Feasibility of a Multi-Pass Thomson Scattering System with Confocal Spherical Mirrors

Transcription

1 Plasma and Fusion Research: Letters Volume 5, ) Feasibility o a Multi-Pass Thomson Scattering System with Conocal Spherical Mirrors Junichi HIRATSUKA, Akira EJIRI, Yuichi TAKASE and Takashi YAMAGUCHI The University o Tokyo, Kashiwa , Japan Received 2 April 200 / Accepted 27 August 200) Multi-pass Thomson scattering is an attractive method or measuring electron temperature and density in low density plasmas. The easibility o multi-pass laser optics consisting o two conocal spherical mirrors has been studied, and analytic expressions or the path have been obtained. These can be used to optimize the design under the given conditions. The eects o aberration, as well as positioning and alignment errors, were ound to be critical or obtaining large transit numbers. Beam expansion resulting rom aberration, surace roughness o the mirrors and diraction is considered. Under practical conditions, transit numbers o about 37 i.e., 9 round trips) are possible. c 200 The Japan Society o Plasma Science and Nuclear Fusion Research Keywords: Thomson scattering, multi-pass, conocal mirror, ECH start-up, low-density plasma DOI: 0.585/pr Multi-pass Thomson scattering has the advantage o enhancing the scattered signal by accumulating signals rom multiple passes through the plasma. On the TST-2 spherical tokamak device, non inductive start-up experiments were conducted using ECH 2.45 GHz/5 kw) []. According to intererometric measurements, typical electron densities are n e 0 7 m 3, about two orders o magnitude lower than typical densities in ohmic discharges. Thereore, enhancement o the scattered signal by a multipass scheme is necessary. On TEXTOR, multi-pass measurements were carried out using two spherical mirrors. The mirrors can be aligned or dierent number o passes, such as double- and sixpass. In this six-pass system, the laser beam passes through the plasma six times and returns to the laser rod. Compared with the double-pass system on TEXTOR, the pulse probing energy was increased by a actor o 2.6 and the total probing energy during each 9 ms pumping pulse was increased by a actor o 4. In this system, the angles o the mirrors must be aligned with a precision o 0 5 rad [2]. Although their achievement is remarkable, it uses an intracavity coniguration in which the mirrors or the multi-pass system enclose a laser cavity and the laser oscillates along the scattering path. Such a scheme may not be useul or devices in which the laser system is independent rom the rest o the optical system. Furthermore, the dependence o the pass and transit numbers on the optical parameters and the theoretical limits o multi-pass optics have not been clariied. In this paper, theoretical aspects o the conocal mirror system are studied. General expressions or the beam path and transit number o a conocal spherical mirror system are obtained or a general coniguration. author s hiratsuka@usion.k.u-tokyo.ac.jp 044- Several schemes are possible or multi-pass optics. i) The irst is coaxial coninement using a ast gate such as a Pockels cell. By switching the Pockels cell, the laser polarization can be changed. Laser light can be conined by combining a Pockels cell and a polarizer. An advantage o this scheme is that the alignment procedure is simple because the optical axis is unique. ii) A system with two parallel mirrors is the simplest. While laser light travels between the two mirrors, the relection point moves slowly along each mirror. Since the beam axis moves, the maximum transit number is determined by the allowable motion o the axis. iii) A conocal spherical mirror system is as simple as a two parallel mirror system, but the beam paths tend to converge to the axis o the mirrors, and a large transit number is assumed to be possible. This paper presents the results o a theoretical analysis o the conocal mirror system, particularly the maximum transit number. The beam path is ormulated analytically. In addition, the eects o positioning and alignment errors are ormulated. A conocal mirror system consists o two concave mirrors with a common ocal point and a common axis Fig. ). Incident light parallel to the mirror axis is relected by the irst mirror at height z > 0), passes through the ocal point, and hits the second mirror at height z 2 < 0). The light is relected by the second mirror, becomes parallel to the mirror axis, and hits the irst mirror at height z 3 < 0). This process is repeated, and the light gradually approaches the mirror axis. How it approaches the axis depends on the ocal lengths o the two mirrors. When parabolic mirrors are used, the light converges to the axis; whereas in the case o spherical mirrors, the light deviates rom the axis ater a certain number o transits because o aberration. In general, the surace accuracy o parabolic mirrors is not as good as that o spherical mirrors. Morec 200 The Japan Society o Plasma Science and Nuclear Fusion Research

2 Plasma and Fusion Research: Letters Volume 5, ) The dierence is Oz 2 ) L. Using the distance between the two hit points and φ 2, the position o the second hit point is ẑ 2 = l r z L r z3 + L φ. 2) r Fig. Schematic drawing o a conocal spherical mirror system. Important variables are deined in the igure. Here, ẑ i z i /r is the normalized i-th hit point. In the same manner, φ 3 = 2θ l φ 2 + r ) z 3 r φ, 3) l l ẑ 3 = l r z + + r ) z 3 + l 2 l 2 r r ) φ. 4) l Fig. 2 Deinitions o the injection angles φ and φ 2 and hit positions z, z 2 and θ r, θ l. over, spherical mirrors with various ocal lengths are readily available. Thereore, in this paper, we study conocal mirror systems consisting o spherical mirrors. In addition to aberration, we need to consider the accuracy o alignment and positioning. Note that a parabolic mirror has a unique optical axis, whereas the optical axis o a spherical mirror is not unique. Thereore, the alignment is much more diicult or parabolic mirrors than that or spherical mirrors. The system is identiied by the ollowing ive variables: injection angle φ ; hit position z on the irst mirror; ocal lengths, 2 2 ); and the distance between the mirrors L. The radii o curvature or the mirrors are r = 2 and l = 2 2 Fig. 2). The angles θ r = arctanz /r) and θ l = arctan z 2 /l) determine the hit points. The incident condition i.e., the initial condition) is deined by the incident angle φ φ and the normalized irst hit point z z /r. In the ollowing, we assume z, because z is smaller than the window radius and r and l are larger than the diameter o the device. In particular, z is several tens o millimeters, whereas r and l are several meters. We derive analytic expressions to determine the path as a unction o φ and z. In the derivation, we keep terms up to Oz 3 ) and neglect the higher-order terms. We assume φ = Oz 3 ). The angle o the relected light φ 2 rom the irst mirror is φ 2 = 2θ r φ 2z + z3 φ. ) 3 Here, we use the approximation θ r z + z 3 /6. Since the mirrors have a inite thickness, the distance between the two hit points diers rom L by about z 2 /r + z2 2 /l)/ Here, we use the approximation θ l z 2 /l z 3 2 /6l3. In these expressions, the terms that include z 3 are minute, because z. Typical z values are on the order o 0 2. These terms relect the thicknesses and aberration eects o the mirrors. When terms including z 3 are neglected, φ 3 becomes larger than φ by a actor r/l >, while z 3 becomes smaller than z by a actor l/r <. Thus, one may expect z > z 3 > z 5 >... Ater several relections between the mirrors, however, the terms including z 3 become signiicant, as shown below. Using Eqs. 3) and 4), φ 3 and z 3 can be derived rom φ and z. Thereore, these equations orm a recursive set. The general ormulae or φ 2n and z 2n are φ 2n = ) n P 2n x)z 3 + x) n φ, 5) ẑ 2n = x) n z + 2 )n P 2n x)z )n x n x n )φ, 6) 4n 7 n 3 P 2n x) = x n+i x 4 n+4i n 3 x 3 n+4i, 7) where x l/r <. During the initial phase, while the transit number is small, ẑ 2n decreases with increasing n. However, Eq. 6) includes x n, so this term becomes signiicant or large n. As a result, ẑ 2n increases with n once n exceeds a certain number. Figure 3 shows an example o z 2n as a unction o n. Here, we assume a coniguration with the ollowing parameters: = r/2 = 762 mm, 2 = l/2 = 508 mm, L = + 2 = 270 mm, z = 35 mm, and φ = 0.32 mrad. These parameters are chosen to duplicate the experiment described below. In Fig. 3, z 2n calculated by Eq. 6) and that calculated by a ray tracing code are plotted. This code tracks a ray with a given starting position and vector. When the ray hits one o the spherical mirrors, it is relected according to the law o relection. The z-coordinates o the relection points are compared with the results o the analytical expression. The dierence between the values obtained by Eq. 6) and the code is as small as 0.3% when 2n = 25. z 2n exhibits a minimum at around 2n = 3.

3 Plasma and Fusion Research: Letters Volume 5, ) Fig. 3 Hit positions or each transit number 2n derived by the analytic ormula +) and a ray tracing code circles). Fig. 4 Photograph o the irst mirror o the conocal mirror system. Hit spots are numbered sequentially rom to. Considering x <, Eq. 6) or large n can be approximated by the leading term ẑ 2n = 2 )n x n z 3 φ), 8) because this term includes x n. Note that the next leading term, which includes x 2 n, becomes important when z 3 φ. Given the size o the irst mirror, we can express the maximum transit number or this conocal mirror system. The radius o the irst mirror should be larger than z. We deine the maximum transit number as the largest 2n satisying ẑ 2n < z. Hereater, the maximum transit number is reerred to as 2N max. Taking the logarithm o Eq. 8), N max is N max = + log2z/ z3 φ ) logx). 9) When z 3 = φ, we cannot use approximation 8), and we should extract the terms that include x 2 n rom Eq. 6). Then, the expression or N max is N max = 2 + log2/z2 ) logx). 0) As we remove the x n term by setting z 3 = φ, we can remove the x 2 n term by adjusting φ. In this way, N max can be ininity. However, the dierence between the optimum φ and φ = z 3 is very small and inite errors in L and φ may reduce the maximum transit number. The results o the detailed analysis on the eects o errors in L and φ are described below. Equations 9) and 0) show that z should be small, r/l = /x should be close to, and φ should be close to z 3 in order to obtain a large N max. Figure 4 shows a photograph o the irst mirror o the conocal mirror system with the same parameters as the example described above. We used a He-Ne laser. Relection spots are visible in the photograph. Ater 2 transits, the beam diameter increases to more than 40 mm and the spot becomes aint; hence, it becomes impossible to identiy urther relections. Although we used two convex lenses to suppress the beam expansion, the spot size increases. We will discuss this phenomenon later. The positions o the relection spots agree with the results o analytic expression 6) within the margin o reading error. In this experiment, we set φ = 0.32 mrad in order to compare the experiment and the analytic ormula. By adjusting the angle, the experimentally obtained maximum transit number was 28, whereas the maximum transit number was 43 at φ = 0 using ormula 9). The dierence is partly due to diiculties in identiying relection spots, as the beam becomes aint ater a ew tens o transits. Another reason or the dierence is the eect o positioning and alignment errors. These eects are described below. In practice, these errors cannot be ignored, and they may signiicantly modiy Eqs. 5) and 6) and N max. Since the eect o angular error is already included in φ, we describe here the eect caused by the dierence ɛ L 2. As in the preceding analysis, we keep the terms o Oz 3 ) and neglect the higher-order terms. We assume φ = Oz 3 ) and q = Oz 2 ), where q ɛ/l is the normalized dierence. Replacing L with L + ɛ, the modiied φ 2n and z 2n are φ 2n = 4q x 2 [ x)n x) n ]z + ) n P 2n x)z 3 + x) n φ, ) ẑ 2n = { x) n + 2xq x 2 [ x)n x) n ]}z + 2 )n P 2n x)z )n x n x n )φ. 2) Equations ) and 2) include new terms proportional to qz, which do not appear in Eqs. 5) and 6). From these ormulae, the modiied maximum transit number 2N max is derived as N max = + log2z/ z3 φ + 4xq z ) x 2. 3) logx)

4 Plasma and Fusion Research: Letters Volume 5, ) We extract the terms including x n as beore. The additional term 4xqz/ x 2 ) appears, which do not appear in Eq. 9). Equation 3) indicates that z and q should be small, r/l = /x should be close to, and φ should be close to z 3 in order to obtain a large N max. In practice, N max is oten limited by positioning and alignment errors, which are expressed by q and a inite dierence in φ rom the optimum value. In a practical system, the beam has a inite size, and the second mirror should be small enough not to cut the incident beam propagating toward the irst mirror. This condition imposes a maximum limit on x < ); thus, we cannot choose x very close to one. Obviously, z 2 xz ) should be larger than the beam radius. Furthermore, the window sizes o the laser injection and exit ports must be larger than z. Thereore, z and x must be chosen to satisy these experimental conditions. L = + 2 )isthe distance between the mirrors, which should be larger than the plasma size, so that the mirrors are located outside the plasma. In addition, in order to distinguish the orwardand backward-scattered signals, L should be large enough to separate these signals in the time domain. Here, we discuss the mechanism o beam expansion, which can be seen in Fig. 4. The eects o aberration, surace roughness, and diraction can cause beam expansion. First, we estimate the eect o aberration by using Eq. 2). By linearizing the inal position z 2n with respect to the deviation in the initial condition δz, δφ, we can estimate the beam expansion. The inal beam diameter D a is D a = 2 ẑ 2n δz + r ẑ 2n z φ δφ. 4) Substituting δz = 2.5 mm, δφ = 0.7 mrad and ɛ = 5 mm, D a becomes 39 mm ater 2 transits. Here, δz is the initial beam radius and δφ is the beam divergence o the incident laser light under the experimental conditions. We used the terms that include x n and x 2 n, because they are dominant in the present coniguration. Second, we estimate the eect o angular scattering o the rays due to the surace roughness o the mirror. I the angular error generated at the 2i )th relection is denoted by θ 2i,the 2n )th normalized position o the beam spot ẑ 2n is n ẑ 2n = ẑ 2n + i=2 θ 2i 2 )n i+ x n i x i n ) ẑ 2n + θ 3 2 )n x 2 n, 5) where ẑ 2n is the ideal position calculated by Eq. 2). θ 3 is the dominant term because x < and θ can be made zero by adjusting the initial angle φ. When the surace roughness i.e., deviation rom a sphere) is λ/4, the corresponding θ is less than 0.0 mrad and the resultant beam diameter is D r < 0.3 mm. Thus, this eect is negligible. Third, we estimate the eect o diraction assuming a Gaussian beam. The ormulae or Gaussian beam imaging Fig. 5 Spot size on the mirrors or each transit number 2n assuming a Gaussian beam. = 762 mm, 2 = 508 mm, ɛ = 0 mm, d = 0, 000 mm, and w = mm. by using an optical component with a ocal length are ) πw 2 2 ) λ d d d 2 = w2 w πw 2 λ = πw 2 λ + d, 6) ). 7) + d 2 Here, d and d 2 are the anterior and posterior ocal lengths, respectively, and w and w 2 are the waist sizes at these ocal points. Since the initial injected beam is nearly collimated, d and w λ. Ater the irst relection, the beam has a waist near the ocal point o the conocal mirror system. Ater the second relection, the beam is nearly collimated. These initial behaviors are similar to those in geometrical optics. However, the Gaussian beam shows dierent behavior ater several transits. Figure 5 shows the calculated spot size on the mirrors. The coniguration parameters are as ollows: = 762 mm, 2 = 508 mm, the initial anterior ocal length is d = 0, 000 mm, w = mm, and ɛ = 0. When we consider a inite error in the distance between the mirrors, ɛ = 30 mm, the spot diameter ater 2 transits is about D d = 34 mm. Thereore, both aberration and diraction can cause beam expansion. We present as an example the conocal spherical mirror system design or multi-pass Thomson scattering in the QUEST spherical tokamak device [3]. The radius o the vacuum vessel is.4 m. However, we choose L = 5500 mm. With this L, orward- and backward-scattered signals are separated by about 20 ns, which is suicient or a ast detection system [4]. To pass a laser beam with a diameter o 8 mm, we choose the ollowing parameters: mirror radii D = 50 mm, D 2 = 42 mm; ocal lengths = 300 mm, 2 = 2400 mm, and z = 2 mm z = ); and φ = 0. In this case, the maximum transit number 2N max = 95 by using Eq. 9). To ensure a transit number o more than 40, the allowable position-

5 Plasma and Fusion Research: Letters Volume 5, ) ing and alignment errors are ɛ < Oz 2 ) l 0.06 mm and φ < rad rom Eq. 3). For large ɛ Oz) l, the approximate expression 3) cannot be applied. Using Eq. 2), the allowable positioning error is ɛ < 7.4mmin contrast, using Eq. 3), the allowable positioning error is ɛ < 8.4 mm). When we take parameters d = 0, 000 mm and w 0 = 5 mm, the beam diameter expands to D > 40 mm owing to diraction. Thereore, the maximum transit number or practical conditions will be about 37. In summary, the easibility o multi-pass laser optics consisting o two conocal spherical mirrors has been studied. We obtained analytic expressions or the path and the maximum transit number, φ 2n, z 2n, and 2N max. The eects o aberration, positioning and alignment errors, and beam expansion are considered. An example o a conocal spherical mirror system design was described. In the example, the maximum transit number can be more than 37, considering typical positioning and alignment errors and beam expansion. This study was supported by Japan Society or the Promotion o Science JSPS) Grant-in-Aid or Scientiic Research No and National Institute or Fusion Science NIFS) Collaborative Research Program No. NIFS09KUTR039. [] A. Ejiri et al., Nucl. Fusion 49, ). [2] M. Yu. Kantor et al., Plasma Phys. Control. Fusion 5, ). [3] K. Hanada et al., IEEJ Trans. FM 29, ). [4] A. Ejiri et al., to be published in Plasma Fusion Res. Special Issue ITC-9)