^{1,a)}and J. D. D. Martin

^{1}

### Abstract

We have developed a senior undergraduate experiment that illustrates frequency stabilization techniques using radio-frequency electronics. The primary objective is to frequency stabilize a voltage controlled oscillator to a cavity resonance at 800 MHz using the Pound-Drever-Hall method. This technique is commonly applied to stabilize lasers at optical frequencies. By using only radio-frequency equipment, it is possible to systematically study aspects of the technique more thoroughly, inexpensively, and free from eye hazards. Students also learn about modular radio-frequency electronics and basic feedback control loops. By varying the temperature of the resonator, students can determine the thermal expansion coefficients of copper, aluminum, and super invar.

We gratefully acknowledge the assistance of Zhenwen Wang, J. Szubra, and H. Haile of the University of Waterloo Science Technical Services. We thank C. Bennett, J. Carter, S. De Young, and A. Lupascu for comments on the manuscript. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

I. INTRODUCTION

II. RESONATING CAVITY

III. RESONANCE

IV. REFLECTION COEFFICIENT

V. MODULATION OF THE VOLTAGE CONTROLLED OSCILLATOR

VI. POUND-DREVER-HALL LOCKING

VII. MEASUREMENT OF LINEAR THERMAL EXPANSION COEFFICIENTS

VIII. IMPLEMENTATION

IX. CONCLUDING REMARKS

### Key Topics

- Oscillators
- 44.0
- Reflection coefficient
- 16.0
- Thermal expansion
- 12.0
- Oscilloscopes
- 8.0
- Undergraduates
- 8.0

##### G01B

##### G01K

##### H01S3/00

##### H01S3/098

##### H01S3/10

##### H01S3/14

##### H03B

## Figures

*λ*/4 coaxial transmission line resonator. (a) Relevant dimensions. (b) The coupling loops inserted in the cartridges. (c) View from the open end (current node). (d) View of top of cavity (voltage node). The tops of the cartridges are seen here.

*λ*/4 coaxial transmission line resonator. (a) Relevant dimensions. (b) The coupling loops inserted in the cartridges. (c) View from the open end (current node). (d) View of top of cavity (voltage node). The tops of the cartridges are seen here.

Cavity resonance under critical coupling conditions. (a) Transmitted signal. (b) Reflected signal. The loaded quality factor *Q* _{ L } is determined by fitting a Lorentzian to the reflected power [see Eq. (9)]. A linearly varying incident power has been included in the fit to accommodate for the frequency dependent losses of components other than the cavity.

Cavity resonance under critical coupling conditions. (a) Transmitted signal. (b) Reflected signal. The loaded quality factor *Q* _{ L } is determined by fitting a Lorentzian to the reflected power [see Eq. (9)]. A linearly varying incident power has been included in the fit to accommodate for the frequency dependent losses of components other than the cavity.

A lumped element circuit model of the input transmission line, coupling, and the resonating cavity.

A lumped element circuit model of the input transmission line, coupling, and the resonating cavity.

Experimental setup to observe the real and imaginary parts of the cavity reflection coefficient. Depending on the bandwidth of the oscilloscope, it may be necessary to insert a low-pass filter after the mixer output. Key: ISO (isolator), SP (splitter), ADL (adjustable delay line), CR (circulator), MX (mixer), AMP (amplifier), DD (zero-bias Schottky diode), M-M (male-to-male connector), FG (function generator).

Experimental setup to observe the real and imaginary parts of the cavity reflection coefficient. Depending on the bandwidth of the oscilloscope, it may be necessary to insert a low-pass filter after the mixer output. Key: ISO (isolator), SP (splitter), ADL (adjustable delay line), CR (circulator), MX (mixer), AMP (amplifier), DD (zero-bias Schottky diode), M-M (male-to-male connector), FG (function generator).

Observation of cavity transmission and reflection using the setup of Fig. 4. (a) Transmission through the cavity. (b) Observation of the real part of the cavity reflection coefficient (to within a positive scale factor). (c) Observation of the imaginary part of the reflection coefficient (to within a positive scale factor). The curve labeled “delay line effect” is a calculation accounting for the variation in phase shift of the delay line with frequency. The calculations are vertically scaled for the best least squares fits.

Observation of cavity transmission and reflection using the setup of Fig. 4. (a) Transmission through the cavity. (b) Observation of the real part of the cavity reflection coefficient (to within a positive scale factor). (c) Observation of the imaginary part of the reflection coefficient (to within a positive scale factor). The curve labeled “delay line effect” is a calculation accounting for the variation in phase shift of the delay line with frequency. The calculations are vertically scaled for the best least squares fits.

Experimental setup for measuring Fourier components of the modulated voltage controlled oscillator. The relation between *β* and the power in each Fourier component can be measured using this setup. The computer triggers the function generator to begin scanning and the oscilloscope to begin measuring data. Key: ISO: isolator, SP: splitter, CR: circulator, MX: mixer, AMP: amplifier, DD: zero-bias Schottky diode, M-M: male-to-male connector, BP: band pass filter for 10 MHz.

Experimental setup for measuring Fourier components of the modulated voltage controlled oscillator. The relation between *β* and the power in each Fourier component can be measured using this setup. The computer triggers the function generator to begin scanning and the oscilloscope to begin measuring data. Key: ISO: isolator, SP: splitter, CR: circulator, MX: mixer, AMP: amplifier, DD: zero-bias Schottky diode, M-M: male-to-male connector, BP: band pass filter for 10 MHz.

Carrier and sideband powers as a function of the frequency modulation index *β* observed using the cavity filter method, with Ω/(2*π*) = 6 MHz, and an RF spectrum analyzer with Ω/(2*π*) = 10 MHz. The values of *β* are determined from the measured voltage controlled oscillator dc tuning curve. Also shown are the theoretically expected relations *P* _{ c } = [*J* _{0}(*β*)]^{2} *P* _{0} and *P* _{ s } = [*J* _{1}(*β*)]^{2} *P* _{0} (see text).

Carrier and sideband powers as a function of the frequency modulation index *β* observed using the cavity filter method, with Ω/(2*π*) = 6 MHz, and an RF spectrum analyzer with Ω/(2*π*) = 10 MHz. The values of *β* are determined from the measured voltage controlled oscillator dc tuning curve. Also shown are the theoretically expected relations *P* _{ c } = [*J* _{0}(*β*)]^{2} *P* _{0} and *P* _{ s } = [*J* _{1}(*β*)]^{2} *P* _{0} (see text).

The RF equivalent Pound-Drever-Hall locking method. The configuration shown is for locking the voltage controlled oscillator. The relative phases of the 10MHz modulating and demodulating signals are set using coaxial cable lengths, which depend on the phase shifts of various components. To examine the Pound-Drever-Hall error signal as the voltage controlled oscillator frequency is tuned, as in Fig. 9, the feedback control circuit is omitted. The voltage controlled oscillator offset is scanned by a function generator, and modulation applied through a bias T. The output of the mixer (I) is low-pass filtered and displayed on an oscilloscope. Key: SP: splitter, CR: circulator, MX:- mixer, AMP: amplifier, BP: band pass filter for 10 MHz.

The RF equivalent Pound-Drever-Hall locking method. The configuration shown is for locking the voltage controlled oscillator. The relative phases of the 10MHz modulating and demodulating signals are set using coaxial cable lengths, which depend on the phase shifts of various components. To examine the Pound-Drever-Hall error signal as the voltage controlled oscillator frequency is tuned, as in Fig. 9, the feedback control circuit is omitted. The voltage controlled oscillator offset is scanned by a function generator, and modulation applied through a bias T. The output of the mixer (I) is low-pass filtered and displayed on an oscilloscope. Key: SP: splitter, CR: circulator, MX:- mixer, AMP: amplifier, BP: band pass filter for 10 MHz.

Pound-Drever-Hall error signal as the voltage controlled oscillator frequency is tuned. (a) Transmission through the cavity. (b) Pound-Drever-Hall error signal for a voltage controlled oscillator modulation frequency of Ω/(2*π*) = 10 MHz (see text for details).

Pound-Drever-Hall error signal as the voltage controlled oscillator frequency is tuned. (a) Transmission through the cavity. (b) Pound-Drever-Hall error signal for a voltage controlled oscillator modulation frequency of Ω/(2*π*) = 10 MHz (see text for details).

Experimental data for determining the expansion coefficient of copper, aluminum, and super invar. By measuring the frequency change as the temperature of the cavity changes, we can determine the expansion of the metal which composes the inner cylinder. Also shown are the linear fits used for the determination of the linear coefficients of thermal expansion.

Experimental data for determining the expansion coefficient of copper, aluminum, and super invar. By measuring the frequency change as the temperature of the cavity changes, we can determine the expansion of the metal which composes the inner cylinder. Also shown are the linear fits used for the determination of the linear coefficients of thermal expansion.

## Tables

Values of linear thermal expansion coefficients.

Values of linear thermal expansion coefficients.

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