it includes an infrared imaging system and a low temperature black body. All measurements in our work express the value of thermal sensitivity of the butterfly wing, used as a focal point array.

Anyway, we think that significant improvements can be made to the photophoretically active surface. Quite large wing portions with better uniformity can be selected (especially on Morpho butterflies), cut and tiled to make a larger detection surface. In that respect, we have demonstrated that butterfly wings can be precisely cut and scribed by a femto-second laser. It could be expected that the photophoretic effect largely depends on the butterfly species and will be maximized for insects with intricate nanostructures (day flying butterflies, e.g. lycaenidae, nymphalidae). Alternatively, wing scales can be harvested, manipulated and attached to an artificial substrate in an orderly manner. Finally, photophoretic principle can be used in a completely artificial system imitating to some degree wing scales - see structures described in [26].

The spatial resolution of the described butterfly-wing-based sensor is determined by the length of the scales, because the maximum deflection is at the tip. For regions closer to the petal deflection is much lower and drops below the detection capabilities of holographic interferometry. We found that the resolving power of the holographic setup is 19.5 µm x 29.5 µm, according to Rayleigh criterion. This was defined by the detector chip size (13.2 × 8.8 mm) and its distance to the butterfly wing (800 mm). The resolution can be significantly improved by reducing the distance in a holographic setup and by using a larger detector chip size.

There is also an important question of efficient absorption of electro-magnetic radiation by the wing scales. We used completely unmodified wing with naturally present, chitin, pigments and proteins. Presence of melanin is well established in Morpho wing scales [27] resulting in much higher absorption in the blue-UV, compared to IR spectral range. We really observed that the detection sensitivity at 405 nm is much higher than at 980 nm. We used a Fourier-transform infrared (FTIR) spectrometer to measure the far infrared spectrum of the Morpho wing, where we found two prospective regions (2800-3300 nm and 5700-6600 nm) with absorption as high as 50%. With proper sensitization (e.g. single-walled carbon nanotubes or water), detection of far IR [28] and terahertz radiation is possible [29], too. Therefore, these nature-based structures can be modified into a thermal detector sensitive within an extremely wide spectral range of electro-magnetic radiation.

With regard to photophoresis, we see additional possibilities to amplify the effect by changing the chemical composition of the gas and its pressure or increase the temperature and thermal gradient, by proper sensitization of the base material. So far, photophoresis was studied in simple systems like AFM cantilevers [21] or spherical particles [23], and adequate models should be developed for butterfly-like structures, with a number of Bragg layers or photonic crystals.

We emphasize that the radiation detection, within the IR power range used in this research, is completely nondestructive to the wing. We used the same wing for months without any deterioration or damage. The fact is supported by hundred years old butterfly specimens in entomological collections, still retaining their structural color.

It should be added that the principle of electrophoretic radiation detection is universal and applicable to other types of invisible radiation (UV, terahertz). There are a number of things that can be optimized to improve the sensitivity and spatial resolution of the system, such as: building an artificial micro-cantilever array, changing dimensions and material composition of micro-cantilevers, adjusting chemical composition of gaseous environment, designing other, nonoptical, readout methods (e.g. capacitive, piezo or resistive). Findings of this experiment and results of the model can be an inspiration for fabrication of butterfly-inspired thermal chip. For example, by using elastomeric cantilever made of PDMS, whose modulus of elasticity is three orders of magnitude lower compared to wing-scales used in this research, we can expect three orders of magnitude larger thermal sensitivity. Even the existing systems based on bi-material cantilevers could be improved by nano-patterning to introduce additional photophoretic effect.

5. Conclusion

A novel method of detecting low level thermal radiation, using scales of the Morpho menelaus butterfly wing, is presented. Thermally induced displacements of scales were measured using digital holography. We have found that characteristics of this, nature-given sensing elements are quite promising, even when compared with the state-of-the-art thermal cameras. Temperature detection sensitivity is 2.9 mK, with the detection threshold of several µJ/cm², depending on the wavelength. The detection frequency reaches over 200 Hz, for SNR of 20 dB. We emphasized a strong mechanical stability and durability of butterfly wings, and completely nondestructive nature of measurements.

We have found that the photophoresis at atmospheric pressure is the underlying mechanism, as verified by measurements and a simplified mechanical model of the wing-scale. The resulting, photophoretically induced, displacement was 1.16 µm/K, in agreement with the experiment.

The described method of thermal radiation detection is universally applicable and will function with other manmade materials and structures, as long as the characteristic size of structures is comparable to the mean free path of molecules of the surrounding gas.

Appendix

A.1 Holographic setup and phase measurements

A simple setup described previously [30] and shown in Fig. 6. was used throughout this research. A beam from the unmodulated laser L1 (Torus 532 laser, manufactured by Laser Quantum, with 532 nm wavelength, 43.1 mW power and 10 m coherence length) is spatially filtered by focusing the laser beam (with a biconvex lens L with 50 mm focal length) through the 25 μm diameter pinhole P. The resulting laser beam is directed towards the concave mirror CM with 75.3 mm aperture diameter and 23 mm focal length. Part of the divergent laser beam directly illuminates the wing section W mounted in front of the concave mirror CM, thus generating an object beam O. The rest of the beam, which misses the object W, generates the reference beam R. A mirrorless CMOS camera C (Nikon 1v3, detector size 13.2 × 8.8 mm, 18.4 MPixel, 60 frames per second) is placed in a zone where the reference beam R interferes with the object beam O, scattered from the wing section W. Laser L2 at 980 nm (with elliptical, divergent beam and 30 mW maximum power) was used, as a source of NIR radiation, to irradiate the butterfly wing section W. Its power was adjusted by a variable beam-attenuator, and irradiation time was controlled by switching laser on and off, using Arduino Mega 2560 Rev3 microcontroller. Laser (L2) produced thermal fingerprint on the butterfly wing which was detected holographically. The setup is mechanically extremely stable and we needed no vibration isolation.

 

Fig. 6 Scheme of a holographic device used to detect photophoretic displacement of a butterfly’s wing. L1 – laser at 532 nm, L2 - laser at 980 nm, L – biconvex lens, CM – concave mirror, C – CMOS camera, W – butterfly wing section, R – reference beam, O – object beam, P – a pinhole used for spatial filtering of the laser beam, M – a flat mirror used to deflect the laser beam.

Download Full Size | PDF

Holographic measurements were performed as follows. A camera C was turned-on for approximately 50 ms (corresponding to about 3 frames) before the infrared laser L2 was turned-on for 128 ms. Recording was continued even after turning the laser off, in order to monitor the wing cooling process. Total number of holograms was 40, during the recording time of 670 ms. Holograms were then numerically reconstructed using shifted Fresnel transform algorithm, implemented on a CUDA-enabled graphics card (ASUS Expedition GeForce GTX 1050 Ti) [13]. As a result, a time series of butterfly wing images was produced, containing both amplitude [see Fig. 1(c)] and phase information. Phase values of the first reconstructed image were taken as a reference. A phase difference between the reference and subsequent images were calculated and depicted in Fig. 2(c).

A.2 Thermal gradient across the butterfly’s scale

Radiation is partially absorbed by the wing scales leading to the temperature increase. Here we establish a connection between the average temperature increase of the wing scale (which is measurable) and the thermal gradient across the scale (which is not measurable, due to smallness of the scale). A butterfly wing scale will be approximated by a number of layers having certain thicknesses D_j, masses M_j, and absorptivities A_j, as depicted in Fig. 7(a).

 

Fig. 7 (a) Layers of the wing scale with their masses M_j, thicknesses D_j and absorptivities A_j. (b) A single wing scale treated as a bulk body having the mass M and thickness D equal to sum of masses and thicknesses of its layers. I₀ is radiation intensity, A is absorptivity, and Q absorbed energy.

Download Full Size | PDF

First we will treat the scale as a monolithic body [Fig. 7(b)] having thickness D and mass M equal to the sum of thicknesses D_j and masses M_j of its individual layers. Suppose that the incident light has intensity I₀, and that the fraction A of the incident energy is absorbed by the scale, i.e. energy Q = I₀A is dissipated in the scale. This energy is used to increase the temperature (by amount ΔT) of the scale having total mass M and specific heat c. A temperature increase ΔT is actually an average temperature of the wing-scale (the only measurable parameter, due to the thinness of the scale). The process is described by:

(1)$$Q=Mc\Delta T,$$

(2)$$I0A=Mc\Delta T,$$

(3)$$I0=\frac{Mc\Delta T}{A}.$$

We are now going to treat a scale as layered structure, as can be seen in the Fig. 7(b). A fraction A_j of the input energy I₀ is absorbed by the layer j, i.e. Q_j = I₀A_j and the temperature will increase by ΔT_j as:

(4)$$I0Aj=Mjc\Delta Tj.$$

By introducing Eq. (3) into Eq. (4), we get:

(5)$$\Delta Tj=\frac{M}{M_j}\frac{A_j}{A}\Delta T.$$

Let’s analyze the temperature rise of the j-th layer having volume ΔV_j = ΔxΔyD_j (where D_j, Δx, Δy and ρ are thicknesses, width, length and density, respectively). Taking into account that M_j= ρΔV_j=ρΔxΔyD_j and using Eq. (5) we then get:

(6)$$\Delta T_j=\frac{{\displaystyle {\sum }_{j=1}{M}_j}}{M_j}\frac{A_j}{A}\Delta T=\frac{{\displaystyle {\sum }_{j=1}{D}_j}}{D_j}\frac{A_j}{A}\Delta T=\frac{D}{D_j}\frac{A_j}{A}\Delta T.$$

From measured absorptivity of the Morpho wing at 980 nm we calculated the coefficient of absorption as α = 4.5 × 10⁻⁴. Fractions of absorbed energy A and A_j were calculated using Lambert-Beer’s law [I=I₀exp(-αD)]. We are now able to calculate the temperature distribution across the wing scale, assuming 7 lamelae, having equal thickness of 70 nm, and two laminae with 150 nm thickness, each. If the average increase of temperature is ΔT = 1 K, we get the temperature distribution depicted in Fig. 8. As can be seen, the average temperature increase is 1 K, but the temperature difference across the scale is δ_T = 0.31 K.

 

Fig. 8 Temperature gradient along the wing-scale having the average temperature increase of 1K, assuming the coefficient of absorption α = 4.5·10⁻⁴.

Download Full Size | PDF

This motivated us to treat the wing-scale as a nonuniformly heated cantilever, which bends as a result of uneven dilatation. This is a well known problem in mechanics and the maximum deflection Δ₀ of the cantilever tip is given by the following equation (see chapter 5 in [31]):

(7)$${\Delta }_0=\frac{k{L}_0^2}{2}\frac{{\delta }_T}{t},$$

where k is a coefficient of thermal expansion, L₀ is a cantilever length, t is its thickness and δ_T is a temperature difference established between cantilever’s top and bottom surfaces. We found that, for δ_T = 0.31 K, total thickness of the scale of t = 2.7 µm, length L₀ = 100 µm and k = 5·10⁻⁵ 1/K the corresponding maximum deflection is 0.029 µm/K [using Eq. (7)]. This is still much lower than, holographically observed, 1.08µm/K.

A.3 Calculation of mechanical effects due to photophoretic pressure

A.3.1 Photophoretic pressure

A fluid inside the wing scale cannot be treated as continuum, but as free molecules where molecule-surface (of the scale lamellae) collisions dominate molecule-molecule collisions. A simplified model [schematically shown in Fig. 9] was developed in [21,21], where the thermodynamic system consists of a substrate S at temperature T_s, thin membrane M at temperature T_c, embedded in a fluid with temperature T_r, and the distance between the substrate and membrane is of the order of the mean path length of the fluid. It was found that the resulting radiometric pressure P is described by the following expression [21,24]:

(8)$$P=\frac{P_r}{2}\left[\begin{array}{l}\sqrt{\frac{a_s{\tau }_s+a_{cb}\left(1-a_s\right){\tau }_c}{a_s+a_{cb}-a_s{a}_{cb}}}+\\ \sqrt{\frac{a_{cb}{\tau }_c+a_s\left(1-a_{cb}\right){\tau }_s}{a_s+a_{cb}-a_s{a}_{cb}}}\\ -\sqrt{1-a_{ct}+a_{ct}{\tau }_c}-1\end{array}\right],$$

where P_r is environmental pressure, a_s, a_ct, a_cb are accommodation coefficients of the substrate and membrane surfaces [Fig. 9], and τ_c = Τ_c/Τ_r, τ_s = Τ_s/Τ_r are relative temperatures (relative to ambient temperature Τ_r).

 

Fig. 9 A scheme of a radiometric system embedded in a fluid with temperature T_r, having a substrate S at temperature T_s, thin membrane M at temperature T_c, surfaces having accommodation coefficients a_s, a_ct, a_cb, and the spacing between the substrate and membrane is of the order of the mean free path of the gas molecules.

Download Full Size | PDF

For the purpose of this study we will slightly modify the Eq. (8), by assuming that all surfaces are identical and diffusely reflect molecules. In that case accommodation coefficients are all equal to 1 (diffuse scattering) a_s= a_ct= a_cb=1. Also, assuming that the substrate S temperature is slightly above the ambient Τ_s=Τ_r+δ_T, and the Eq. (8) reduces to:

(9)$$\begin{array}{c}P=\frac{P_r}{2}\left[\sqrt{{\tau }_s}-1\right]\\ =\frac{P_r}{2}\left[\sqrt{\frac{T_r+{\delta }_T}{T_r}}-1\right]\\ =\frac{P_r}{2}\left[\sqrt{1+\frac{{\delta }_T}{T_r}}-1\right].\end{array}$$

If the temperature increase δ_T is small, compared to the ambient temperature, square root can be approximated by the first two members of a Taylor series, producing the simple equation for the photophoretic pressure:

(10)$$\begin{array}{c}P=\frac{P_r}{2}\left[\sqrt{1+\frac{{\delta }_T}{T_r}}-1\right]\\ \approx \frac{P_r}{2}\left[1+\frac{1}{2}\frac{{\delta }_T}{T_r}-1\right]\\ =\frac{P_r{\delta }_T}{4T_r}.\end{array}$$

A.3.2 Photophoretic bending of the wing-scale

From the mechanical point of view, a butterfly wing-scale is a thin, hollow, corrugated plate attached to a short cantilever beam, see Fig. 10(a). We will treat it as a classical, simple cantilever beam [shaded gray on the Fig. 10(a)], with additional load on its left and right sides. In this way, we have a cantilever beam as in Fig. 10(b) loaded with the total force acting on the whole wing-scale body. Therefore, the resulting pressure P on the cantilever is increased by the amount equal to the ratio of the wing scale area and the beam area (equal to b₀/r), thus:

 

Fig. 10 (a) Butterfly wing-scale is approximated with a thin, hollow, corrugated plate attached to the short cantilever beam. (b) Approximation of a wing-scale as a beam with increased pressure due to additional force imposed by the whole scale body.

Download Full Size | PDF

(11)$$P=\frac{P_0{b}_0}{r}.$$

In this way, we are left with the well-known and solved problem in mechanics [32] (within Euler-Bernoulli beam theory [33]). A maximum deflection Δ of such beam is defined as:

(12)$$\Delta =\frac{W{L}_0^4}{8EI},$$

where W is the load per unit length, L₀ is cantilever plate length, I is it’s the second moment of cantilever cross-section and E is the modulus of elasticity. It is easy to find load per unit length as:

(13)$$W=P{b}_0.$$

By introducing Eqs. (10), (11) and (13) into Eq. (12) we get the final expression for the maximum deflection of the plate:

(14)$$\Delta =\frac{1}{32E}\frac{P_r}{T_r}\frac{b_0^2{L}_0^4}{rI}{\delta }_T.$$

The second moment I of the wing scale cross-section (with respect to area centroid) was calculated referring to the Fig. 10. Parameters of the theoretical model are presented in Table 1. Results of analytical expression (14) show good agreement with the FEM model [Fig. 11].

 

Fig. 11 Comparison between analytic and finite element methods applied to the problem of photophoretic deflection of the butterfly wing-scale. (a) Maximum deflection as a function of modulus of elasticity. (b) Maximum deflection as a function of the wing-scale length.

Download Full Size | PDF

The second moment I of the wing scale cross-section (with respect to area centroid) was calculated referring to the Fig. 10. Parameters of the theoretical model are presented in Table 1. Results of analytical expression (14) show good agreement with the FEM model, see. Fig. 11.

Funding

Serbian Ministry of Education, Science and Technological Development (III45016, ON171038).

Acknowledgment

This work is done in partial fulfillment of the requirements for the PhD degree of Dusan Grujić at the University of Belgrade, Faculty of Physics.

Author contributions: D.P. conceived the idea, D.P. and D.G. constructed the holographic device and wrote the software, D.G. performed holographic measurements, D.V. and Lj.T. performed thermal camera measurements, D.V. and Z. S. modeled mechanical structure, B.J. supervised the research. D.P., D.G., B.J. and D.V. prepared the manuscript.

References and links

1. A. Rogalski, “Infrared detectors: an overview,” Infrared Phys. Technol. 43(3-5), 187–210 (2002). [CrossRef]  

2. A. Rogalski, “History of infrared detectors,” Opto-Electron. Rev. 20(3), 279–308 (2012). [CrossRef]  

3. S. Goossens, G. Navickaite, C. Monasterio, S. Gupta, J. J. Piqueras, R. Pérez, G. Burwell, T. Ivan Nikitskiy, T. Lasanta, T. Galán, E. Puma, A. Centeno, A. Pesquera, A. Zurutuza, G. Konstantatos, and F. Koppens, “Broadband image sensor array based on graphene–CMOS integration,” Nat. Photonics 11(6), 366–371 (2017). [CrossRef]  

4. E. Theocharous, R. Deshpande, A. C. Dillon, and J. Lehman, “Evaluation of a pyroelectric detector with a carbon multiwalled nanotube black coating in the infrared,” Appl. Opt. 45(6), 1093–1097 (2006). [CrossRef]   [PubMed]  

5. L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometre-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photonics 2(4), 226–229 (2008). [CrossRef]  

6. F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013). [CrossRef]  

7. M. Wu, D. N. Congreve, M. W. B. Wilson, J. Jean, N. Geva, M. Welborn, T. Van Voorhis, V. Bulović, M. G. Bawendi, and M. A. Baldo, “Solid-state infrared-to-visible upconversion sensitized by colloidal nanocrystals,” Nat. Photonics 10(1), 31–34 (2016). [CrossRef]  

8. P. W. Kruse, Uncooled Thermal Imaging - Arrays, Systems, and Applications (SPIE, 2001).

9. M. T. Mueller, A. P. Pisano, R. Azevedo, D. C. Walther, D. R. Myers, and M. Wasilik, Patent No. WO 2009039193 A1 (2009).

10. A. D. Pris, Y. Utturkar, C. Surman, W. G. Morris, A. Vert, S. Zalyubovskiy, T. Deng, H. T. Ghiradella, and R. A. Potyrailo, “Towards high-speed imaging of infrared photons with bio-inspired nanoarchitectures,” Nat. Photonics 6(3), 195–200 (2012). [CrossRef]  

11. F. Zhang, Q. Shen, X. Shi, S. Li, W. Wang, Z. Luo, G. He, P. Zhang, P. Tao, C. Song, W. Zhang, D. Zhang, T. Deng, and W. Shang, “Infrared Detection Based on Localized Modification of Morpho Butterfly Wings,” Adv. Mater. 27(6), 1077–1082 (2015). [CrossRef]   [PubMed]  

12. S. Berthier, Photonique des Morphos (Springer Science & Business Media, 2010).

13. T. Shimobaba, J. T. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183(5), 1124–1138 (2012). [CrossRef]  

14. D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly Calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012). [CrossRef]   [PubMed]  

15. C. L. Phillips, J. M. Parr, and E. A. Riskin, Signals, Systems and Transforms (Pearson Education, Inc., 2008).

16. Y. Ogawa, R. Hori, U.-J. Kim, and M. Wada, “Elastic modulus in the crystalline region and the thermal expansion coefficients of α-chitin determined using synchrotron radiated X-ray diffraction,” Carbohydr. Polym. 83(3), 1213–1217 (2011). [CrossRef]  

17. J. F. Vincent and U. G. K. Wegst, “Design and mechanical properties of insect cuticle,” Arthropod Struct. Dev. 33(3), 187–199 (2004). [CrossRef]   [PubMed]  

18. S. G. Jennings, “The mean free path in air,” J. Aerosol Sci. 19(2), 159–166 (1988). [CrossRef]  

19. A. Ventura, N. Gimelshein, S. Gimelshein, and A. Ketsdever, “Effect of vane thickness on radiometric force,” J. Fluid Mech. 735, 684–704 (2013). [CrossRef]  

20. M. R. Cardenas, I. Graur, P. Perrier, and J. G. Meolans, “Thermal transpiration flow: A circular cross-section microtube submitted to a temperature gradient,” Phys. Fluids 23(3), 031702 (2011). [CrossRef]  

21. A. Passian, A. Wig, F. Meriaudeau, T. L. Ferrell, and T. Thundat, “Knudsen forces on microcantilevers,” J. Appl. Phys. 92(10), 6326–6333 (2002). [CrossRef]  

22. N. K. Gupta and Y. B. Gianchandani, “Thermal transpiration in zeolites: A mechanism for motionless gas pumps,” Appl. Phys. Lett. 93(19), 193511 (2008). [CrossRef]  

23. V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010). [CrossRef]   [PubMed]  

24. B. Gotsmann and U. Duerig, “Experimental observation of attractive and repolsive thermal forces on microcantilevers,” Appl. Phys. Lett. 87(19), 194102 (2005). [CrossRef]  

25. T. Kreis, Handbook of Holographic Interferometry Optical and Digital Methods (WILEY-VCH, 2005).

26. S. Zhang and Y. Chen, “Nanofabrication and coloration study of artificial Morpho butterfly wings with aligned lamellae layers,” Sci. Rep. 5(1), 16637 (2015). [CrossRef]   [PubMed]  

27. M. A. Giraldo and D. G. Stavenga, “Brilliant iridescence of Morpho butterfly wing scales is due to both a thin film lower lamina and a multilayered upper lamina,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 202(5), 381–388 (2016). [CrossRef]   [PubMed]  

28. T. Kampfrath, K. von Volkmann, C. M. Aguirre, P. Desjardins, R. Martel, M. Krenz, C. Frischkorn, M. Wolf, and L. Perfetti, “Mechanism of the Far-Infrared Absorption of Carbon-Nanotube Films,” Phys. Rev. Lett. 101(26), 267403 (2008). [CrossRef]   [PubMed]  

29. X. Xin, H. Altan, A. Sainta, D. Matten, and R. R. Alfano, “Terahertz absorption spectrum of para and ortho water vapors at different humidities at room temperature,” J. Appl. Phys. 100(9), 094905 (2006). [CrossRef]  

30. D. V. Pantelić, D. Ž. Grujić, and D. M. Vasiljević, “Single-beam, dual-view digital holographic interferometry for biomechanical strain measurements of biological objects,” J. Biomed. Opt. 19(12), 127005 (2014). [CrossRef]   [PubMed]  

31. R. B. Hetnarski and M. R. Eslami, Thermal Stresses - Advanced Theory and Applications (Springer, 2009).

32. C. Y. Young and R. G. Budynas, Roark’s Formulas for Stress and Strain, 7th ed. (McGraw-Hill, 2002).

33. O. A. Bauchau and J. I. Craig, Structural Analysis with Applications to Aerospace Structures (Springer, 2009).