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Propagation properties of broadband terahertz pulses through a bounded magnetized thermal plasma
Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Propagation properties of broadband terahertz pulses through a bounded magnetized thermal plasma Chengxun Yuan a,⇑, Zhongxiang Zhou a, Xiaoli Xiang b, Hongguo Sun a, He Wang a Mengda Xing a, Zhengjun Luo a a b
Physics Department, Harbin Institute of Technology, Harbin 150001, PR China Shanghai Academy of Spaceflight Technology, Shanghai 201108, PR China
a r t i c l e
i n f o
Article history: Received 15 September 2010 Received in revised form 5 October 2010 Available online 16 November 2010 Keywords: THz waves Plasma Propagation properties Resonance absorption
a b s t r a c t An analysis of THz waves propagation in dense, collisional, thermal, magnetized and bounded plasma is presented. By introducing the dielectric constant of a warm magnetoplasma and using the method of impedance transformation with multiple dielectrics, the coefficients of power reflection (R) and absorption (A) are derived for a bounded plasma model by a lossless plate and a conductor plate. The effects of electron temperature, collision frequency, external magnetic field, electron density and thickness of the plasma slab on the absorption coefficient are analyzed numerically. It is found that these plasma parameters can cause significant change in the value of A. Some phenomena, for example negative power absorption, upper-hybrid resonance absorption and geometric resonances absorption, are observed and the behavior of the THz wave propagation inside the plasma model is explained numerically and physically. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The terahertz region (THz) of the electromagnetic (EM) spectrum, wedged between the infrared and microwave end, has received considerable attention in recent years, due to their widespread scientific and technological applications in medical diagnosis, security screening, military detection, radio astronomy, atmospheric studies, high speed communication, chemical and biological sensing. Despite the great importance of terahertz signals, the propagation characteristics of THz waves have not yet been studied in detail, especially THz waves propagation in plasma. It is a common knowledge that an EM wave propagates above electron plasma frequency xp =2p in a non-magnetized plasma. The frequency is determined by plasma electron density Ne [1]. The EM wave can transmit through a plasma and interact with the plasma when the wave frequency x=2p is much higher than xp =2p. Below, xp =2p the EM wave cannot propagate into the bulk region and will be reflected by a thin evanescent layer on the surface [1]. According to the formula xp ¼ ðN e e2 =me e0 Þ1=2 56:4N 1=2 rad s1 , we can see that the broadband terahertz pulses e with the frequency range (0.1–10 THz) can transmit through the plasma with electron density range 1020–1024 m3, where me is the electron mass, e is the electron charge, and e0 is the vacuum ⇑ Corresponding author. Tel./fax: +86 0451 86414141. E-mail address: [email protected] (C. Yuan). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.10.003
permittivity. The electron density of lots of plasmas, such as fusion devices [2] plasma, arc discharge plasma, and dielectric barrier discharges (DBD) plasma, ranges from 1020 to 1024 m3. The characteristic cut-off frequencies, resonant frequencies in plasma are in THz waves ranges. Optical detection in this spectral region is a modern and promising technique for measuring number densities of stable molecules and radicals [3], which can be used in plasma diagnostics [4]. Hot density plasma diagnostic is a challenging task for laser produced plasmas and fusion plasmas [5]; therefore, it is important to study the propagation characteristics of THz wave in plasma. The investigations on the characteristics of EM waves propagation in critical density plasma have been widely presented due to its robustness and widespread applications in plasma physics, radio wave propagation, plasma diagnostics with microwaves, laser induced fusion, laser wakefield accelerator, nuclear physics, etc. [6,7]. The investigations on the interactions between plasmas and EM wave have been limited to the microwave frequency (<100 GHz) due to lack of strong tabletop THz (0.1–10 THz) sources [8]. Recent technical advances in developing intensive THz sources [9–11] have provided us the new opportunities to investigate such interactions in THz range and its potential applications in the characterization and monitoring of industrial plasma, tokomaks, wakefield accelerators, radiation sources based laser–plasma interactions, and conventional gas lasers plasma [12]. Recently, several experimental investigations on the interaction of THz waves with plasmas have been performed. For example, Jamison and his
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colleagues have applied terahertz time-domain spectral techniques to study the characterization of a He discharge plasma using a simple theoretical analysis [12]. Kolner et al. have used broadband terahertz pulses to characterize the time evolution of electron density and collision rate of rapidly evolving argon plasmas [13,14]. The previous works mainly focused on experimental investigation and simple theoretical analysis; however, the detailed reflection, absorption and transmission characteristics of the THz wave while propagating in the plasma, and the effects of the plasma parameters on the characteristics have not been studied. Further more, strong magnetic field and thermal effects have not been considered as all the plasmas studied in these literatures are non-magnetized and cold plasma. In this paper, we have done an analytical study of THz waves propagation in a bounded plasma model. The electromagnetic property of plasma in THz band is investigated with Boltzmann equation [15]. The effects of the plasma density, plasma slab width, the effective collision frequency, electron temperature, external magnetic field strength, and polarization state of the incident wave on the linear propagation, reflection, and absorption of a THz wave in plasma are discussed. A model of THz wave propagation in plasma is presented, an expression for the conductivity of warm magnetoplasma and equations of EM wave propagation in plasma are derived in Section 2. In Section 3, a numerical method is used to study the propagation properties of THz waves in the plasma model, and the power absorption coefficient for incident wave is drawn with various electron temperature, electron densities, collision frequencies, and plasma layer widths. The conclusion based on the above study is conducted in Section 4.
Fig. 1. Schematic diagram of a bounded plasma slab in an external magnetic field with a conducting boundary at z = d.
perature in the plasmas. Therefore, it is necessary to describe a THz wave propagation in a high-density plasma with dielectric constant tensor of a warm magnetoplasma. Under the assumption of the proposed model that most thermal electrons are confined in the plasma by the sheath potential, a perfectly reflecting specular boundary condition is adopted for the electrons at the plasma boundaries. This allows us to use the non-local conductivity of an infinitely homogeneous plasma instead of a bounded plasma [19,20]. For the circularly polarized incident wave, the conductivity tensor in a magnetic field can be derived from the Boltzmann equation [21,22], ^
r ¼ rxx irxy 2. Model and formulations of the problem 2.1. Establishment of physical model Real-life plasmas may be bounded by walls [16–18] and the model of plasma-covered plane conductors is often used to study the propagation properties of an EM wave in plasma both experimentally [17] and theoretically [7,19]. Based on actual cases, a bounded plasma model is proposed in this paper. Two movable thin plates are placed in parallel: the left one is a lossless plate transparent to THz waves and the right one is a perfect conductor. The volume bordered by the plates is filled with the plasma, by which it creates the homogeneous plasma [18]. The left plate is lossless plastic and very thin. Its main function is to slightly modify the phase of waves. Hence the effects of the plate on THz wave propagation can be neglected. Fig. 1 shows the schematic diagram of the THz wave propagation in the bounded plasma model, where 1, 2, and 3 denote medium 1 (air), medium 2 (plasma slab), and medium 3 (conductor), respectively. The incident wave is circularly polarized and propagates along the Z-axis. The external magnetic field aligns with wave propagation. 2.2. Dielectric constant of a warm magnetoplasma The interaction of a transverse EM wave with plasma electrons is an interesting project in plasma physics. For a low frequency incident wave, the plasma is usually considered to be cold, and its dielectric constant is assumed to be Drude type for non-magnetized plasma and Appleton type for magnetized plasma. For a THz wave, the corresponding plasma should be described as high-density. Most high density plasmas are thermal plasmas. Their elecffi pffiffiffiffiffiffiffiffiffiffiffi tron velocity tth ¼ T=me is large enough so that thermal motion of the electrons cannot be neglected, where T is the electron tem-
ð1Þ
The + sign and sign refer to the left-handed and the right-handed circularly polarized wave, respectively. For propagation along the field, the conductivity rxx and rxy in [15] become:
e0 x2p Z 1 cosðYsÞ exp½UðsÞds x 0 Z e0 x2p 1 rxy ¼ sinðYsÞ exp½UðsÞds x 0
rxx ¼
ð2Þ ð3Þ ^
where UðsÞ ¼ ið1 im=xÞs 12 ds2 ; d ¼ e kB T=mc2 ; Y ¼ xc =x; m is ^ the effective collision frequency, e is equivalent complex dielectric constant, kB is Boltzmann’s constant, xc is the electron cyclotron frequency, and c is the velocity of light. Substituting (2) and (3) into (1), we can obtain the conductivity ^
e0 x2p x e0 x2p ¼ x e0 x2p ¼ x e0 x2p ¼ x
r¼
Z
1
exp½UðsÞðcos½Ys i sin½YsÞds
0
Z
1
exp½UðsÞ expðiYsÞds
0
xc jm 1 s ds2 ds exp i 1 2 x x 0 Z 1 1 2 1 2 4 xc im s ds 1 ds þ d s exp i 1 2 8 x x 0 ( ie0 x2p x2 3x4 ¼ 1þ dþ d2 2 x im xc ð x i m xc Þ ðx im xc Þ4 ) 15x6 3 d þ ð4Þ þ ð x i m xc Þ 6 Z
1
In general, the maximum plasma temperature T is less than kB T kB T 100 keV; therefore, the term mc 2 satisfies the condition mc2 1. Hence the asymptotic expression for the conductivity can be simplified as
C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
r
ie0 x2p x im xc
( ," 1
1
The dielectric constant following expression:
"
^
e ¼ =1
^ kB T x2 e 2 2 mc ð x i m xc Þ
^
1
^
2 p
x xðx im xc Þ
#,"
2
1
#
^
1
ð5Þ
e ¼ 1 i e0rx can be evaluated by using the
Using the approximation condition
,"
#)
^
x e kB T ðx im xc Þ2 mc2 kB T mc2
ð6Þ
^
e ¼ 1
x2p xðx im xc Þ
#," 1
xx2p kB T ðx im xc Þ3 mc2
ð7Þ
# ð8Þ
We can see that if T = 0, then (8) goes to Appleton model ^
e ¼1
x2p xðx im xc Þ
ð9Þ
2.3. The propagation properties of broadband terahertz pulses in this model Further we consider a broadband terahertz (0.1–1 THz) pulse propagation in the bounded plasma described in the model. The origin of coordinates is on the back surface of the plasma layer. Some parts of the EM wave are reflected at z = 0 and the rest permeate into the plasma. Because of the existence of the two interfaces at z = 0 and z = d, parts of the waves bounce back and forth between the two bounding surfaces. Some are absorbed by the plasma and some penetrate back into medium 1. The method of impedance transformation with multiple dielectrics will be used to analyze the effects of multiple reflections and derive the reflectivity and absorptivity. The process can be briefly described in the following. We firstly consider the parallel (Ex) components of the incident wave. The Maxwell’s equations describing the transverse THz wave are
lr @Hy ðzÞ @Ex ðzÞ c
@t
lr @Hy ðzÞ @z
þ
¼
@z
¼0
ð10Þ
er @Ex ðzÞ 4p c
@t
þ
c
J x ðzÞ
^
J x ðzÞ ¼ rxx Ex ðzÞ
ð11Þ ð12Þ
where Ex ðzÞ and Hy ðzÞ are the electric and magnetic fields, er and lr are the relative dielectric constant and relative magnetic permeability, and J x ðzÞ is the current density. The solution of Maxwell equations in medium 1 is given by:
E1 ¼ E1x ¼ Eþ1 eik1 z þ E1 eik1 z H1 ¼ H1y ¼
ðEþ1 eik1 z
E1 eik1 z Þ=Z 1
H0 ð0Þ ¼ H1 ð0Þ
ð18Þ
Then the reflection coefficient r can be obtained by the solution under boundary conditions
^
Then
"
ð17Þ
And the boundary conditions at z = 0 can be described as:
E0 ð0Þ ¼ E1 ð0Þ;
1, we obtain
x2 e kB T x2 e kB T 1þ 2 mc 2 ðx im xc Þ ðx im xc Þ2 mc2
For boundary at z ¼ d, the impedance of the conductor Zc by the boundary conditions is given: Z c ¼ E2x =H2y . For a perfect conductor, the impedance Zc is 0, which means E2x jz¼d ¼ 0. Hence,
Eþ2 eik2 d ¼ E2 eik2 d
#
25
ð13Þ ð14Þ
The solution of Maxwell equations in medium 2 is given by:
E2 ¼ E2x ¼ Eþ2 eik2 z þ E2 eik2 z
ð15Þ
H2 ¼ H2y ¼ ðEþ2 eik2 z E2 eik2 z Þ=Z 2
ð16Þ
þ where Eþ j and Hj are the electric and magnetic fields in z direction in medium j (j = 1, 2), respectively, E j and Hj are electric and magnetic fields in z direction, respectively, kjz is the wave vector in medium j (j = 1, 2), and Z j is the intrinsic impedance of j layer.
E1 Z 2 tanhðik2 dÞ Z 1 ¼ Eþ1 Z 2 tanhðik2 dÞ þ Z 1 pffiffiffiffiffiffiffiffiffiffi k2 ¼ 2p=k2 ¼ 2pf e2 l2 =c
r
ð19Þ ð20Þ
Eq. (19) is obtained by only considering the parallel (Ex) components of the incident wave; however, it in fact suits for arbitrarily polarized EM wave [23]. For the circular1y polarized incident wave studied in this paper, we can just employ the corresponding impedance and obtain the reflection coefficient of the incident described in this paper. For the perpendicular components, the impedance in different dielectric layer is given by [30]
Zi ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi l0 li =e0 ei ¼ Z 0 li =ei
ð21Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi So, the impedances become Z 1 ¼ Z 0 ; Z 2 ¼ Z 0 l2 =e2 . Now we can consider the permeability l2 ¼ 1, and complex ^ dielectric constant e2 ¼ e . The reflectance R of the THz wave is directly related to the coefficients of reflection:
pffiffiffiffi p^ ffiffiffiffi2 tanhði2pfd ^ e =cÞ e ffi pffiffiffi p ffiffiffiffi R ¼ jrj2 ¼ ^ tanhði2pfd ^ e =cÞ þ e
ð22Þ
The absorptivity can be derived from the power-balance relation:
A¼1R
ð23Þ
3. Numerical results and discussions From (8) and (22), we can see that the plasma parameters such as plasma density, plasma slab width, the effective collision frequency, electron temperature, external magnetic field strength, and polarization state of the incident wave will have effects on the propagation of THz wave in this model; therefore, the effects of these parameters on the absorptivity are discussed in this section. The following parameters are chosen as they are typical in THz-plasma interaction experiments: N e 1021 m3 , B 1–10 T, T 0–10 keV, v 0.01–1 THz, the incident THz pulses frequency range 0.1–1 THz. 3.1. Effects of temperature on the absorptivity Temperatures in magnetic confinement devices may range from several eV in the scrape-off layer to tens of keV in the plasma core [24]. Therefore, the electron temperatures are chosen as 0, 10, 100, and 500 keV, respectively, for the theoretical simulation. Other parameters are chosen as follows: plasma density, N e ¼ 1:2 1020 m3 ; magnetic field strength, B = 1 T, collision frequency, v = 0.01 THz; and plasma thickness, d = 0.1 m. Fig. 2a and b illustrates the power absorption spectra for the right- and lefthand polarization mode, respectively. The results show that the absorptivity increases for both modes with the increase of electron
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C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
Fig. 3. The attenuation coefficient versus electron temperatures in the case of incident wave frequency f = 1 THz.
Fig. 2. The power absorption spectra versus wave frequency for various electron temperatures. (a) Right- and (b) left-hand polarization mode.
temperature. It’s worth noting that the absorptivity has an obvious change only when the temperature rises above 10 keV. The complex propagation coefficient can be expressed as: ^
^
c ¼ a þ ib ¼ i e 1=2 x=c;
ð24Þ
where a and b are the attenuation and phase coefficient, respectively. Fig. 3 presents the attenuation coefficient for 1 THz wave while the electron temperature varying from 0 to 500 keV. For both modes, the attenuation coefficient a is directly proportional to the temperature. Thus, the absorptivity gets larger with the increase of temperature. The reason of this effect is that there are relatively more high-velocity electrons at the higher temperature which induce larger resonant effects [25]. Comparing Fig. 2a with b, we can see that the positions of peak absorption are different for different polarization modes. The peak for the right-hand polarization mode lies between 0.1 and 0.4 THz, while the peak for the left-hand polarization mode lies between 0.1 and 0.2 THz. This is mainly due to the difference of resonance absorption frequency between the two modes. 3.2. Effects of collision frequency on the absorptivity In the presence of high density and collisionality at high gas pressures where the collision frequency (v) is of the order of both
the plasma (xp ) and the wave frequency of the THz wave xðm xp ; xÞ [26], it is necessary to analyze the effects of collision frequency on the absorptivity. The parameters used in the calculation are chosen as follows: plasma density N e ¼ 1021 m3 , magnetic field strength B = 10 T, temperature T = 10 keV, plasma thickness d = 0.1 m and variable collision frequency v = 0, 0.01, 0.5 and 5 THz. Fig. 4a and b shows the attenuation versus incident wave frequency at the four collision frequencies m for the right- and left-hand polarization mode, respectively. We can find that with the increase of collision frequency, the absorptivity will increase. When the collision frequency is of the order of the THz wave frequency, almost all the THz wave will be absorbed. The attenuation behavior can be explained as follows: the transferred momentum from the electron to the neutral particle increases with the increase of the collision frequency, so the absorbed energy of the THz wave will increase accordingly. For the right-hand polarization mode, we can see that a valley appears and the phenomenon of negative power absorption takes place [20,27]. This effect is known as the spatial dispersion of plasma conductivity caused by electron thermal motion and is typical for the anomalous skin effect [20]. In this case, a bremsstrahlung radiation occurred from the elastic collisions of electrons with atoms [27]. Fig. 5 shows the propagation coefficient versus incident wave frequency for the right-hand polarization mode. It clearly shows that the value of attenuation coefficient a becomes negative when the frequency of the incident wave is among the range 0.3– 0.4 GHz. 3.3. Effects of electron density on the absorptivity In magnetic confinement devices, the density range covers from 1017 to 1023 m3, and the electron density profile is flat in the core [28]. We take some typical parameters to analyze the absorptivity: plasma density, N e ¼ 0:5 1020 ; 5 1020 ; 1021 and 5 1021 m3 ; magnetic field strength, B = 10 T; temperature, T = 10 keV; plasma thickness, d = 0.1 m; and collision frequency, v = 0.01 THz. Fig. 6a and b shows the attenuation versus incident wave frequency at the four plasma densities for the right- and left-hand polarization mode, respectively. The results shown in Fig. 6 indicate that the broadband power absorption increases with the increases of electron density; meanwhile, the peak power absorptions shifts to the higher frequency range, and the reflection bandwidth increases with the increases of electron density. This is mainly due to the res-
C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
Fig. 4. The power absorption spectra versus wave frequency for various collision frequencies. (a) Right- and (b) left-hand polarization mode.
27
Fig. 6. The power absorption spectra versus wave frequency for various electron densities. (a) Right- and (b) left-hand polarization mode.
onance absorption in the upper hybrid frequency range [29,30]. The upper-hybrid resonance condition is x2 ¼ x2c þ x2p [29], so the plasma frequency xp ðxp ¼ 56:4N 1=2 rad s1 Þ increases with e the increases of electron density, and the resonance absorption peaks move to the higher frequency range. From Fig. 6a, we can see that with the increase of electron density, the phenomenon of negative power absorption is more obvious. We can conclude that the anomalous skin effect is more evident in the frequency range of 0.1–1 THz when plasma density increases. 3.4. Effects of magnetic field on the absorptivity
Fig. 5. The propagation coefficient versus wave frequency for the right-hand polarization mode.
Plasma in a strong external magnetic field is an interesting topic in laser-produced plasma [31], magnetic fusion devices and arc discharge. The magnetic field strength in these devices can range from a few tenths to tens of Teslas, corresponding to an electron cyclotron frequency in the order of THz, by the equation xp ¼ eB=me 1:76 1011 B rad s1 . We then investigate the effects of magnetic field on the absorptivity while uniform magnetic field B = 0.1, 1, 10, and 50 T, respectively. The plasma density, collision frequency, temperature and plasma thickness are 1021 m3, 0.01 THz, 10 keV and 0.1 m, respectively. Fig. 7a and b shows the absorption versus incident wave frequency for the right- and
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C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
Fig. 7. The power absorption spectra versus wave frequency for various magnetic field strengths. (a) Right- and (b) left-hand polarization mode.
left-hand polarization mode, respectively. It clearly shows that the absorption level and the position of peaks and valleys have obviously been affected by the magnetic field for both polarization modes. The existence of the external magnetic field increases the peak power absorption and moves it to a higher frequency range. The reason is still that the upper-hybrid resonance frequency increases with the increase of electron cyclotron frequency xc which is directly proportional to magnetic field strength xc ¼ eB=me 1:76 1011 B rad s1 . From Fig. 7a we can see that, the negative power absorption appears for the right-hand polarization mode when B = 10 T, which indicates that a proper magnetic field can influence the spatial dispersion of plasma conductivity and cause the anomalous skin effect.
3.5. Effects of plasma layer thickness on the absorptivity For the bounded plasma model presented in this paper, the thickness of the plasma is an important factor. Fig. 8a and b shows the absorptivity versus the frequency of the incident THz wave at different widths of the plasma layers for the right- and left-hand polarization mode, respectively; where N e ¼ 1021 m3 ; B ¼ 10 T; m ¼ 0:01 THz and T ¼ 10 keV. The results show that the average attenuation level always increases with the increase of the plasma
Fig. 8. The power absorption spectra versus wave frequency for various plasma layer thicknesses. (a) Right- and (b) left-hand polarization mode.
thickness d, because the thicker the plasma layer, the more EM wave energy it can absorb through the collision damping of the electrons and the neutrons. From Fig. 8, we can see that the thin plasma has higher absorption than the thick one in some frequency bands. It is because that the THz wave transmission through the limited thick plasma slab will exhibit a series of geometric, i.e. Fabry–Perot type, resonances as the result of constructive interference of multiple waves scattered from the front and back surfaces. When the thickness is large enough (d p=k), the presence of second boundary has very little influence on the electromagnetic field. The field at the second boundary is much smaller than that at the first boundary; therefore multiple refection effects can be neglected. The analysis is consistent with the curve shown in Fig. 8 while plasma thickness d = 1 m, the absorption curve is nearly flat in the higher frequency range. From Fig. 8, we also note that, the thickness of the plasma has no effect on the position and value of the absorptivity valley. 4. Conclusion In this paper, the reflectance and absorptivity of a THz wave interacting with a magnetized bounded plasma have been theoretically analyzed by introducing the conductivity tensor of a
C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
warm magnetized plasma. Numerical results show that power absorption of the THz wave strongly depends on the plasma temperature, collision frequency, external magnetic field, electron density, and thickness of the plasma slab. The attenuation of the THz wave in the warm and magnetized bounded plasma is mainly caused by the collisional heating effect, electron cyclotron resonance absorption, and geometric resonances absorption. The negative power absorption is observed when the plasma parameters meet the non-local conductivity condition. The strong effects of plasma parameters on the absorption in the THz frequency range also makes broadband terahertz pulses a very direct and practical tool of measuring collisional properties, electron temperature, and electron density of the warm plasma. Thus this conclusion may find applications in plasma diagnoses for a dense discharge plasma, laser-induced plasma, and fusion plasma in strong magnetic field.
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Propagation properties of broadband terahertz pulses through a bounded magnetized thermal plasma Chengxun Yuan a,⇑, Zhongxiang Zhou a, Xiaoli Xiang b, Hongguo Sun a, He Wang a Mengda Xing a, Zhengjun Luo a a b
Physics Department, Harbin Institute of Technology, Harbin 150001, PR China Shanghai Academy of Spaceflight Technology, Shanghai 201108, PR China
a r t i c l e
i n f o
Article history: Received 15 September 2010 Received in revised form 5 October 2010 Available online 16 November 2010 Keywords: THz waves Plasma Propagation properties Resonance absorption
a b s t r a c t An analysis of THz waves propagation in dense, collisional, thermal, magnetized and bounded plasma is presented. By introducing the dielectric constant of a warm magnetoplasma and using the method of impedance transformation with multiple dielectrics, the coefficients of power reflection (R) and absorption (A) are derived for a bounded plasma model by a lossless plate and a conductor plate. The effects of electron temperature, collision frequency, external magnetic field, electron density and thickness of the plasma slab on the absorption coefficient are analyzed numerically. It is found that these plasma parameters can cause significant change in the value of A. Some phenomena, for example negative power absorption, upper-hybrid resonance absorption and geometric resonances absorption, are observed and the behavior of the THz wave propagation inside the plasma model is explained numerically and physically. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The terahertz region (THz) of the electromagnetic (EM) spectrum, wedged between the infrared and microwave end, has received considerable attention in recent years, due to their widespread scientific and technological applications in medical diagnosis, security screening, military detection, radio astronomy, atmospheric studies, high speed communication, chemical and biological sensing. Despite the great importance of terahertz signals, the propagation characteristics of THz waves have not yet been studied in detail, especially THz waves propagation in plasma. It is a common knowledge that an EM wave propagates above electron plasma frequency xp =2p in a non-magnetized plasma. The frequency is determined by plasma electron density Ne [1]. The EM wave can transmit through a plasma and interact with the plasma when the wave frequency x=2p is much higher than xp =2p. Below, xp =2p the EM wave cannot propagate into the bulk region and will be reflected by a thin evanescent layer on the surface [1]. According to the formula xp ¼ ðN e e2 =me e0 Þ1=2 56:4N 1=2 rad s1 , we can see that the broadband terahertz pulses e with the frequency range (0.1–10 THz) can transmit through the plasma with electron density range 1020–1024 m3, where me is the electron mass, e is the electron charge, and e0 is the vacuum ⇑ Corresponding author. Tel./fax: +86 0451 86414141. E-mail address: [email protected] (C. Yuan). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.10.003
permittivity. The electron density of lots of plasmas, such as fusion devices [2] plasma, arc discharge plasma, and dielectric barrier discharges (DBD) plasma, ranges from 1020 to 1024 m3. The characteristic cut-off frequencies, resonant frequencies in plasma are in THz waves ranges. Optical detection in this spectral region is a modern and promising technique for measuring number densities of stable molecules and radicals [3], which can be used in plasma diagnostics [4]. Hot density plasma diagnostic is a challenging task for laser produced plasmas and fusion plasmas [5]; therefore, it is important to study the propagation characteristics of THz wave in plasma. The investigations on the characteristics of EM waves propagation in critical density plasma have been widely presented due to its robustness and widespread applications in plasma physics, radio wave propagation, plasma diagnostics with microwaves, laser induced fusion, laser wakefield accelerator, nuclear physics, etc. [6,7]. The investigations on the interactions between plasmas and EM wave have been limited to the microwave frequency (<100 GHz) due to lack of strong tabletop THz (0.1–10 THz) sources [8]. Recent technical advances in developing intensive THz sources [9–11] have provided us the new opportunities to investigate such interactions in THz range and its potential applications in the characterization and monitoring of industrial plasma, tokomaks, wakefield accelerators, radiation sources based laser–plasma interactions, and conventional gas lasers plasma [12]. Recently, several experimental investigations on the interaction of THz waves with plasmas have been performed. For example, Jamison and his
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colleagues have applied terahertz time-domain spectral techniques to study the characterization of a He discharge plasma using a simple theoretical analysis [12]. Kolner et al. have used broadband terahertz pulses to characterize the time evolution of electron density and collision rate of rapidly evolving argon plasmas [13,14]. The previous works mainly focused on experimental investigation and simple theoretical analysis; however, the detailed reflection, absorption and transmission characteristics of the THz wave while propagating in the plasma, and the effects of the plasma parameters on the characteristics have not been studied. Further more, strong magnetic field and thermal effects have not been considered as all the plasmas studied in these literatures are non-magnetized and cold plasma. In this paper, we have done an analytical study of THz waves propagation in a bounded plasma model. The electromagnetic property of plasma in THz band is investigated with Boltzmann equation [15]. The effects of the plasma density, plasma slab width, the effective collision frequency, electron temperature, external magnetic field strength, and polarization state of the incident wave on the linear propagation, reflection, and absorption of a THz wave in plasma are discussed. A model of THz wave propagation in plasma is presented, an expression for the conductivity of warm magnetoplasma and equations of EM wave propagation in plasma are derived in Section 2. In Section 3, a numerical method is used to study the propagation properties of THz waves in the plasma model, and the power absorption coefficient for incident wave is drawn with various electron temperature, electron densities, collision frequencies, and plasma layer widths. The conclusion based on the above study is conducted in Section 4.
Fig. 1. Schematic diagram of a bounded plasma slab in an external magnetic field with a conducting boundary at z = d.
perature in the plasmas. Therefore, it is necessary to describe a THz wave propagation in a high-density plasma with dielectric constant tensor of a warm magnetoplasma. Under the assumption of the proposed model that most thermal electrons are confined in the plasma by the sheath potential, a perfectly reflecting specular boundary condition is adopted for the electrons at the plasma boundaries. This allows us to use the non-local conductivity of an infinitely homogeneous plasma instead of a bounded plasma [19,20]. For the circularly polarized incident wave, the conductivity tensor in a magnetic field can be derived from the Boltzmann equation [21,22], ^
r ¼ rxx irxy 2. Model and formulations of the problem 2.1. Establishment of physical model Real-life plasmas may be bounded by walls [16–18] and the model of plasma-covered plane conductors is often used to study the propagation properties of an EM wave in plasma both experimentally [17] and theoretically [7,19]. Based on actual cases, a bounded plasma model is proposed in this paper. Two movable thin plates are placed in parallel: the left one is a lossless plate transparent to THz waves and the right one is a perfect conductor. The volume bordered by the plates is filled with the plasma, by which it creates the homogeneous plasma [18]. The left plate is lossless plastic and very thin. Its main function is to slightly modify the phase of waves. Hence the effects of the plate on THz wave propagation can be neglected. Fig. 1 shows the schematic diagram of the THz wave propagation in the bounded plasma model, where 1, 2, and 3 denote medium 1 (air), medium 2 (plasma slab), and medium 3 (conductor), respectively. The incident wave is circularly polarized and propagates along the Z-axis. The external magnetic field aligns with wave propagation. 2.2. Dielectric constant of a warm magnetoplasma The interaction of a transverse EM wave with plasma electrons is an interesting project in plasma physics. For a low frequency incident wave, the plasma is usually considered to be cold, and its dielectric constant is assumed to be Drude type for non-magnetized plasma and Appleton type for magnetized plasma. For a THz wave, the corresponding plasma should be described as high-density. Most high density plasmas are thermal plasmas. Their elecffi pffiffiffiffiffiffiffiffiffiffiffi tron velocity tth ¼ T=me is large enough so that thermal motion of the electrons cannot be neglected, where T is the electron tem-
ð1Þ
The + sign and sign refer to the left-handed and the right-handed circularly polarized wave, respectively. For propagation along the field, the conductivity rxx and rxy in [15] become:
e0 x2p Z 1 cosðYsÞ exp½UðsÞds x 0 Z e0 x2p 1 rxy ¼ sinðYsÞ exp½UðsÞds x 0
rxx ¼
ð2Þ ð3Þ ^
where UðsÞ ¼ ið1 im=xÞs 12 ds2 ; d ¼ e kB T=mc2 ; Y ¼ xc =x; m is ^ the effective collision frequency, e is equivalent complex dielectric constant, kB is Boltzmann’s constant, xc is the electron cyclotron frequency, and c is the velocity of light. Substituting (2) and (3) into (1), we can obtain the conductivity ^
e0 x2p x e0 x2p ¼ x e0 x2p ¼ x e0 x2p ¼ x
r¼
Z
1
exp½UðsÞðcos½Ys i sin½YsÞds
0
Z
1
exp½UðsÞ expðiYsÞds
0
xc jm 1 s ds2 ds exp i 1 2 x x 0 Z 1 1 2 1 2 4 xc im s ds 1 ds þ d s exp i 1 2 8 x x 0 ( ie0 x2p x2 3x4 ¼ 1þ dþ d2 2 x im xc ð x i m xc Þ ðx im xc Þ4 ) 15x6 3 d þ ð4Þ þ ð x i m xc Þ 6 Z
1
In general, the maximum plasma temperature T is less than kB T kB T 100 keV; therefore, the term mc 2 satisfies the condition mc2 1. Hence the asymptotic expression for the conductivity can be simplified as
C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
r
ie0 x2p x im xc
( ," 1
1
The dielectric constant following expression:
"
^
e ¼ =1
^ kB T x2 e 2 2 mc ð x i m xc Þ
^
1
^
2 p
x xðx im xc Þ
#,"
2
1
#
^
1
ð5Þ
e ¼ 1 i e0rx can be evaluated by using the
Using the approximation condition
,"
#)
^
x e kB T ðx im xc Þ2 mc2 kB T mc2
ð6Þ
^
e ¼ 1
x2p xðx im xc Þ
#," 1
xx2p kB T ðx im xc Þ3 mc2
ð7Þ
# ð8Þ
We can see that if T = 0, then (8) goes to Appleton model ^
e ¼1
x2p xðx im xc Þ
ð9Þ
2.3. The propagation properties of broadband terahertz pulses in this model Further we consider a broadband terahertz (0.1–1 THz) pulse propagation in the bounded plasma described in the model. The origin of coordinates is on the back surface of the plasma layer. Some parts of the EM wave are reflected at z = 0 and the rest permeate into the plasma. Because of the existence of the two interfaces at z = 0 and z = d, parts of the waves bounce back and forth between the two bounding surfaces. Some are absorbed by the plasma and some penetrate back into medium 1. The method of impedance transformation with multiple dielectrics will be used to analyze the effects of multiple reflections and derive the reflectivity and absorptivity. The process can be briefly described in the following. We firstly consider the parallel (Ex) components of the incident wave. The Maxwell’s equations describing the transverse THz wave are
lr @Hy ðzÞ @Ex ðzÞ c
@t
lr @Hy ðzÞ @z
þ
¼
@z
¼0
ð10Þ
er @Ex ðzÞ 4p c
@t
þ
c
J x ðzÞ
^
J x ðzÞ ¼ rxx Ex ðzÞ
ð11Þ ð12Þ
where Ex ðzÞ and Hy ðzÞ are the electric and magnetic fields, er and lr are the relative dielectric constant and relative magnetic permeability, and J x ðzÞ is the current density. The solution of Maxwell equations in medium 1 is given by:
E1 ¼ E1x ¼ Eþ1 eik1 z þ E1 eik1 z H1 ¼ H1y ¼
ðEþ1 eik1 z
E1 eik1 z Þ=Z 1
H0 ð0Þ ¼ H1 ð0Þ
ð18Þ
Then the reflection coefficient r can be obtained by the solution under boundary conditions
^
Then
"
ð17Þ
And the boundary conditions at z = 0 can be described as:
E0 ð0Þ ¼ E1 ð0Þ;
1, we obtain
x2 e kB T x2 e kB T 1þ 2 mc 2 ðx im xc Þ ðx im xc Þ2 mc2
For boundary at z ¼ d, the impedance of the conductor Zc by the boundary conditions is given: Z c ¼ E2x =H2y . For a perfect conductor, the impedance Zc is 0, which means E2x jz¼d ¼ 0. Hence,
Eþ2 eik2 d ¼ E2 eik2 d
#
25
ð13Þ ð14Þ
The solution of Maxwell equations in medium 2 is given by:
E2 ¼ E2x ¼ Eþ2 eik2 z þ E2 eik2 z
ð15Þ
H2 ¼ H2y ¼ ðEþ2 eik2 z E2 eik2 z Þ=Z 2
ð16Þ
þ where Eþ j and Hj are the electric and magnetic fields in z direction in medium j (j = 1, 2), respectively, E j and Hj are electric and magnetic fields in z direction, respectively, kjz is the wave vector in medium j (j = 1, 2), and Z j is the intrinsic impedance of j layer.
E1 Z 2 tanhðik2 dÞ Z 1 ¼ Eþ1 Z 2 tanhðik2 dÞ þ Z 1 pffiffiffiffiffiffiffiffiffiffi k2 ¼ 2p=k2 ¼ 2pf e2 l2 =c
r
ð19Þ ð20Þ
Eq. (19) is obtained by only considering the parallel (Ex) components of the incident wave; however, it in fact suits for arbitrarily polarized EM wave [23]. For the circular1y polarized incident wave studied in this paper, we can just employ the corresponding impedance and obtain the reflection coefficient of the incident described in this paper. For the perpendicular components, the impedance in different dielectric layer is given by [30]
Zi ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi l0 li =e0 ei ¼ Z 0 li =ei
ð21Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi So, the impedances become Z 1 ¼ Z 0 ; Z 2 ¼ Z 0 l2 =e2 . Now we can consider the permeability l2 ¼ 1, and complex ^ dielectric constant e2 ¼ e . The reflectance R of the THz wave is directly related to the coefficients of reflection:
pffiffiffiffi p^ ffiffiffiffi2 tanhði2pfd ^ e =cÞ e ffi pffiffiffi p ffiffiffiffi R ¼ jrj2 ¼ ^ tanhði2pfd ^ e =cÞ þ e
ð22Þ
The absorptivity can be derived from the power-balance relation:
A¼1R
ð23Þ
3. Numerical results and discussions From (8) and (22), we can see that the plasma parameters such as plasma density, plasma slab width, the effective collision frequency, electron temperature, external magnetic field strength, and polarization state of the incident wave will have effects on the propagation of THz wave in this model; therefore, the effects of these parameters on the absorptivity are discussed in this section. The following parameters are chosen as they are typical in THz-plasma interaction experiments: N e 1021 m3 , B 1–10 T, T 0–10 keV, v 0.01–1 THz, the incident THz pulses frequency range 0.1–1 THz. 3.1. Effects of temperature on the absorptivity Temperatures in magnetic confinement devices may range from several eV in the scrape-off layer to tens of keV in the plasma core [24]. Therefore, the electron temperatures are chosen as 0, 10, 100, and 500 keV, respectively, for the theoretical simulation. Other parameters are chosen as follows: plasma density, N e ¼ 1:2 1020 m3 ; magnetic field strength, B = 1 T, collision frequency, v = 0.01 THz; and plasma thickness, d = 0.1 m. Fig. 2a and b illustrates the power absorption spectra for the right- and lefthand polarization mode, respectively. The results show that the absorptivity increases for both modes with the increase of electron
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C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
Fig. 3. The attenuation coefficient versus electron temperatures in the case of incident wave frequency f = 1 THz.
Fig. 2. The power absorption spectra versus wave frequency for various electron temperatures. (a) Right- and (b) left-hand polarization mode.
temperature. It’s worth noting that the absorptivity has an obvious change only when the temperature rises above 10 keV. The complex propagation coefficient can be expressed as: ^
^
c ¼ a þ ib ¼ i e 1=2 x=c;
ð24Þ
where a and b are the attenuation and phase coefficient, respectively. Fig. 3 presents the attenuation coefficient for 1 THz wave while the electron temperature varying from 0 to 500 keV. For both modes, the attenuation coefficient a is directly proportional to the temperature. Thus, the absorptivity gets larger with the increase of temperature. The reason of this effect is that there are relatively more high-velocity electrons at the higher temperature which induce larger resonant effects [25]. Comparing Fig. 2a with b, we can see that the positions of peak absorption are different for different polarization modes. The peak for the right-hand polarization mode lies between 0.1 and 0.4 THz, while the peak for the left-hand polarization mode lies between 0.1 and 0.2 THz. This is mainly due to the difference of resonance absorption frequency between the two modes. 3.2. Effects of collision frequency on the absorptivity In the presence of high density and collisionality at high gas pressures where the collision frequency (v) is of the order of both
the plasma (xp ) and the wave frequency of the THz wave xðm xp ; xÞ [26], it is necessary to analyze the effects of collision frequency on the absorptivity. The parameters used in the calculation are chosen as follows: plasma density N e ¼ 1021 m3 , magnetic field strength B = 10 T, temperature T = 10 keV, plasma thickness d = 0.1 m and variable collision frequency v = 0, 0.01, 0.5 and 5 THz. Fig. 4a and b shows the attenuation versus incident wave frequency at the four collision frequencies m for the right- and left-hand polarization mode, respectively. We can find that with the increase of collision frequency, the absorptivity will increase. When the collision frequency is of the order of the THz wave frequency, almost all the THz wave will be absorbed. The attenuation behavior can be explained as follows: the transferred momentum from the electron to the neutral particle increases with the increase of the collision frequency, so the absorbed energy of the THz wave will increase accordingly. For the right-hand polarization mode, we can see that a valley appears and the phenomenon of negative power absorption takes place [20,27]. This effect is known as the spatial dispersion of plasma conductivity caused by electron thermal motion and is typical for the anomalous skin effect [20]. In this case, a bremsstrahlung radiation occurred from the elastic collisions of electrons with atoms [27]. Fig. 5 shows the propagation coefficient versus incident wave frequency for the right-hand polarization mode. It clearly shows that the value of attenuation coefficient a becomes negative when the frequency of the incident wave is among the range 0.3– 0.4 GHz. 3.3. Effects of electron density on the absorptivity In magnetic confinement devices, the density range covers from 1017 to 1023 m3, and the electron density profile is flat in the core [28]. We take some typical parameters to analyze the absorptivity: plasma density, N e ¼ 0:5 1020 ; 5 1020 ; 1021 and 5 1021 m3 ; magnetic field strength, B = 10 T; temperature, T = 10 keV; plasma thickness, d = 0.1 m; and collision frequency, v = 0.01 THz. Fig. 6a and b shows the attenuation versus incident wave frequency at the four plasma densities for the right- and left-hand polarization mode, respectively. The results shown in Fig. 6 indicate that the broadband power absorption increases with the increases of electron density; meanwhile, the peak power absorptions shifts to the higher frequency range, and the reflection bandwidth increases with the increases of electron density. This is mainly due to the res-
C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
Fig. 4. The power absorption spectra versus wave frequency for various collision frequencies. (a) Right- and (b) left-hand polarization mode.
27
Fig. 6. The power absorption spectra versus wave frequency for various electron densities. (a) Right- and (b) left-hand polarization mode.
onance absorption in the upper hybrid frequency range [29,30]. The upper-hybrid resonance condition is x2 ¼ x2c þ x2p [29], so the plasma frequency xp ðxp ¼ 56:4N 1=2 rad s1 Þ increases with e the increases of electron density, and the resonance absorption peaks move to the higher frequency range. From Fig. 6a, we can see that with the increase of electron density, the phenomenon of negative power absorption is more obvious. We can conclude that the anomalous skin effect is more evident in the frequency range of 0.1–1 THz when plasma density increases. 3.4. Effects of magnetic field on the absorptivity
Fig. 5. The propagation coefficient versus wave frequency for the right-hand polarization mode.
Plasma in a strong external magnetic field is an interesting topic in laser-produced plasma [31], magnetic fusion devices and arc discharge. The magnetic field strength in these devices can range from a few tenths to tens of Teslas, corresponding to an electron cyclotron frequency in the order of THz, by the equation xp ¼ eB=me 1:76 1011 B rad s1 . We then investigate the effects of magnetic field on the absorptivity while uniform magnetic field B = 0.1, 1, 10, and 50 T, respectively. The plasma density, collision frequency, temperature and plasma thickness are 1021 m3, 0.01 THz, 10 keV and 0.1 m, respectively. Fig. 7a and b shows the absorption versus incident wave frequency for the right- and
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C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
Fig. 7. The power absorption spectra versus wave frequency for various magnetic field strengths. (a) Right- and (b) left-hand polarization mode.
left-hand polarization mode, respectively. It clearly shows that the absorption level and the position of peaks and valleys have obviously been affected by the magnetic field for both polarization modes. The existence of the external magnetic field increases the peak power absorption and moves it to a higher frequency range. The reason is still that the upper-hybrid resonance frequency increases with the increase of electron cyclotron frequency xc which is directly proportional to magnetic field strength xc ¼ eB=me 1:76 1011 B rad s1 . From Fig. 7a we can see that, the negative power absorption appears for the right-hand polarization mode when B = 10 T, which indicates that a proper magnetic field can influence the spatial dispersion of plasma conductivity and cause the anomalous skin effect.
3.5. Effects of plasma layer thickness on the absorptivity For the bounded plasma model presented in this paper, the thickness of the plasma is an important factor. Fig. 8a and b shows the absorptivity versus the frequency of the incident THz wave at different widths of the plasma layers for the right- and left-hand polarization mode, respectively; where N e ¼ 1021 m3 ; B ¼ 10 T; m ¼ 0:01 THz and T ¼ 10 keV. The results show that the average attenuation level always increases with the increase of the plasma
Fig. 8. The power absorption spectra versus wave frequency for various plasma layer thicknesses. (a) Right- and (b) left-hand polarization mode.
thickness d, because the thicker the plasma layer, the more EM wave energy it can absorb through the collision damping of the electrons and the neutrons. From Fig. 8, we can see that the thin plasma has higher absorption than the thick one in some frequency bands. It is because that the THz wave transmission through the limited thick plasma slab will exhibit a series of geometric, i.e. Fabry–Perot type, resonances as the result of constructive interference of multiple waves scattered from the front and back surfaces. When the thickness is large enough (d p=k), the presence of second boundary has very little influence on the electromagnetic field. The field at the second boundary is much smaller than that at the first boundary; therefore multiple refection effects can be neglected. The analysis is consistent with the curve shown in Fig. 8 while plasma thickness d = 1 m, the absorption curve is nearly flat in the higher frequency range. From Fig. 8, we also note that, the thickness of the plasma has no effect on the position and value of the absorptivity valley. 4. Conclusion In this paper, the reflectance and absorptivity of a THz wave interacting with a magnetized bounded plasma have been theoretically analyzed by introducing the conductivity tensor of a
C. Yuan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 23–29
warm magnetized plasma. Numerical results show that power absorption of the THz wave strongly depends on the plasma temperature, collision frequency, external magnetic field, electron density, and thickness of the plasma slab. The attenuation of the THz wave in the warm and magnetized bounded plasma is mainly caused by the collisional heating effect, electron cyclotron resonance absorption, and geometric resonances absorption. The negative power absorption is observed when the plasma parameters meet the non-local conductivity condition. The strong effects of plasma parameters on the absorption in the THz frequency range also makes broadband terahertz pulses a very direct and practical tool of measuring collisional properties, electron temperature, and electron density of the warm plasma. Thus this conclusion may find applications in plasma diagnoses for a dense discharge plasma, laser-induced plasma, and fusion plasma in strong magnetic field.
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