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The application of the fractional calculus model for dispersion and absorption in dielectrics II. Infrared waves
International Journal of Engineering Science 104 (2016) 62–74
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The application of the fractional calculus model for dispersion and absorption in dielectrics II. Infrared waves Andrew W. Wharmby∗ 711th Human Performance Wing, Human Effectiveness Directorate, Bioeffects Division, Optical Radiation Branch, 4141 Petroleum Rd., JBSA, Fort Sam Houston, TX 78234, USA
a r t i c l e
i n f o
Article history: Received 10 July 2015 Revised 22 February 2016 Accepted 8 April 2016 Available online 22 April 2016 Keywords: Infrared waves Maxwell’s equations Dielectrics Fractional calculus Generalized derivatives Wave equation Viscoelasticity
a b s t r a c t In the first paper of this series, an empirical formula based on viscoelastic analysis techniques that employs concepts from the fractional calculus originally used to model the dielectric behavior of materials exposed to oscillating electromagnetic fields in the radiofrequency band was applied to do the same for electromagnetic fields oscillating in the terahertz frequency range. The empirical formula was integrated into Maxwell’s equations producing a fractional order Ampere’s law whereof a fractional order wave equation was derived. This wave equation was used to describe the absorption and dispersion of terahertz waves in a dielectric medium. In this work, the empirical formula is extended again for application in the infrared frequency spectrum. The fractional calculus dielectric model is adapted to curve fit the complex refractive index data of a variety of semiconductors and insulators. Following the same procedure used in the first paper of this series, the fractional calculus dielectric model is again integrated in Maxwell’s equations with the same dispersion and absorption analysis performed using the newly derived fractional order wave equation. The mathematical consequences of extending this model into infrared frequencies are also discussed. Published by Elsevier Ltd.
1. Introduction It was shown in the first article of this series (Wharmby & Bagley, 2015a) that an empirical model used to predict the dielectric response of materials exposed to radiofrequency electromagnetic radiation could be easily applied to predict the dielectric response of materials exposed to electromagnetic radiation in the terahertz band. The extension of this model into the terahertz frequency range from the radiofrequency range was shown to be mathematically straightforward, requiring no significant change in the development or application of the model aside from the adjustment of the parameter α . This parameter represented the value of the exponent in the model that was shown to translate into the order of the fractional derivative in time that was observed after the model was integrated into Maxwell’s equations (Wharmby & Bagley, 2014, 2015a). This article continues the investigation by applying the empirical model to the higher frequency range of infrared (IR). The empirical model is used to model the dielectric behavior of materials when exposed to electromagnetic fields oscillating at infrared frequencies and then incorporated into Maxwell’s equations to produce a fractional order wave equation that will be used to model the dispersion and absorption of infrared waves in a dielectric. ∗
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http://dx.doi.org/10.1016/j.ijengsci.2016.04.003 0020-7225/Published by Elsevier Ltd.
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Aluminum Antimonide
2
Complex Refractive Index
10
0
10
−2
10
−4
10
13.3
10
13.5
13.7
10 10 Angular Frequency ω, rad/s
Fig. 1. The complex refractive index of aluminum antimonide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
2
Beryllium Oxide
Complex Refractive Index
10
0
10
−2
10
−4
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 2. The complex refractive index of beryllium oxide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
2. The fractional calculus empirical model The equation that will be used to model the dielectric response to electromagnetic radiation at infrared frequencies is written as
∗ ( ω ) D 1 + A0 ( j ω )α = r∗ ( jω ) = , ∗ A1 + A2 ( j ω )α E (ω ) →∗
→∗
(1)
where D (ω ) and E (ω ) are the Laplace transformed magnitudes or Euclidian norms of the electric flux density and electric field intensity respectively that have been brought into the complex s-plane (s = jω). The variables A0 , A1 , A2 , and α are the parameters of the dielectric model that approximates the material’s complex permittivity (r∗ ) whose development has been discussed in the first article of this series (Wharmby & Bagley, 2015a) and elsewhere (Bagley & Torvik, 1986; Wharmby & Bagley, 2014). Figs. 1–4 and 10–20 (found in the Appendix) show the generated values of complex refractive index n, where n2 = r∗ , using the model given in Eq. (1) as lines plotted against measured complex refractive index (plotted points) in the infrared range (Palik, 1998). The parameters used are given in Table 1 and 2. These figures show very high agreement between the experimental data and the prediction made by the four-parameter model, particularly within the
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Calcium Fluoride
2
Complex Refractive Index
10
1
10
0
10
−1
10
−2
10
−3
10 13 10
14
15
10 Angular Frequency ω, rad/s
10
Fig. 3. The complex refractive index of calcium fluoride: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Magnesium Fluoride
2
Complex Refractive Index
10
0
10
−2
10
−4
10
−6
10 11 10
12
13
14
15
10 10 10 Angular Frequency ω, rad/s
10
Fig. 4. The complex refractive index of magnesium fluoride: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Table 1 Parameters of fractional derivative model for 4 dielectrics. Material Aluminum antimonide Beryllium oxide Calcium fluoride Magnesium fluoride
A0 (Sec)α
A1 (Dimensionless) −28
2.732 × 10 3.010 × 10−29 3.039 × 10−28 1.440 × 10−28
−2
9.100 × 10 1.380 × 10−1 1.320 × 10−1 1.900 × 10−1
A2 (Sec)α
α (Dimensionless) −29
2.848 × 10 9.889 × 10−30 1.302 × 10−28 6.171 × 10−29
1.996 1.992 1.972 1.980
resonant frequency region where the behavior appears to be asymptotic. However, in some cases there appears to be strong deviations from the behavior predicted by the four-parameter model at the borders of the resonant frequency regions. While this can be interpreted as an indication of the invalidity of the model, the reader is reminded that most materials exhibit multiple resonant frequency regions (Saleh & Teich, 1991). With this in mind, the deviations from the model prediction on either border of the resonant frequency region are most likely indications of nearby resonant frequency regions.
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3. Modifying Maxwell’s equations The procedure that is used to modify Maxwell’s equations is the same in the radiofrequency and terahertz cases. However, a value of α greater than one, which is often the case as seen in Table 1 and 2, will produce a slightly different result than those seen for the radiofrequency and terahertz bandwidths. Though this difference is slight, it brings certain mathematical implications into question. Therefore, the modification of Maxwell’s equations will be revisited here for the case (t ) = E (t ) and applying a Laplace transform in the of α > 1, beginning, as before, by using the constitutive relationship D complex s-plane (s = jω ), as well as assuming a frequency-dependent complex permittivity. This allows Ampere’s Law to be written as a function of frequency
∗ (ω ) = ∗ (ω )0 • ( jω )E ∗ (ω ). ∇ ×H r The four-parameter fractional derivative model found in Eq. (1) is then substituted in for
∗ (ω ) = ∇ ×H
1 + A0 ( j ω )α 0 • ( jω )E ∗ (ω ). A1 + A2 ( j ω )α
(2)
r∗ (ω )
which produces (3)
Multiplying both sides of Eq. (3) by the denominator of the expression for r∗ (ω ) gives
∗ (ω )) + A2 ( jω )α (∇ × H ∗ (ω )) = 0 ( jω )E ∗ (ω ) + A0 0 ( jω )1+α E ∗ (ω ). A1 ( ∇ × H
(4)
However, α now has a value greater than one, which prompts Eq. (4) to be written as
∗ (ω )) + A2 ( jω )1+α (∇ × H ∗ (ω )) = 0 ( jω )E ∗ (ω ) + A0 0 ( jω )2+α E ∗ (ω ). A1 ( ∇ × H
(5)
where α now represents the non-integer portion of its initial value. Taking the inverse Laplace transform of Eq. (5) yields
A1 ( ∇
(t )) + A2 ×H
2+α ∂ ∂ 1+α ∂ E (t ) + A0 0 E (t ) . ∇ × H (t ) = 0 ∂t ∂ t 1+α ∂ t 2+α
(6)
Through the composition property of fractional order and integer order derivatives (Podlubny, 1999), it can be seen that the inclusion of fractional derivatives within the constitutive relationship now results in a modified version of Ampere’s Law shown in Eq. (6) which is seen to exhibit fractional derivatives of order 1 + α and 2 + α . 4. The wave equation With Ampere’s Law modified to accommodate for the inclusion of the fractional derivative model whose α parameter is above one, the wave equation stemming from that result will need to be re-derived to take this change into account. The method use to derive the modified wave equation is carried out in the same way one would derive the traditional wave equation from Ampere’s Law and Faraday’s Law. Assuming a plane electromagnetic wave traveling in the x direction as shown in reference (Wharmby & Bagley, 2014), the first derivative with respect to time of the one-dimensional modified Ampere’s Law (adapted from Eq. (6)) is taken yielding
A1
∂ 2 Ey ∂ 3+α Ey ∂ ∂ Hz ∂ 2+α ∂ Hz + A2 2+α = −0 − A0 0 2 ∂t ∂ x ∂x ∂t ∂t ∂ t 3+α
(7)
which can be rewritten using the composition property of fractional order and integer order derivatives (Podlubny, 1999) as
∂ 2 Ey ∂ ∂ Hz ∂ 1+α ∂ ∂ Hz ∂ 1+α ∂ 2 Ey A1 + A2 1+α = −0 − A0 0 1+α . ∂t ∂ x ∂t ∂ x ∂t ∂t2 ∂t ∂t2
(8)
Next our attention is redirected towards Faraday’s Law, which is compactly written in one dimension as
∂ Ey ∂ Hz = −μr μ0 . ∂x ∂t
(9)
Once again, it is important to note that the permittivity term r does not appear in Faraday’s Law as it is written here and will therefore remain unmodified for the derivation of the modified wave equation. The derivation of the modified wave equation is continued by taking the first derivative in space of Faraday’s Law which is written here as
∂ 2 Ey ∂ ∂ Hz − = . μr μ0 ∂ x 2 ∂ x ∂t 1
(10)
Since it does not matter in Eq. (10) whether we differentiate first with respect to x and then with respect to t, or vice versa, we can substitute the left hand side of Eq. (10) into Eq. (8) where appropriate which produces
A1 −
∂ 2 Ey μr μ0 ∂ x 2 1
+ A2
∂ 2 Ey 1 ∂ 2 Ey ∂ 1+α ∂ 1+α ∂ 2 Ey − = − − A . 0 0 0 μr μ0 ∂ x 2 ∂ t 1+α ∂t2 ∂ t 1+α ∂ t 2
(11)
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Finally, multiplying both sides by −μr μ0 gives us
A1
∂ 2 Ey ∂ 2 Ey ∂ 1+α ∂ 2 Ey ∂ α ∂ 3 Ey + A = μ μ + A μ μ . r 0 0 2 0 r 0 0 ∂tα ∂t3 ∂ x2 ∂ t 1+α ∂ x2 ∂t2
(12)
Here, μr = μ/μ0 , where μr is the relative permeability of the media, μ is the absolute permeability of the media, and μ0 is the permeability of vacuum (μ0 ≈ 1.257 × 10−6 H/m). It is now apparent that employing the four-parameter fractional derivative model as the parameter for complex permittivity results in a modified version of the wave equation shown in Eq. (12) which is seen to exhibit fractional derivatives of order 1 + α and 3 + α . Alternatively, the same result can be obtained by applying the elastic-viscoelastic correspondence principle to the one-dimensional wave equation (i.e., incorporating the four-parameter model through the use of integral transforms). As in the previous cases, this modification is easily extended to all the three dimensions of the wave equation using the same method. Additionally, Eq. (12) can be solved analytically using the same procedure detailed in reference (Wharmby & Bagley, 2014) as long as the appropriate initial waveform and initial conditions are assumed as the increased value of α brings about certain mathematical implications which are discussed elsewhere (Wharmby & Bagley, 2015b). With a boundary waveform of 1 − cos(ω0 t ), the solution was again found to be
Ey (x, t ) = −
1
π
∞
Im 0
Ey (x, re
jπ
)e
−rt
dr −
e
√ x jω0 c−1 μr
1+A0 (− jω0 )α A1 +A2 ( − j ω0 )α
− j ω0 t
−
4
e
√ −x jω0 c−1 μr
1+A0 ( jω0 )α A1 +A2 ( j ω0 )α
4
+ j ω0 t
+
1 . 2
(13)
1
for infrared waves. Here, c is the speed of light equal to (μ0 0 )− 2 . It is again important to note here that this solution has three parts: first, the integral term representing the transient response that decays to zero for long time t, second, the steady state response represented by the exponential terms, and finally, the persistent term of 1/2. For large values of x and t the exponential terms disappear as does the transient response leaving this value of 1/2. This term corresponds to the electrostatic field (or the electric field at zero frequency) present in the electromagnetic field. Performing the same derivation for the solution of the wave traveling in the opposite direction produces a similar result, with the solution reducing to 1/2 for large x and t. Adding these steady state responses together results in the value of 1 which reflects the value of 1 in the expression for the boundary waveform 1 − cos(ω0 t ) which is the only term not dependent on frequency, distance, or time. 5. Absorption The absorption analysis is performed using the same method previously applied in Wharmby and Bagley (2015a) and Wharmby and Bagley (2014). The steady-state part of the solution to the modified wave equation seen as Eq. (13) is plotted for the dielectric materials of beryllium oxide and calcium fluoride in Figs. 5 and 6 whose parameters are found in Table 1. These two materials were selected for this analysis due to their difference in values of α relative to the other materials
Dissipation of an Electromagnetic Wave in Beryllium Oxide 1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
1
2
3 Distance (m)
4
5
Fig. 5. Absorption of an electromagnetic wave in beryllium oxide.
6 −6
x 10
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Dissipation of an Electromagnetic Wave in Calcium Fluoride 1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
1
2
3 Distance (m)
4
5
6 −6
x 10
Fig. 6. Absorption of an electromagnetic wave in calcium fluoride.
observed in this study as well as their absorption bands (asymptotic regions) spanning a range of similar frequencies. An V electric field of E0 = 1 m , applied by multiplying Eq. (13) through by E0 , at a frequency of 1.5 × 1014 Hz for beryllium oxide and a frequency of 7.74 × 1013 Hz for calcium fluoride and a relative permeability of μr = 1.0 were used to calculate the solution at increments of x = 4 × 10−9 meters through a total of 6 × 10−6 meters at time t = 5 seconds. It is of great importance to note that the absorption profiles displayed here differ significantly from those shown in references (Wharmby & Bagley, 2015a) and (Wharmby & Bagley, 2014). As expected, the infrared waves traversing through a dielectric are absorbed almost immediately compared to waves oscillating in the radiofrequency and terahertz bands. The infrared waves shown in Figs. 5 and 6 appear to be absorbed quickly enough that they do not have a chance to oscillate as radiofrequency and terahertz waves do. The role the parameter of α has in this analysis must once again be noted. Even though the infrared waves do not oscillate as they are absorbed almost immediately, a difference in their absorption rates can be perceived. The wave propagating through beryllium oxide, though at a higher frequency, is attenuating faster than the wave propagating through calcium fluoride that is oscillating at relatively lower frequency. As shown in reference (Wharmby & Bagley, 2015a) and (Wharmby & Bagley, 2014), this can be attributed to difference in the value of α between these two materials. Beryllium oxide whose α value was found to be 1.992 is higher than the α value for calcium fluoride, found to be 1.972, and hence attenuates at a faster rate. This effect has been found to be universal in all the materials analyzed in this investigation just as has been observed in Wharmby and Bagley (2015a) and Wharmby and Bagley (2014).
6. Dispersion The evolution of the dispersive effect is readily apparent in Figs. 7–9. Once again, the steady state portion of the analytical solution to the modified wave equation seen as Eq. (13) was plotted at two different frequencies (4 × 1013 Hz and 8 × 1013 Hz) at three different distances away from the origin of propagation at time steps of t = 1 × 10−17 seconds through calcium fluoride. The values of 0, 1.5 × 10−4 , and 3.0 × 10−4 meters were chosen for x in order to show the change in waveform resulting from dispersion, beginning with none (x = 0), to an increasingly exaggerated effect (x = 1.5 × 10−4 and x = 3.0 × 10−4 ). Figs. 7–9 show that the wave at both frequencies is experiencing not only attenuation but a shift in its phase, with the wave experiencing a larger attenuation and phase shift farther away from the origin. However, unlike what was observed in the radiofrequency and terahertz frequency bands, the figures are showing a more pronounced attenuation and shift at the lower frequency rather than the high frequency. This behavior is indicative of anomalous dispersion, a phenomenon that accompanies absorption bands that exhibit shapes reminiscent of asymptotic behavior like those seen in Figs. 1–4 and 10–20. This analysis again demonstrates results that are consistent with electromagnetic theory in that waves at different frequencies travel through a dielectric medium at different speeds.
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Dispersion Analysis of Electromagnetic Wave in Calcium Fluoride
1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
Time (sec)
1 x 10
−13
Fig. 7. Dispersion Analysis of electromagnetic wave at x = 0 meters in calcium fluoride at frequencies of 4 × 1013 Hz (−) and 8 × 1013 Hz (−−).
Dispersion Analysis of Electromagnetic Wave in Calcium Fluoride
1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
Time (sec) Fig. 8. Dispersion analysis of electromagnetic wave at x = 1.5 × 10
−5
0.8
1 x 10
−13
meters in calcium fluoride at frequencies of 4 × 1013 Hz (−) and 8 × 1013 Hz.
7. Conclusions It has been shown that the model used to predict the dielectric response of a material to terahertz waves that was based off of a viscoelastic model (Wharmby & Bagley, 2015a) also has the ability to predict a material’s dielectric response to infrared waves without any major modification. In addition to showing high agreement with experimental observation seen in Figs. 1–4 and 10–20, it was demonstrated that the procedure for the application of the developed model was the same not only in the radiofrequency and terahertz bandwidths, but the infrared bandwidth as well. Just as with terahertz frequencies, the model remained fundamentally unmodified for its use in the infrared frequency band, exhibiting only a change in the value of α , which was again seen to translate into the order of the fractional derivative in the time domain. It is important to note here that while the values of α were seen to range between zero and one-half for dielectric materials exposed to radiofrequency radiation (Wharmby & Bagley, 2014) and between one-half and one for terahertz radiation
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Dispersion Analysis of Electromagnetic Wave in Calcium Fluoride
1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
Time (sec)
1 x 10
−13
Fig. 9. Dispersion Analysis of electromagnetic wave at x = 3.0 × 10−5 meters in calcium fluoride at frequencies of 4 × 1013 Hz (−) and 8 × 1013 Hz.
(Wharmby & Bagley, 2015a), the majority of the values that α took on for infrared radiation are well above one and approaching the value of two. However, it was noted that once the value of α goes beyond one, certain mathematical implications arise when attempting to incorporate the fractional calculus dielectric model into Maxwell’s equations with the goal of obtaining analytical solutions to the modified wave equation that they produce. These implications manifest themselves as terms representing initial conditions that are not always simple to handle, and that assuming an appropriate input waveform and/or initial conditions that mitigate this problem must be considered (Wharmby & Bagley, 2015b). With this consequence addressed, it was shown that the resulting absorption and dispersion analyses are consistent with electromagnetic theory. Figs. 5–9 produce results that reflect the behaviors of their respective curve-fits of their complex refractive index plots. These observations further suggest that viscoelastic models employing the fractional calculus may, in many cases, be applied to dielectric theory for an exceptionally wide range of electromagnetic wave frequencies. Appendix Table 2 contains the values of the parameters used in Eq. (1) to curve-fit the complex refractive index data of the materials shown within this section. All complex refractive index data analyzed in this investigation were taken from the consolidated work of E.D. Palik found in reference (Palik, 1998). It is important to note that in the case of low-density polyethylene found in Fig. 16, α is seen to be a comparatively low 0.548. It was found that electromagnetic radiation in this frequency
Table 2 Parameters of fractional derivative model for 11 dielectrics. Material Aluminum arsenide Aluminum oxide Cesium iodide Cuprous oxide (A) Cuprous oxide (B) Gallium antimonide Low-density polyethylene Magnesium oxide Potassium bromide Sodium fluoride Water
A0 (Sec )α
A1 (Dimensionless ) −28
2.806 × 10 7.031 × 10−28 1.258 × 10−26 1.025 × 10−28 1.530 × 10−27 5.749 × 10−28 8.157 × 10−11 1.082 × 10−28 3.607 × 10−27 3.959 × 10−28 1.409 × 10−25
−1
1.090 × 10 2.755 × 10−1 1.690 × 10−1 1.100 × 10−1 1.190 × 10−1 9.0 0 0 × 10−2 4.330 × 10−1 1.100 × 10−1 2.010 × 10−1 1.780 × 10−1 5.490 × 10−1
A2 (Sec )α
α (Dimensionless ) −29
3.754 × 10 6.172 × 10−28 3.775 × 10−27 1.220 × 10−29 1.878 × 10−28 5.749 × 10−29 1.359 × 10−12 3.785 × 10−29 1.247 × 10−27 2.014 × 10−28 2.063 × 10−25
1.986 1.639 1.962 1.988 1.994 1.994 0.548 1.980 1.996 1.972 1.500
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1
Aluminum Arsenide
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 10. The complex refractive index of aluminum arsenide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
1
Aluminum Oxide
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10 16 10
17
10 Angular Frequency ω, rad/s
18
10
Fig. 11. The complex refractive index of aluminum oxide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
range has a relatively high (∼80%) transmission (Krimm, Liang, & Sutherland, 1956). This suggests that a resonant frequency region exhibiting high absorption as observed in most other materials analyzed in this work will be absent in this frequency range which is indeed the case. However, the data and accompanying curve for absorption is seen to be steadily increasing as the frequency increases. As previously discussed, this can be an indication of a resonant frequency region adjacent to the window of frequencies shown in Fig. 16. Upon further inspection of the infrared spectrum for polyethylene, there is indeed a sharp decrease in transmission at a slightly higher frequency (Krimm et al., 1956) than what is visible in Fig. 16 suggesting that a resonant frequency region which contains a rise in absorption is nearby.
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Cesium Iodide
1
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10
−4
10 12 10
13
10 Angular Frequency ω, rad/s
14
10
Fig. 12. The complex refractive index of cesium iodide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
1
Cuprous Oxide (A)
Complex Refractive Index
10
0
10
−1
10
−2
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 13. The complex refractive index of cuprous oxide (A): (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
1
Cuprous Oxide (B)
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10
−4
10
13.4
13.5
10 10 Angular Frequency ω, rad/s
13.6
10
Fig. 14. The complex refractive index of cuprous oxide (B): (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
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Gallium Antimonide
2
Complex Refractive Index
10
0
10
−2
10
−4
10
13.3
13.5
10
13.7
10 10 Angular Frequency ω, rad/s
Fig. 15. The complex refractive index of gallium antimonide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Low−Density Polyethylene
1
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10
−4
10
12.1
10
12.3
12.5
10 10 Angular Frequency ω, rad/s
12.7
10
Fig. 16. The complex refractive index of low-density polyethylene: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1). 2
Magnesium Oxide
Complex Refractive Index
10
0
10
−2
10
−4
10
−6
10
−8
10 12 10
13
14
10 10 Angular Frequency ω, rad/s
15
10
Fig. 17. The complex refractive index of magnesium oxide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
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Potassium Bromide
1
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10 12 10
13
10 Angular Frequency ω, rad/s
14
10
Fig. 18. The complex refractive index of potassium bromide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Sodium Fluoride
2
Complex Refractive Index
10
0
10
−2
10
−4
10
−6
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 19. The complex refractive index of sodium fluoride: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Water
5
Complex Refractive Index
10
0
10
−5
10
−10
10
15
10
16
17
10 10 Angular Frequency ω, rad/s
18
10
Fig. 20. The complex refractive index of water: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
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References Bagley, R. L., & Torvik, P. J. (1986). On the fractional calculus model of viscoelastic behavior. Journal of Rheology, 30, 133–155. Krimm, S., Liang, C. Y., & Sutherland, G. B. B. M. (1956). Infrared spectra of high polymers II. Polyethylene. Journal Of Chemical Physics, 25(3), 549–562. Palik, E. D. (1998). Handbook of optical constants of solids II. San Diego: Academic Press. Podlubny, I. (1999). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1st ed.). San Diego: Academic Press. Saleh, B. E. A., & Teich, M. C. (1991). Fundamentals of photonics. New York: Wiley. Wharmby, A. W., & Bagley, R. L. (2014). Modifying Maxwell’s equations for dielectric materials based on techniques from viscoelasticity and concepts from fractional calculus. International Journal of Engineering Science, 79, 59–80. Wharmby, A. W., & Bagley, R. L. (2015a). The application of the fractional calculus model for dispersion and absorption in dielectrics I. Terahertz waves. International Journal of Engineering Science, 93, 1–12. Wharmby, A. W., & Bagley, R. L. (2015b). Necessary conditions to solve fractional order wave equations using traditional Laplace transforms. Fractional Calculus and Applied Analysis, 18(6).
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The application of the fractional calculus model for dispersion and absorption in dielectrics II. Infrared waves Andrew W. Wharmby∗ 711th Human Performance Wing, Human Effectiveness Directorate, Bioeffects Division, Optical Radiation Branch, 4141 Petroleum Rd., JBSA, Fort Sam Houston, TX 78234, USA
a r t i c l e
i n f o
Article history: Received 10 July 2015 Revised 22 February 2016 Accepted 8 April 2016 Available online 22 April 2016 Keywords: Infrared waves Maxwell’s equations Dielectrics Fractional calculus Generalized derivatives Wave equation Viscoelasticity
a b s t r a c t In the first paper of this series, an empirical formula based on viscoelastic analysis techniques that employs concepts from the fractional calculus originally used to model the dielectric behavior of materials exposed to oscillating electromagnetic fields in the radiofrequency band was applied to do the same for electromagnetic fields oscillating in the terahertz frequency range. The empirical formula was integrated into Maxwell’s equations producing a fractional order Ampere’s law whereof a fractional order wave equation was derived. This wave equation was used to describe the absorption and dispersion of terahertz waves in a dielectric medium. In this work, the empirical formula is extended again for application in the infrared frequency spectrum. The fractional calculus dielectric model is adapted to curve fit the complex refractive index data of a variety of semiconductors and insulators. Following the same procedure used in the first paper of this series, the fractional calculus dielectric model is again integrated in Maxwell’s equations with the same dispersion and absorption analysis performed using the newly derived fractional order wave equation. The mathematical consequences of extending this model into infrared frequencies are also discussed. Published by Elsevier Ltd.
1. Introduction It was shown in the first article of this series (Wharmby & Bagley, 2015a) that an empirical model used to predict the dielectric response of materials exposed to radiofrequency electromagnetic radiation could be easily applied to predict the dielectric response of materials exposed to electromagnetic radiation in the terahertz band. The extension of this model into the terahertz frequency range from the radiofrequency range was shown to be mathematically straightforward, requiring no significant change in the development or application of the model aside from the adjustment of the parameter α . This parameter represented the value of the exponent in the model that was shown to translate into the order of the fractional derivative in time that was observed after the model was integrated into Maxwell’s equations (Wharmby & Bagley, 2014, 2015a). This article continues the investigation by applying the empirical model to the higher frequency range of infrared (IR). The empirical model is used to model the dielectric behavior of materials when exposed to electromagnetic fields oscillating at infrared frequencies and then incorporated into Maxwell’s equations to produce a fractional order wave equation that will be used to model the dispersion and absorption of infrared waves in a dielectric. ∗
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http://dx.doi.org/10.1016/j.ijengsci.2016.04.003 0020-7225/Published by Elsevier Ltd.
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63
Aluminum Antimonide
2
Complex Refractive Index
10
0
10
−2
10
−4
10
13.3
10
13.5
13.7
10 10 Angular Frequency ω, rad/s
Fig. 1. The complex refractive index of aluminum antimonide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
2
Beryllium Oxide
Complex Refractive Index
10
0
10
−2
10
−4
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 2. The complex refractive index of beryllium oxide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
2. The fractional calculus empirical model The equation that will be used to model the dielectric response to electromagnetic radiation at infrared frequencies is written as
∗ ( ω ) D 1 + A0 ( j ω )α = r∗ ( jω ) = , ∗ A1 + A2 ( j ω )α E (ω ) →∗
→∗
(1)
where D (ω ) and E (ω ) are the Laplace transformed magnitudes or Euclidian norms of the electric flux density and electric field intensity respectively that have been brought into the complex s-plane (s = jω). The variables A0 , A1 , A2 , and α are the parameters of the dielectric model that approximates the material’s complex permittivity (r∗ ) whose development has been discussed in the first article of this series (Wharmby & Bagley, 2015a) and elsewhere (Bagley & Torvik, 1986; Wharmby & Bagley, 2014). Figs. 1–4 and 10–20 (found in the Appendix) show the generated values of complex refractive index n, where n2 = r∗ , using the model given in Eq. (1) as lines plotted against measured complex refractive index (plotted points) in the infrared range (Palik, 1998). The parameters used are given in Table 1 and 2. These figures show very high agreement between the experimental data and the prediction made by the four-parameter model, particularly within the
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Calcium Fluoride
2
Complex Refractive Index
10
1
10
0
10
−1
10
−2
10
−3
10 13 10
14
15
10 Angular Frequency ω, rad/s
10
Fig. 3. The complex refractive index of calcium fluoride: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Magnesium Fluoride
2
Complex Refractive Index
10
0
10
−2
10
−4
10
−6
10 11 10
12
13
14
15
10 10 10 Angular Frequency ω, rad/s
10
Fig. 4. The complex refractive index of magnesium fluoride: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Table 1 Parameters of fractional derivative model for 4 dielectrics. Material Aluminum antimonide Beryllium oxide Calcium fluoride Magnesium fluoride
A0 (Sec)α
A1 (Dimensionless) −28
2.732 × 10 3.010 × 10−29 3.039 × 10−28 1.440 × 10−28
−2
9.100 × 10 1.380 × 10−1 1.320 × 10−1 1.900 × 10−1
A2 (Sec)α
α (Dimensionless) −29
2.848 × 10 9.889 × 10−30 1.302 × 10−28 6.171 × 10−29
1.996 1.992 1.972 1.980
resonant frequency region where the behavior appears to be asymptotic. However, in some cases there appears to be strong deviations from the behavior predicted by the four-parameter model at the borders of the resonant frequency regions. While this can be interpreted as an indication of the invalidity of the model, the reader is reminded that most materials exhibit multiple resonant frequency regions (Saleh & Teich, 1991). With this in mind, the deviations from the model prediction on either border of the resonant frequency region are most likely indications of nearby resonant frequency regions.
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3. Modifying Maxwell’s equations The procedure that is used to modify Maxwell’s equations is the same in the radiofrequency and terahertz cases. However, a value of α greater than one, which is often the case as seen in Table 1 and 2, will produce a slightly different result than those seen for the radiofrequency and terahertz bandwidths. Though this difference is slight, it brings certain mathematical implications into question. Therefore, the modification of Maxwell’s equations will be revisited here for the case (t ) = E (t ) and applying a Laplace transform in the of α > 1, beginning, as before, by using the constitutive relationship D complex s-plane (s = jω ), as well as assuming a frequency-dependent complex permittivity. This allows Ampere’s Law to be written as a function of frequency
∗ (ω ) = ∗ (ω )0 • ( jω )E ∗ (ω ). ∇ ×H r The four-parameter fractional derivative model found in Eq. (1) is then substituted in for
∗ (ω ) = ∇ ×H
1 + A0 ( j ω )α 0 • ( jω )E ∗ (ω ). A1 + A2 ( j ω )α
(2)
r∗ (ω )
which produces (3)
Multiplying both sides of Eq. (3) by the denominator of the expression for r∗ (ω ) gives
∗ (ω )) + A2 ( jω )α (∇ × H ∗ (ω )) = 0 ( jω )E ∗ (ω ) + A0 0 ( jω )1+α E ∗ (ω ). A1 ( ∇ × H
(4)
However, α now has a value greater than one, which prompts Eq. (4) to be written as
∗ (ω )) + A2 ( jω )1+α (∇ × H ∗ (ω )) = 0 ( jω )E ∗ (ω ) + A0 0 ( jω )2+α E ∗ (ω ). A1 ( ∇ × H
(5)
where α now represents the non-integer portion of its initial value. Taking the inverse Laplace transform of Eq. (5) yields
A1 ( ∇
(t )) + A2 ×H
2+α ∂ ∂ 1+α ∂ E (t ) + A0 0 E (t ) . ∇ × H (t ) = 0 ∂t ∂ t 1+α ∂ t 2+α
(6)
Through the composition property of fractional order and integer order derivatives (Podlubny, 1999), it can be seen that the inclusion of fractional derivatives within the constitutive relationship now results in a modified version of Ampere’s Law shown in Eq. (6) which is seen to exhibit fractional derivatives of order 1 + α and 2 + α . 4. The wave equation With Ampere’s Law modified to accommodate for the inclusion of the fractional derivative model whose α parameter is above one, the wave equation stemming from that result will need to be re-derived to take this change into account. The method use to derive the modified wave equation is carried out in the same way one would derive the traditional wave equation from Ampere’s Law and Faraday’s Law. Assuming a plane electromagnetic wave traveling in the x direction as shown in reference (Wharmby & Bagley, 2014), the first derivative with respect to time of the one-dimensional modified Ampere’s Law (adapted from Eq. (6)) is taken yielding
A1
∂ 2 Ey ∂ 3+α Ey ∂ ∂ Hz ∂ 2+α ∂ Hz + A2 2+α = −0 − A0 0 2 ∂t ∂ x ∂x ∂t ∂t ∂ t 3+α
(7)
which can be rewritten using the composition property of fractional order and integer order derivatives (Podlubny, 1999) as
∂ 2 Ey ∂ ∂ Hz ∂ 1+α ∂ ∂ Hz ∂ 1+α ∂ 2 Ey A1 + A2 1+α = −0 − A0 0 1+α . ∂t ∂ x ∂t ∂ x ∂t ∂t2 ∂t ∂t2
(8)
Next our attention is redirected towards Faraday’s Law, which is compactly written in one dimension as
∂ Ey ∂ Hz = −μr μ0 . ∂x ∂t
(9)
Once again, it is important to note that the permittivity term r does not appear in Faraday’s Law as it is written here and will therefore remain unmodified for the derivation of the modified wave equation. The derivation of the modified wave equation is continued by taking the first derivative in space of Faraday’s Law which is written here as
∂ 2 Ey ∂ ∂ Hz − = . μr μ0 ∂ x 2 ∂ x ∂t 1
(10)
Since it does not matter in Eq. (10) whether we differentiate first with respect to x and then with respect to t, or vice versa, we can substitute the left hand side of Eq. (10) into Eq. (8) where appropriate which produces
A1 −
∂ 2 Ey μr μ0 ∂ x 2 1
+ A2
∂ 2 Ey 1 ∂ 2 Ey ∂ 1+α ∂ 1+α ∂ 2 Ey − = − − A . 0 0 0 μr μ0 ∂ x 2 ∂ t 1+α ∂t2 ∂ t 1+α ∂ t 2
(11)
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Finally, multiplying both sides by −μr μ0 gives us
A1
∂ 2 Ey ∂ 2 Ey ∂ 1+α ∂ 2 Ey ∂ α ∂ 3 Ey + A = μ μ + A μ μ . r 0 0 2 0 r 0 0 ∂tα ∂t3 ∂ x2 ∂ t 1+α ∂ x2 ∂t2
(12)
Here, μr = μ/μ0 , where μr is the relative permeability of the media, μ is the absolute permeability of the media, and μ0 is the permeability of vacuum (μ0 ≈ 1.257 × 10−6 H/m). It is now apparent that employing the four-parameter fractional derivative model as the parameter for complex permittivity results in a modified version of the wave equation shown in Eq. (12) which is seen to exhibit fractional derivatives of order 1 + α and 3 + α . Alternatively, the same result can be obtained by applying the elastic-viscoelastic correspondence principle to the one-dimensional wave equation (i.e., incorporating the four-parameter model through the use of integral transforms). As in the previous cases, this modification is easily extended to all the three dimensions of the wave equation using the same method. Additionally, Eq. (12) can be solved analytically using the same procedure detailed in reference (Wharmby & Bagley, 2014) as long as the appropriate initial waveform and initial conditions are assumed as the increased value of α brings about certain mathematical implications which are discussed elsewhere (Wharmby & Bagley, 2015b). With a boundary waveform of 1 − cos(ω0 t ), the solution was again found to be
Ey (x, t ) = −
1
π
∞
Im 0
Ey (x, re
jπ
)e
−rt
dr −
e
√ x jω0 c−1 μr
1+A0 (− jω0 )α A1 +A2 ( − j ω0 )α
− j ω0 t
−
4
e
√ −x jω0 c−1 μr
1+A0 ( jω0 )α A1 +A2 ( j ω0 )α
4
+ j ω0 t
+
1 . 2
(13)
1
for infrared waves. Here, c is the speed of light equal to (μ0 0 )− 2 . It is again important to note here that this solution has three parts: first, the integral term representing the transient response that decays to zero for long time t, second, the steady state response represented by the exponential terms, and finally, the persistent term of 1/2. For large values of x and t the exponential terms disappear as does the transient response leaving this value of 1/2. This term corresponds to the electrostatic field (or the electric field at zero frequency) present in the electromagnetic field. Performing the same derivation for the solution of the wave traveling in the opposite direction produces a similar result, with the solution reducing to 1/2 for large x and t. Adding these steady state responses together results in the value of 1 which reflects the value of 1 in the expression for the boundary waveform 1 − cos(ω0 t ) which is the only term not dependent on frequency, distance, or time. 5. Absorption The absorption analysis is performed using the same method previously applied in Wharmby and Bagley (2015a) and Wharmby and Bagley (2014). The steady-state part of the solution to the modified wave equation seen as Eq. (13) is plotted for the dielectric materials of beryllium oxide and calcium fluoride in Figs. 5 and 6 whose parameters are found in Table 1. These two materials were selected for this analysis due to their difference in values of α relative to the other materials
Dissipation of an Electromagnetic Wave in Beryllium Oxide 1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
1
2
3 Distance (m)
4
5
Fig. 5. Absorption of an electromagnetic wave in beryllium oxide.
6 −6
x 10
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67
Dissipation of an Electromagnetic Wave in Calcium Fluoride 1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
1
2
3 Distance (m)
4
5
6 −6
x 10
Fig. 6. Absorption of an electromagnetic wave in calcium fluoride.
observed in this study as well as their absorption bands (asymptotic regions) spanning a range of similar frequencies. An V electric field of E0 = 1 m , applied by multiplying Eq. (13) through by E0 , at a frequency of 1.5 × 1014 Hz for beryllium oxide and a frequency of 7.74 × 1013 Hz for calcium fluoride and a relative permeability of μr = 1.0 were used to calculate the solution at increments of x = 4 × 10−9 meters through a total of 6 × 10−6 meters at time t = 5 seconds. It is of great importance to note that the absorption profiles displayed here differ significantly from those shown in references (Wharmby & Bagley, 2015a) and (Wharmby & Bagley, 2014). As expected, the infrared waves traversing through a dielectric are absorbed almost immediately compared to waves oscillating in the radiofrequency and terahertz bands. The infrared waves shown in Figs. 5 and 6 appear to be absorbed quickly enough that they do not have a chance to oscillate as radiofrequency and terahertz waves do. The role the parameter of α has in this analysis must once again be noted. Even though the infrared waves do not oscillate as they are absorbed almost immediately, a difference in their absorption rates can be perceived. The wave propagating through beryllium oxide, though at a higher frequency, is attenuating faster than the wave propagating through calcium fluoride that is oscillating at relatively lower frequency. As shown in reference (Wharmby & Bagley, 2015a) and (Wharmby & Bagley, 2014), this can be attributed to difference in the value of α between these two materials. Beryllium oxide whose α value was found to be 1.992 is higher than the α value for calcium fluoride, found to be 1.972, and hence attenuates at a faster rate. This effect has been found to be universal in all the materials analyzed in this investigation just as has been observed in Wharmby and Bagley (2015a) and Wharmby and Bagley (2014).
6. Dispersion The evolution of the dispersive effect is readily apparent in Figs. 7–9. Once again, the steady state portion of the analytical solution to the modified wave equation seen as Eq. (13) was plotted at two different frequencies (4 × 1013 Hz and 8 × 1013 Hz) at three different distances away from the origin of propagation at time steps of t = 1 × 10−17 seconds through calcium fluoride. The values of 0, 1.5 × 10−4 , and 3.0 × 10−4 meters were chosen for x in order to show the change in waveform resulting from dispersion, beginning with none (x = 0), to an increasingly exaggerated effect (x = 1.5 × 10−4 and x = 3.0 × 10−4 ). Figs. 7–9 show that the wave at both frequencies is experiencing not only attenuation but a shift in its phase, with the wave experiencing a larger attenuation and phase shift farther away from the origin. However, unlike what was observed in the radiofrequency and terahertz frequency bands, the figures are showing a more pronounced attenuation and shift at the lower frequency rather than the high frequency. This behavior is indicative of anomalous dispersion, a phenomenon that accompanies absorption bands that exhibit shapes reminiscent of asymptotic behavior like those seen in Figs. 1–4 and 10–20. This analysis again demonstrates results that are consistent with electromagnetic theory in that waves at different frequencies travel through a dielectric medium at different speeds.
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Dispersion Analysis of Electromagnetic Wave in Calcium Fluoride
1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
Time (sec)
1 x 10
−13
Fig. 7. Dispersion Analysis of electromagnetic wave at x = 0 meters in calcium fluoride at frequencies of 4 × 1013 Hz (−) and 8 × 1013 Hz (−−).
Dispersion Analysis of Electromagnetic Wave in Calcium Fluoride
1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
Time (sec) Fig. 8. Dispersion analysis of electromagnetic wave at x = 1.5 × 10
−5
0.8
1 x 10
−13
meters in calcium fluoride at frequencies of 4 × 1013 Hz (−) and 8 × 1013 Hz.
7. Conclusions It has been shown that the model used to predict the dielectric response of a material to terahertz waves that was based off of a viscoelastic model (Wharmby & Bagley, 2015a) also has the ability to predict a material’s dielectric response to infrared waves without any major modification. In addition to showing high agreement with experimental observation seen in Figs. 1–4 and 10–20, it was demonstrated that the procedure for the application of the developed model was the same not only in the radiofrequency and terahertz bandwidths, but the infrared bandwidth as well. Just as with terahertz frequencies, the model remained fundamentally unmodified for its use in the infrared frequency band, exhibiting only a change in the value of α , which was again seen to translate into the order of the fractional derivative in the time domain. It is important to note here that while the values of α were seen to range between zero and one-half for dielectric materials exposed to radiofrequency radiation (Wharmby & Bagley, 2014) and between one-half and one for terahertz radiation
A.W. Wharmby / International Journal of Engineering Science 104 (2016) 62–74
69
Dispersion Analysis of Electromagnetic Wave in Calcium Fluoride
1
Ey (V/m)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
Time (sec)
1 x 10
−13
Fig. 9. Dispersion Analysis of electromagnetic wave at x = 3.0 × 10−5 meters in calcium fluoride at frequencies of 4 × 1013 Hz (−) and 8 × 1013 Hz.
(Wharmby & Bagley, 2015a), the majority of the values that α took on for infrared radiation are well above one and approaching the value of two. However, it was noted that once the value of α goes beyond one, certain mathematical implications arise when attempting to incorporate the fractional calculus dielectric model into Maxwell’s equations with the goal of obtaining analytical solutions to the modified wave equation that they produce. These implications manifest themselves as terms representing initial conditions that are not always simple to handle, and that assuming an appropriate input waveform and/or initial conditions that mitigate this problem must be considered (Wharmby & Bagley, 2015b). With this consequence addressed, it was shown that the resulting absorption and dispersion analyses are consistent with electromagnetic theory. Figs. 5–9 produce results that reflect the behaviors of their respective curve-fits of their complex refractive index plots. These observations further suggest that viscoelastic models employing the fractional calculus may, in many cases, be applied to dielectric theory for an exceptionally wide range of electromagnetic wave frequencies. Appendix Table 2 contains the values of the parameters used in Eq. (1) to curve-fit the complex refractive index data of the materials shown within this section. All complex refractive index data analyzed in this investigation were taken from the consolidated work of E.D. Palik found in reference (Palik, 1998). It is important to note that in the case of low-density polyethylene found in Fig. 16, α is seen to be a comparatively low 0.548. It was found that electromagnetic radiation in this frequency
Table 2 Parameters of fractional derivative model for 11 dielectrics. Material Aluminum arsenide Aluminum oxide Cesium iodide Cuprous oxide (A) Cuprous oxide (B) Gallium antimonide Low-density polyethylene Magnesium oxide Potassium bromide Sodium fluoride Water
A0 (Sec )α
A1 (Dimensionless ) −28
2.806 × 10 7.031 × 10−28 1.258 × 10−26 1.025 × 10−28 1.530 × 10−27 5.749 × 10−28 8.157 × 10−11 1.082 × 10−28 3.607 × 10−27 3.959 × 10−28 1.409 × 10−25
−1
1.090 × 10 2.755 × 10−1 1.690 × 10−1 1.100 × 10−1 1.190 × 10−1 9.0 0 0 × 10−2 4.330 × 10−1 1.100 × 10−1 2.010 × 10−1 1.780 × 10−1 5.490 × 10−1
A2 (Sec )α
α (Dimensionless ) −29
3.754 × 10 6.172 × 10−28 3.775 × 10−27 1.220 × 10−29 1.878 × 10−28 5.749 × 10−29 1.359 × 10−12 3.785 × 10−29 1.247 × 10−27 2.014 × 10−28 2.063 × 10−25
1.986 1.639 1.962 1.988 1.994 1.994 0.548 1.980 1.996 1.972 1.500
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1
Aluminum Arsenide
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 10. The complex refractive index of aluminum arsenide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
1
Aluminum Oxide
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10 16 10
17
10 Angular Frequency ω, rad/s
18
10
Fig. 11. The complex refractive index of aluminum oxide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
range has a relatively high (∼80%) transmission (Krimm, Liang, & Sutherland, 1956). This suggests that a resonant frequency region exhibiting high absorption as observed in most other materials analyzed in this work will be absent in this frequency range which is indeed the case. However, the data and accompanying curve for absorption is seen to be steadily increasing as the frequency increases. As previously discussed, this can be an indication of a resonant frequency region adjacent to the window of frequencies shown in Fig. 16. Upon further inspection of the infrared spectrum for polyethylene, there is indeed a sharp decrease in transmission at a slightly higher frequency (Krimm et al., 1956) than what is visible in Fig. 16 suggesting that a resonant frequency region which contains a rise in absorption is nearby.
A.W. Wharmby / International Journal of Engineering Science 104 (2016) 62–74
71
Cesium Iodide
1
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10
−4
10 12 10
13
10 Angular Frequency ω, rad/s
14
10
Fig. 12. The complex refractive index of cesium iodide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
1
Cuprous Oxide (A)
Complex Refractive Index
10
0
10
−1
10
−2
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 13. The complex refractive index of cuprous oxide (A): (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
1
Cuprous Oxide (B)
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10
−4
10
13.4
13.5
10 10 Angular Frequency ω, rad/s
13.6
10
Fig. 14. The complex refractive index of cuprous oxide (B): (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
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Gallium Antimonide
2
Complex Refractive Index
10
0
10
−2
10
−4
10
13.3
13.5
10
13.7
10 10 Angular Frequency ω, rad/s
Fig. 15. The complex refractive index of gallium antimonide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Low−Density Polyethylene
1
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10
−4
10
12.1
10
12.3
12.5
10 10 Angular Frequency ω, rad/s
12.7
10
Fig. 16. The complex refractive index of low-density polyethylene: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1). 2
Magnesium Oxide
Complex Refractive Index
10
0
10
−2
10
−4
10
−6
10
−8
10 12 10
13
14
10 10 Angular Frequency ω, rad/s
15
10
Fig. 17. The complex refractive index of magnesium oxide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
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73
Potassium Bromide
1
Complex Refractive Index
10
0
10
−1
10
−2
10
−3
10 12 10
13
10 Angular Frequency ω, rad/s
14
10
Fig. 18. The complex refractive index of potassium bromide: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Sodium Fluoride
2
Complex Refractive Index
10
0
10
−2
10
−4
10
−6
10 13 10
14
10 Angular Frequency ω, rad/s
15
10
Fig. 19. The complex refractive index of sodium fluoride: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
Water
5
Complex Refractive Index
10
0
10
−5
10
−10
10
15
10
16
17
10 10 Angular Frequency ω, rad/s
18
10
Fig. 20. The complex refractive index of water: (◦) measured refractive index, (•) measured absorption coefficient, (−) real component of Eq. (1), (−−) imaginary component of Eq. (1).
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References Bagley, R. L., & Torvik, P. J. (1986). On the fractional calculus model of viscoelastic behavior. Journal of Rheology, 30, 133–155. Krimm, S., Liang, C. Y., & Sutherland, G. B. B. M. (1956). Infrared spectra of high polymers II. Polyethylene. Journal Of Chemical Physics, 25(3), 549–562. Palik, E. D. (1998). Handbook of optical constants of solids II. San Diego: Academic Press. Podlubny, I. (1999). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1st ed.). San Diego: Academic Press. Saleh, B. E. A., & Teich, M. C. (1991). Fundamentals of photonics. New York: Wiley. Wharmby, A. W., & Bagley, R. L. (2014). Modifying Maxwell’s equations for dielectric materials based on techniques from viscoelasticity and concepts from fractional calculus. International Journal of Engineering Science, 79, 59–80. Wharmby, A. W., & Bagley, R. L. (2015a). The application of the fractional calculus model for dispersion and absorption in dielectrics I. Terahertz waves. International Journal of Engineering Science, 93, 1–12. Wharmby, A. W., & Bagley, R. L. (2015b). Necessary conditions to solve fractional order wave equations using traditional Laplace transforms. Fractional Calculus and Applied Analysis, 18(6).