Frequency Modulation

What is Frequency Modulation?

A Frequency Modulated wave is a sine wave with a periodically varying instantaneous frequency and a constant amplitude.

The average frequency is called the carrier frequency and the instantaneous frequency changes at the modulation frequency . The maximum excursion of the instantaneous frequency from the average is related to the modulation depth .

If this were an FM radio transmission, the carrier frequency would be the station you tune to, and you would hear a pure audio tone at the modulation frequency, with a loudness derived from the modulation depth.

In our laser spectroscopy, the carrier frequency is the frequency of red light (about 4 x 10¹⁴ Hz) and the modulation frequency is a radio frequency (about 200 MHz). If drawn to scale, you wouldn't be able to see the modulation in a graph like the one above. In the drawing above, the modulation frequency is only ten times lower than the carrier frequency and and the modulation depth is very high.

Fixed frequency components

A Frequency Modulated wave can also be considered as a sum of several waves with constant frequencies. The frequency modulated wave has frequency components at the carrier frequency and at sidebands, spaced above and below the carrier frequency at integer multiples of the modulation frequency.

The sidebands must have a particular balance of phases and amplitudes in order to add to the carrier wave and maintain a constant amplitude. Upper and lower sidebands of the same order are matched in amplitude. The odd order sidebands must be exactly out of phase, and the even order sidebands must be exactly in phase, shown schematically in the figure at the left.

How can you talk about the phase of multiple frequencies?

The moving purple wave on the left shows the sum of three sine waves, equally spaced in frequency. Its three components are shown below. The stationary black wave is the strong carrier frequency; the weaker red and blue ones are lower and higher frequencies. The carrier wave is stationary in a movie if we capture each frame just as the black wave begins a new cycle. In this rotating frame the lower frequency red wave will appear to go forwards, and the blue wave will appear to go backwards.

Notice that when the red and blue waves momentarily are in phase, they line up 90 degrees out of phase with the carrier. At this moment, the sum of all the waves is shifted in time from the carrier. The result is a sum that appears to have a near-constant frequency and amplitude, but is alternately advanced or retarded compared to the phase of the carrier wave.

The three frequencies are not quite enough to make a pure frequency modulated wave. When the first order sidebands are 15% of the carrier, the second order sidebands must be about 1% of the carrier amplitude. These would appear to travel twice as fast as the first order sidebands in this movie. If included, they would cancel the small amplitude modulation at twice the modulation frequency that is present in the displayed purple wave.

FM absorption spectroscopy

If a frequency modulated beam goes through a sample that attenuates one sideband more than the other, the balance is broken and the result is an amplitude modulation. Here, the red sideband is depleted compared to the blue one. The total wave now "breathes" at the modulation frequency. A fast light detector could measure this radio frequency component to the light intensity, providing a sensitive means of measuring an absorption spectrum.

More about Phase and FM line shapes
Some FM spectroscopy References
Back to High-Resolution Laser Spectroscopy and Dynamics

Rev: 23 Mar 00