Broadband W-band Rapid Frequency Sweep Considerations for Fourier Transform EPR

Abstract

A multi-arm W-band (94 GHz) electron paramagnetic resonance spectrometer that incorporates a loop-gap resonator with high bandwidth is described. A goal of the instrumental development is detection of free induction decay following rapid sweep of the microwave frequency across the spectrum of a nitroxide radical at physiological temperature, which is expected to lead to a capability for Fourier transform electron paramagnetic resonance. Progress toward this goal is a theme of the paper. Because of the low Q-value of the loop-gap resonator, it was found necessary to develop a new type of automatic frequency control, which is described in an appendix. Path-length equalization, which is accomplished at the intermediate frequency of 59 GHz, is analyzed. A directional coupler is favored for separation of incident and reflected power between the bridge and the loop-gap resonator. Microwave leakage of this coupler is analyzed. An oversize waveguide with hyperbolic-cosine tapers couples the bridge to the loop-gap resonator, which results in reduced microwave power and signal loss. Benchmark sensitivity data are provided. The most extensive application of the instrument to date has been the measurement of T₁ values using pulse saturation recovery. An overview of that work is provided.

Appendices

Appendix 1

The AFC for Broadband EPR

The block diagram of the AFC system shown in Fig. 8, and the accompanying basic timing diagram in Fig. 9, illustrate the operation of the AFC circuit. They are referred to in the following description. The time intervals on the timing diagram are not shown to scale for ease of display. The 1 GHz IF signal from the Q-band multi-arm bridge shown in Fig. 1 feeds the AFC system. The Q-channel of the 1 GHz I/Q signal mixer functions as a phase discriminator in the AFC system when the 1 GHz LO phase is adjusted to give a 90 degree phase difference between the RF and LO inputs of the mixer and the incident power is at the resonant frequency of the sample resonator. The complex nature of the sample resonator reflection coefficient results in a change in the phase of the reflected signal incident on the input of the I/Q signal mixer as the source frequency and resonator frequency drift apart.

Fig. 8

AFC system for broadband EPR. The S/H amplifier is within the Baseband I/Q Signal Preamp

Fig. 9

AFC system timing diagram

The down-converted reflected signal from the resonator at the 1 GHz IF is first gated by the receiver disable PIN switch. A low off-time duty cycle ratio (250 ns off) transistor-transistor logic pulse from the Stanford Research DG645 Pulse Delay Generator controls the PIN switch. The DG645 is triggered by the Stanford Research DS345, the AFC clock source. These components are shown in Fig. 8. The switch control waveform is shown by trace a in Fig. 9. In the SR mode of operation, this gating pulse is synchronized and timed with the pump pulse to protect the LNA and successive stages in the receiver chain from the higher power pump pulse by preventing pump transmission into the receiver. Signal gating is at a frequency up to 100 kHz settable by the user, and should be at a frequency far removed from the field modulation frequency in the field modulation mode of bridge operation so as to avoid interference. In the SR mode of operation, this frequency is the repetition frequency of the pulse experiment. The gated signal is down-converted to baseband in the 1 GHz I/Q signal mixer. The post mixer preamplifier circuit interfaces the I/Q signal mixer with other signal conditioning circuitry both for the EPR signal and for the AFC signal. Both of these components are also shown in Fig. 8.

The signal output of the Q channel of the preamplifier is shown by trace b in Fig. 9. The output waveform of the preamp shows the effect of the signal gating. The zero amplitude portion of the waveform is the time interval when the receiver disable switch is in the isolation state. The portion of the mixer output sensitive to the frequency/phase of the reflected signal from the sample resonator is the interval when the receiver disable switch is transmitting. The polarity and amplitude of this portion of the signal determine the frequency of the microwave source relative to the resonator when the phase of the LO of the I/Q signal mixer is properly adjusted.

Logic in the preamplifier produces the control waveform shown by trace c in Fig. 9 from the input AFC sample clock supplied by the DS345. The pulse width of the sample portion of the sample/hold (S/H) control waveform is 100 ns, the acquisition time for the S/H amplifier, also located in the preamplifier. The hold time of the S/H amplifier is the logic zero portion of the S/H control waveform. The logic waveform in trace c shows that sampling occurs in both the on and off portions of the gated signal of trace b, converting the shorter (250 ns) switch-off portion of the signal waveform and the longer remaining portion into a square-wave, shown in trace d. This is the AFC signal from the preamplifier that feeds the AFC lock-in amplifier (Stanford Research, SR510). The signal from the S/H amplifier is then synchronously detected by the AFC lock-in amplifier at the same gating frequency as the PIN switch. The frequency range of the lock-in amplifier determines the maximum clock set frequency of the AFC system (100 kHz). After synchronous detection, the signal is integrated, low-pass filtered, and interfaced to the YTO FM coil.

This AFC system provides a very stable and dependable lock, even at low (<−30 dBm) microwave power levels with the low-Q W-band LGR (Sample loaded Q < 100). Because of its robust performance, this AFC is also used in continuous wave EPR experiments. The chopping frequency is adjusted so as not to interfere with the EPR field modulation signal. For most experiments, the AFC clock is 40 kHz, and due to the low duty cycle of the off-time sampling, has minimal effect on the S/N of the EPR signal.

Appendix 2

Path Equalization Analysis

An analysis of the phase shift at the signal mixer output due to LO frequency changes follows. The analysis incorporates the dispersive nature of waveguide transmission media. Consider the general two-path circuit shown in Fig. 4a. The small “loops” in the connecting lines between components represent delays due to transmission line lengths. Component time delays beyond the path length time delay are not considered in this first-order analysis. The microwave source output at frequency f ₁ is divided, and the upper (reference) arm is fed directly to the signal mixer. The frequency in the signal arm (with the UC and DC mixers), however, is translated to frequency f ₂, routed to and from the sample resonator, and mixed back to f ₁. The details of the signal arm are shown in Fig. 4b.

The path ${\ell _2} + 2{\ell _{{\rm{pr}}}}$ consists of bridge circuit waveguide plus twice the length of the probe, the latter being due to the round-trip path. The waveguide probe partially consists of the same waveguide size as the W-band bridge circuit, WR-10, that is, non-oversize waveguide (non-OS WG), over the ${\ell _{{\rm{N}}a}}$ and ${\ell _{{\rm{N}}b}}$ lengths, where ${\ell _{\rm{N}}} = {\ell _{{\rm{N}}a}} + {\ell _{{\rm{N}}b}}$, and partially of oversize waveguide (OS WG) [14]. If f ₁ is constant, and if the phase change as a function of the LO frequency f _LO is zero at the output of the DC mixer, then the phase at the signal mixer θ will be constant. The bridge arms are equalized. This condition is theoretically possible over a frequency range for transverse electromagnetic wave transmission lines, but dispersion in waveguide will perturb the arm balance to some extent.

The phase shift of the transmission line from the LO source to the DC mixer is $- {\beta _4}{\ell _4}$, and the phase shift of the transmission line from the LO source to the UC mixer is $- {\beta _3}{\ell _3}$, where β = 2π/λ = 2πf/v is the phase constant (wave number) in rad/m, v is the phase velocity, and λ is the guide wavelength in the waveguide transmission medium. Assume the delay line is length ${\ell _{DL}}$, where ${\ell _4} = {\ell _3} + {\ell _{{\rm{DL}}}}$. The phase shift in path ${\ell _3}$ from the LO to the UC mixer adds to whatever occurs over the path ${\ell _2} + 2{\ell _{{\rm{pr}}}}$. The phase shift in path ${\ell _4}$ from the LO to the DC mixer is subtracted. Hence, the phase at the signal mixer output is the difference of the phases of the two input signals to the DC mixer (assuming f ₁ is constant),

$$\begin{array}{ccccc}\\ {\it{\theta }} = & - {\beta _3}{\ell _3} - {\beta _2}{\ell _2} - 2{\beta _{\rm{2}}}{\ell _{\rm{N}}} - 2{\beta _{{\rm{OS}}}}{\ell _{{\rm{OS}}}} - \left( { - {\beta _4}{\ell _4}} \right)\\ \\ = & - {\beta _3}{\ell _3} - {\beta _2}{\ell _2} - 2{\beta _{\rm{2}}}{\ell _{\rm{N}}} - 2{\beta _{{\rm{OS}}}}{\ell _{{\rm{OS}}}} + {\beta _4}\left( {{\ell _3} + {\ell _{{\rm{DL}}}}} \right).\\ \end{array}$$

(12)

The waveguide in the LO circuit is the same in both the UC and DC paths. The phase constants are identical, β ₃ = β ₄ = β _LO, leaving

$${\it{\theta }} = - {\beta _2}{\ell _2} - 2{\beta _{\rm{2}}}{\ell _{\rm{N}}} - 2{\beta _{{\rm{OS}}}}{\ell _{{\rm{OS}}}} + {\beta _{{\rm{LO}}}}{\ell _{{\rm{DL}}}}.$$

(13)

Waveguide is dispersive, so when the operating frequencies relative to cutoff in two different waveguide sizes are unequal, the velocities are different. The criterion for minimal phase variation as a function of frequency, dθ/df = 0, requires differentiation of $\beta \ell$ for waveguide with respect to frequency. The cutoff wavelength λ _c in TE₁₀ rectangular waveguide is λ _c  = 2a = c/f _c , where a is the internal waveguide width, c is the speed of light, and f _c is the cutoff frequency. We start with

$$v = {\lambda _g}f = \frac{{{\lambda _o}}}{{\sqrt {1 - {{\left( {\frac{{{\lambda _o}}}{{2a}}} \right)}^2}} }}f = \frac{c}{{\sqrt {1 - {{\left( {\frac{{{f_c}}}{{f}}} \right)}^2}} }},$$

(14)

where the waveguide wavelength λ _g is designated specifically to distinguish it from the free-space wavelength λ _o . The phase constant is

$$\beta = \frac{{2\pi f}}{v} = \frac{{2\pi }}{c}\sqrt {{f^2} - {f_c}^2} .$$

(15)

Then

$$\frac{{{\rm d}\left( {\beta \ell } \right)}}{{{\rm d}f}} = \frac{{2\pi \,f\ell }}{{c\sqrt {{f^2} - {f_c}^2} }}.$$

(16)

Taking the derivative of Eq. (13) and substituting Eq. (16) results in

$$\begin{array}{ccccc}\displaystyle \frac{{{\rm d}\theta }}{{{\rm d}f}} = & \displaystyle \frac{{ - 2\pi \,{f_2}{\ell _2}}}{{c\sqrt {{f_2}^2 - {f_{c2}}^2} }} - 2\frac{{2\pi \,{f_2}2{\ell _{\rm{N}}}}}{{c\sqrt {{f_2}^2 - {f_{c2}}^2} }} - 2\frac{{2\pi {f_2}2{\ell _{{\rm{OS}}}}}}{{c\sqrt {{f_2}^2 - {f_{c{\rm{OS}}}}^2} }}\hfill \\ & + \displaystyle \frac{{2\pi {f_{{\rm{LO}}}}{\ell _{{\rm{DL}}}}}}{{c\sqrt {{f_{{\rm{LO}}}}^2 - {f_{c{\rm{LO}}}}^2} }}.\hfill \end{array}$$

(17)

In the arm-balanced condition with the delay line, dθ/df = 0. In the arm-unbalanced condition without the delay line $\left( {{\ell _{{\rm{DL}}}} = 0} \right)$, dθ/df  ≠ 0 and can be measured on a vector network analyzer as a function of changing the LO frequency. Subtracting equations that correspond to these two conditions, the delay line length can be determined from the measured dθ/df,

$${\ell _{{\rm{DL}}}} = \frac{{ - c\sqrt {{f_{{\rm{LO}}}}^2 - {f_{c{\rm{LO}}}}^2} }}{{2\pi {f_{{\rm{LO}}}}}}\frac{{{\rm d}\theta }}{{{\rm d}f}},$$

(18)

where the negative sign is present because dθ/df is negative without the delay line per Eq. (17).

Appendix 3

Physical Layout of the Spectrometer

The overall assembly of the microwave components and magnet of the W-band spectrometer is shown in Fig. 10. The mechanical support system for moving the microwave bridge, along with the waveguide probe, resonator, and sample, into the side port of a Magnex/Bruker superconducting magnet consists of a granite slab 250 cm long by 50 cm wide by 20 cm thick, with an air-bearing-supported slide floating on a cushion of dry nitrogen. The system allows vibration-free insertion of the probe without disrupting the sample or resonator, which is adjusted for microwave coupling outside the magnet. An early description of the system developed here is provided in Ref. [27]. Use of a granite slab for achieving stability of a high-frequency microwave bridge was also reported in Ref. [28].

Fig. 10

Overall assembly of the W-band spectrometer

The granite slab was ground flat along the top and two sides to within 250 microns. A box-like aluminum-frame structure was fabricated using 2 by 4 cm aluminum rails; 2 cm linen phenolic plates support the rails and wrap around the sides of the granite, providing directional stability while sliding the bridge assembly along the granite slab.

The granite slab is supported at the three points that were used during grinding. Support at these three points ensures that the slab remains flat. Each support consists of a 3.75 cm diameter brass-threaded sleeve that makes contact with the slab. Stainless steel threaded inserts are attached to vibration-damping swivel leveling mounts. Each mount provides isolation from floor vibrations transmitted to the large mass of the granite slab (about 2 metric tons). The supports are designed for this weight and are stabilized by a 10 cm by 10 cm fir wood subframe on large castors, allowing movement with respect to the magnet. The slab is laser-aligned to the magnet bore. It is raised to the proper level at the three support points, and the wheels are raised. The vibration damping mounts rest on double brass plates that slide sideways by an adjustable screw, allowing precise alignment while raised.

The in-house designed gas-bearing system consists of four 5 × 23 × 30 cm granite plates, one at each corner of the sliding table. A bolt at the center of each plate allows the height to be adjusted. Each granite plate was ground flat, and a pattern of air distribution grooves was machined into this surface. Dry nitrogen gas is fed into each pattern of grooves, which forms a cushion above the smooth granite slab.

The sliding table that rests on the gas bearings is 2 cm-thick plastic-coated fiberboard. The sides of the granite slab align the structure parallel to the magnet bore. Nylon-tipped adjustable screws provide a low-friction alignment guide, since little force is involved. With about 150 kg weight of the microwave assembly and table, vibration-free movement is obtained at a pressure of 15 psi. Tests show an estimated 450 kg of lift at 15 psi with approximate lift height of 0.1 mm.

Special attention has been paid to support of the waveguide probe that slides into the center of the magnet. The sliding table is precision drilled, which provides a mounting surface to which the probe is bolted. The waveguide passes through a fiberglass tube filled with expanded urethane foam to dampen vibrations and provide thermal isolation from the environment. A silicone rubber disc at the end of the probe makes a resilient contact with a smooth fiberglass tube inserted in the magnet bore. A water-cooled modulation coil system is installed between the tubes of a double concentric fiberglass-tube assembly in the magnet bore. The entire coil assembly is encapsulated in epoxy and machined to size. There is a slight press fit to the outer tube that stiffens the assembly. The outer tube is supported by round plates that compress an O-ring seal to give additional acoustic dampening to the tube support.