Doppler shift

The Doppler shift is the change in the frequency of light seen by an observer when the source and observer move toward or away from each other. For speeds $v$ much smaller than the speed of light, the fractional shift is $v/c$, so the frequency changes by $v/\lambda$: 0.645 MHz per meter per second at 1550 nm. Light reflected from a moving target is shifted twice, once on arrival and once on re-emission, giving $2v/\lambda$, which is 1.29 MHz per m/s at 1550 nm and 3.16 kHz per mm/s at 632.8 nm. Approach raises the frequency and recession lowers it. Because $v/c$ is only 3.3 × 10⁻⁹ for 1 m/s, the shift is far too small for a spectrometer to resolve and is almost always measured by beating the light against a reference.

Formulas

For a source moving at speed $v$ along the line of sight, with $\beta = v/c$, the relativistic result is

$$f' = f\sqrt{\frac{1+\beta}{1-\beta}},$$

with $\beta$ positive for approach. To first order in $\beta$,

$$\Delta f = \frac{v}{c}f = \frac{v}{\lambda}.$$

For a mirror or scatterer moving at $v$ and illuminated and observed from the same side,

$$f_D = \frac{2v\cos\theta}{\lambda},$$

where $\theta$ is the angle between the velocity and the beam. Only the component of velocity along the line of sight counts; motion across the beam gives no first-order shift. The second-order (transverse, time-dilation) shift is $\beta^2/2$ in relative terms, 4.7 × 10⁻¹³ for atoms at 290 m/s or about 180 Hz on a 384 THz transition, relevant mainly in atomic clocks. For a target at 30 m/s at 1550 nm the exact and first-order reflection formulas differ by about 4 Hz out of 38.7 MHz.

In a medium of index $n$, the wavelength in the formula is the wavelength in the medium, $\lambda_0/n$. The same relation describes Brillouin scattering, which is the Doppler shift of light reflected by a moving acoustic grating: in silica at 1550 nm, with $n$ = 1.444 and a sound speed of 5960 m/s, $2nv_a/\lambda_0$ = 11.1 GHz, the Brillouin shift. An acousto-optic modulator shifts the diffracted beam by exactly the acoustic frequency for the same reason, and is the usual way to produce a known frequency offset.

Measurement

The shifted light is mixed on a photodiode with a reference beam from the same laser, and the detector output contains the beat at $f_D$. This heterodyne arrangement turns a shift of a few parts per billion into a radio-frequency signal. A plain homodyne beat gives the magnitude of $f_D$ but not its sign, so the reference is offset by an acousto-optic modulator (commonly by tens of megahertz) or the measurement is made with in-phase and quadrature detection. The laser's coherence must cover the round-trip path difference, or the beat spreads into the laser's own linewidth.

Where it matters

• Vibrometry. Laser Doppler vibrometry measures surface velocity from $2v/\lambda$ and displacement from the corresponding phase.
• Coherent lidar. In FMCW lidar the Doppler term adds to the range beat with opposite sign on up and down chirps; a target closing at 30 m/s shifts 1550 nm light by 38.7 MHz, and the two chirps separate range from radial velocity, as described in time-of-flight versus FMCW lidar.
• Spectroscopy. Thermal motion gives each atom in a gas a different shift, so the line acquires a Doppler width, 1.3–1.5 GHz for the neon line of a helium-neon laser; line broadening treats this inhomogeneous width. Saturated-absorption spectroscopy with counter-propagating beams selects atoms with zero velocity along the beams, and two-photon absorption from counter-propagating beams cancels the first-order shift for every atom; both recover the narrow homogeneous line.
• Laser cooling. A rubidium atom moving at 290 m/s toward a 780 nm beam sees the light shifted by 372 MHz, sixty times the 6.07 MHz natural width; tuning the beam below resonance makes atoms absorb preferentially from the beam they move toward, the basis of laser cooling.

Pitfalls

The cosine factor causes the most common error: a beam 10° off the direction of motion underreads the velocity by 1.5 %. Light scattered from a rough moving surface produces speckle, which modulates the amplitude of the beat and adds noise that is often mistaken for velocity fluctuation. A finite transit time of the target through the beam broadens the Doppler line, and in long-range measurements the laser linewidth sets a floor on the frequency resolution.

Common questions

What is the Doppler shift formula for light?

To first order, $\Delta f = v/\lambda$ for a moving source or observer and $2v/\lambda$ for reflection from a moving object, with $v$ the velocity component along the line of sight.

Why is the reflected shift doubled?

The moving target first receives the light at a shifted frequency and then acts as a moving source when it reflects it, so the shift is applied twice.

How is a Doppler shift different from Doppler broadening?

A shift is the frequency change of one source with one velocity. Broadening is the spread of shifts from many emitters with a distribution of velocities, which widens a spectral line.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed. (Springer, 2014); L. E. Drain, The Laser Doppler Technique (Wiley, 1980).