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Comb self-referencing compares two parts of one optical frequency comb. Coherent broadening supplies an octave, a factor of two in frequency. Second-harmonic generation doubles a red tooth; a photodiode detects its beat with an original blue tooth.^[ref] Applying the tooth formula givesThe doubled branch carries $2f_0$, while the original high-frequency branch carries $f_0$: their difference retains exactly one offset.

The displayed difference uses the order doubled tooth minus original tooth. An ordinary electronic frequency counter returns a nonnegative beat magnitude; spectral selection and known electronic shifts must identify whether it represents $f_0$, $f_r-f_0$, or a shifted version. Coherent broadening, overlapping photon polarization, spatial alignment and temporal overlap are required. Merely producing a broad patch of colored light does not establish a usable offset signal.

The New Jersey Institute of Technology's Lab 2: FM Modulation, Demodulation and Bessel Zeros, Part I, prepares a voltage-controlled oscillator in low-frequency mode. A frequency counter reads its output while a buffered constant electric potential difference calibrates its sensitivity. Sweeping the control voltage checks linearity; the calibrated gain is then held fixed.^[ref]

An audio oscillator replaces the constant control voltage. An oscilloscope displays the signal and its FFT frequency spectrum together. Zero modulation supplies a carrier reference. Increasing message amplitude reveals sidebands, compares their heights with the carrier, and locates a carrier minimum. These readouts test the spectrum and infer frequency deviation. This is a student procedure, not a report of measured results.^[ref]

A carrier minimum is a minimum of one Fourier coefficient, not disappearance of the whole signal. In the ideal model the first positive zero of $J_0$ is approximately $\beta =2.4048256$, while the wave envelope remains $A$. The lab's introductory special-cases paragraph agrees with this rounded value; step T12 mistakenly repeats the equal-sideband value for the carrier zero.^[ref]

The interpretation also has practical limits. A finite recording limits frequency resolution; inadequate sampling can cause aliasing; and nonlinear VCO response can introduce extra components. Keeping the gain fixed and first checking linearity helps distinguish those effects from a change in FM modulation index. The diagrams above are independent model calculations, not laboratory measurements.

A Kundt's tube turns an invisible standing wave into a measurable spatial pattern. A driven air column supports a longitudinal wave; measuring its frequency and the spacing of repeated nodes of a standing wave gives the speed of sound. The same drive can compare the wavelengths in a gas and a vibrating solid. August Kundt's 1866 paper introduced acoustic dust figures; its footnote explicitly counts successive nodes of a standing wave as half a wavelength.^[ref]

The attractive feature is the separation of measurements: a clock or frequency counter measures repetition in time, while a ruler measures repetition in space. Their product supplies a propagation speed. The pressure pattern and the fluid-motion pattern are different, however; identifying which one a detector responds to prevents a quarter-wavelength mistake.

Mix an unknown single-frequency laser with a nearby comb tooth on a photodiode. Define the signed beat $b=f_L-f_n$, soThe beat arises from the cross term in light intensity:An electronic detector can follow this slow difference even though it cannot resolve either optical oscillation. The square-law expression also shows why a real-valued intensity trace by itself loses the sign of $b$. A small known upward shift of the chosen tooth decreases $|b|$ when the unknown lies above it and increases $|b|$ when it lies below, provided the tooth assignment remains unchanged.

For an original example, stipulateThen $f_n=500,000.020\mathrm{GHz}$ and $f_L=500,000.050\mathrm{GHz}$. Choosing the opposite sign would give $499,999.990\mathrm{GHz}$ instead. The stipulated optical frequency is countable because the large optical value is expressed using the integer $n$ and three electronic frequencies; the diagram's two controls establish the comb, and this additional beat places the unknown against it.

Both comb readouts use one calibrated microwave reference. Feedback fixes $f_r$ and $f_0$, determining each tooth once $n$ is identified. A frequency counter measures the slow beat, transferring the calibration to the optical frequency.^[ref]