Introduction

NUCLEAR magnetic resonance （NMR） allows the observation of specific quantum mechanical magnetic properties of the atomic nucleus. Many scientific techniques exploit NMR phenomena to study molecular biology, biochemistry, crystals, and non-crystalline materials through nuclear magnetic resonance spectroscopy^[1,2,3,4]. However, low sensitivity and spectral resolution have been a roadblock limiting its applications. Dynamic nuclear polarization （DNP） is an NMR technique used to enhance the sensitivity through irradiation of the electron spins with electromagnetic waves in the neighborhood of their Larmor frequency^[5]. To obtain higher resolution spectra, modern NMR spectroscopy has pushed to much higher magnetic fields. The extension of DNP NMR to high magnetic fields depends on the development of THz sources of radiation producing few tens of watts of output power at the relevant frequencies.

Gyrotrons are the most powerful sources of radiation in the range of millimeter, sub-millimeter and terahertz waves with relatively high power levels, high stability of the output parameters, long lifetimes, and efficient scaling to higher frequency^[6,7,8,9]. Solid-state devices in THz range suffer from scalability and efficiency issues that lead to limited output powers. Classical microwave tubes, e.g., traveling-wave tube （TWT） and the Extended Interaction Klystron （EIK）, can produce high power （hundreds of watts） electromagnetic radiation up to 100 GHz^[10]. However, these slow-wave devicesrequire physical

structures in the interaction region that is much smaller than the wavelength of operation, which produces difficulties with thermal damage and manufacturing of the interaction structure when the operation frequency is extended into THz range. Gyrotrons are the only demonstrated, highly stable devices capable of producing adequate power with an adequate lifetime in the frequency of interest for DNP NMR^[11,12].

At University of Fukui in Japan, a THz gyrotron named Gyrotron FU CW VI with a 15 Tesla superconducting magnet has been developed recently. It operates at TE06 mode with a continuous frequency tuning bandwidth of approximately 1.5 GHz at 395.2 GHz^[13]. This gyrotron is designed for a 600 MHz DNP-NMR spectrometer at Institute for Protein Research, Osaka University. Massachusetts Institute of Technology （MIT） has succeeded in the operation of the gyrotron at 263 GHz by using a 9.7 T magnet and operating at TE03 mode for MAS NMR experiments at ~100 K^[14]. A continuous-wave （CW） tunable second-harmonic 460 GHz gyrotron was developed at MIT for 700 MHz NMR experiments^[15], which operates at TE11,2 mode with a smooth frequency tuning range of 1 GHz.

For a DNP-NMR spectrometer, the power generated by the THz gyrotron should be transmitted to the sample with low loss, However, the THz gyrotrons mentioned above operate at high-order modes, which cannot be transmitted to the sample directly with low loss. A conventional transmission line for the DNP-NMR spectroscopy is shown in Fig.1. The transmission line mainly includes: a quasi-optical mode converter converting an operating gyrotron mode into a fundamental Gaussian beam^[16,17], a large-diameter corrugated waveguide used to transmit the Gaussian beam with low loss, and the mirrors which are utilized to focus the beam and match the beam from a large-diameter corrugated waveguide into a small-diameter corrugated waveguide used in the DNP-NMR probe^[12,18]. In order to enhance the Gaussian beam quality radiated at the sample in a DNP-NMR spectrometer, an improved transmission line for transmitting the radiation from the THz gyrotron to the NMR probe has been proposed. In the transmission line, an angle-adjustable phase-correcting mirror is added to improve the Gaussian beam quality and adjust the direction of output beam.

Fig.1 The layout of the transmission-line for DNP-NMR spectroscopy

图1 DNP-NMR谱仪传输线的布置

This paper is organized as follows. Section 1 presents the design of the improved transmission line. The numerical calculation results and discussions are presented in Section 2. Section 3 is the summary and conclusions.

1 Design of the transmission line

The structure of the improved transmission line for a 263 GHz gyrotron in the 400 MHz DNP-NMR spectrometer is shown in Fig. 2（a）. The system consists of two metallic corrugated waveguides, an angle-adjustable phase-correcting mirror and a parabolic mirror. A Gaussian beam from the Quasi-optical mode converter, as the input of the transmission line, is radiated into a 22 mm diameter metallic corrugated waveguide in which the depth, the width and the period of the corrugation are 0.285 mm, 0.253 mm and 0.38 mm, respectively.

Fig.2 （a） Sketch of the improved transmission and mirror system, and （b） Shape of the phase-correcting mirror

图2 （a）改进的透射镜和反射镜系统的结构，（b）相位校正镜的形状

The beam from the 22 mm diameter corrugated waveguide is focused with the parabolic mirror, and the phase-correcting mirror is used to correct the amplitude and phase distributions of the outgoing wave beam to improve the Gaussian beam content of the beam at the sample. The phase-correcting mirror is shown in Fig. 2（b）. These mirrors also match the beam size from the 22 mm corrugated waveguide into the 8mm corrugated waveguide used in the NMR probe.

The vector diffraction theory of electromagnetic waves is utilized to analysis this process ^[19]. The Stratton-Chu formula presented as follows can be used to calculate the vector diffraction fields E′ and H′ at the observing point P （x′, y′, z′） when the source fields E and H are known.

where G is the Green function for a point source in the free space.

where R is the distance between the source point Q （x, y, z） and the observing point P （x′, y′, z′）, k ₀ is the wave number in free space.

The principle of the phase-correcting mirror is shown in Fig. 3. To obtain the phase of the corrected and adjusted field um(r⃑) , the original incident beam field u(r⃑) at the mirror is multiplied by a phase term

Fig.3 Principle of the phase-correcting method

图3 相位矫正原理

here, r⃑ is the position vector of the mirror surface, Δz(r⃑) is the surface deformation of the mirror, and α is the angle of the incidence or the reflection wave. The phase-correcting mirror is optimized through the iteration formula as follows ^[20].

where, G is the Green function, the superscript "*" represents the complex conjugate, ur(r⃑0) is the field distribution of the wave beam at r⃑0 , and ug(r⃑0) is the desired field distribution.

The scalar Gaussian mode content correlation coefficient ηs and the vector Gaussian mode content correlation coefficient ηυ are usually used to describe the mode purity of the output beam, which are defined as the amplitude and phase correlation coefficient of fields to an ideal fundamental Gaussian distribution ^[21,22].

2 Numerical calculation results

Based on the design and analytic method of the transmission line in Section 1, the beam field distribution at the output window is calculated with Matlab. The normalized power distribution at the output window with the angular adjustment of the phase-correcting mirror is shown in Fig. 4, and the result calculated by FEKO is in Fig. 5. FEKO is a 3D full wave electromagnetic simulation software. Since the MoM is used, FEKO does not need to set boundary conditions during simulation. It is found that the beam can keep high fundamental Gaussian mode content when the direction of the output wave-beam changes within ± 20°. When the angle of the phase-correcting mirror is 0°, the scalar Gaussian mode content is 99.90%, the vector Gaussian mode content is 99.55%, the efficiency is up to 96.20%, and the beam waist is 3.01 mm at the window plane, as shown in Table 1. The results calculated by FEKO at the angle of 0° are listed in Table 2.

Fig.4 Normalized power distribution of the output field versus the angle of the phase-corrected mirror（Plotted by Matlab）

图4 输出场与相位校正镜角度的归一化功率分布（Matlab计算的结果）

Fig.5 Normalized power distribution of the output field versus the angle of the phase-corrected mirror（Plotted byFEKO）

图5 输出场的归一化功率分布与相位校正镜的角度（FEKO计算结果）

The direction of the output beam can be changed by only adjusting the angle of the phase-correcting mirror to match the DNP-NMR sample. As shown in Fig. 6, when the direction of the output beam change from -15° to 15°, the scalar and the vector Gaussian mode content at the output window always keep 99.70% ± 0.2% and 99.3% ± 0.2%, respectively. The electric field distribution at the cross section of the improved transmission line from a 3D simulation software FEKO is presented in Fig.7.

Fig.6 （a） The scalar Gaussian mode content versus. angle of the phase-corrected mirror, and （b） the vector Gaussian mode content vs. angle of the phase-corrected mirror

图6 （a）标量高斯成分与相位校正镜角度关系，（b）矢量高斯成分与相位校正镜角度关系

Fig.7 The electric field distribution of the cross section of the improved transmission line from 3-D simulation software FEKO with: （a） 20°, （b） 0°, and （c）-20°

图7 用FEKO三维模拟软件计算改进后的输电线路截面电场分布（a）20°，（b）0°，（c）-20°

3 Conclusion

The improved transmission and mirror system for a 263 GHz DNP-NMR spectrometer have been designed and numerically simulated in this paper based on the geometric optics theory and the vector diffraction theory. Numerical calculations and the results of FEKO show the phase-correcting mirror added in the improved transmission line can improve the scalar and vector Gaussian mode contents of the output beam. When the angle of the phase-correcting mirror is 0°, the scalar and vector Gaussian mode content are 99.90% and 99.55%, respectively. Meanwhile, the direction of the output beam can be changed by only adjusting the angle of the angle-adjustable phase-correcting mirror. The scalar Gaussian mode content is about 99.57% and the vector Gaussian mode content is about 98.97% when the direction of the output beam changes within ± 15°.This improved transmission and mirror system give an efficient solution to the problem of transmission for a 400 MHz DNP-NMR spectrometer.

References