A low-noise resonator fiber-optic gyroscope, comprising: at least one laser configured to output a reference optical beam used to form at least a first optical beam and a second optical beam; a first-optical-beam frequency controller to modulate the first optical beam at a first-modulation frequency f m to form a first-frequency-modulated optical beam; a second-optical-beam frequency controller to modulate the second optical beam at a second-modulation frequency f n to form a second-frequency-modulated optical beam, wherein at least one of the first optical beam and the second optical beam is frequency-shifted from the reference optical beam; a fiber resonator having a fiber resonator coil with a clockwise-input end and a counter-clockwise-input end, the counter-clockwise-input end configured to input the first-frequency-modulated optical beam and the clockwise-input end configured to input the second-frequency-modulated optical beam; a first-frequency demodulator to demodulate, at a first-demodulation frequency f m, an optical beam output from the clockwise-input end of the fiber resonator; and a second-frequency demodulator to demodulate, at a second-demodulation frequency f n, an optical beam output from the counter-clockwise-input end of the fiber resonator, wherein the first-demodulation frequency equals the first-modulation frequency and the second-demodulation frequency equals the second-modulation frequency, characterised in that the first-modulation frequency and the second-modulation frequency are adjusted with respect to each other so that, on the optical beams output from the fiber resonator: sideband noise spectrum centered on the first-modulation frequency is offset in frequency from the second-modulation frequency, and sideband noise spectrum centered on the second-modulation frequency is offset from the first-modulation frequency. at least one laser configured to output a reference optical beam used to form at least a first optical beam and a second optical beam; a first-optical-beam frequency controller to modulate the first optical beam at a first-modulation frequency f m to form a first-frequency-modulated optical beam; a second-optical-beam frequency controller to modulate the second optical beam at a second-modulation frequency f n to form a second-frequency-modulated optical beam, wherein at least one of the first optical beam and the second optical beam is frequency-shifted from the reference optical beam; a fiber resonator having a fiber resonator coil with a clockwise-input end and a counter-clockwise-input end, the counter-clockwise-input end configured to input the first-frequency-modulated optical beam and the clockwise-input end configured to input the second-frequency-modulated optical beam; a first-frequency demodulator to demodulate, at a first-demodulation frequency f m, an optical beam output from the clockwise-input end of the fiber resonator; and a second-frequency demodulator to demodulate, at a second-demodulation frequency f n, an optical beam output from the counter-clockwise-input end of the fiber resonator, wherein the first-demodulation frequency equals the first-modulation frequency and the second-demodulation frequency equals the second-modulation frequency, characterised in that the first-modulation frequency and the second-modulation frequency are adjusted with respect to each other so that, on the optical beams output from the fiber resonator: sideband noise spectrum centered on the first-modulation frequency is offset in frequency from the second-modulation frequency, and sideband noise spectrum centered on the second-modulation frequency is offset from the first-modulation frequency.