Pound–Drever–Hall technique

The Pound–Drever–Hall (PDH) technique is a widely used and powerful approach for stabilizing the frequency (or wavelength) of light emitted by a laser, and along with very closely related techniques such as frequency-modulation spectroscopy, has a broad range of applications including interferometric gravitational wave detectors, atomic physics, and time measurement standards. A wide range of conditions contribute to determine the spectrum produced by a laser and the PDH technique is a means to control and/or exploit a laser’s spectrum, for instance in tunable lasers. A key element of the technique is its response to the frequency of laser emission independent of intensity because many of the methods that control a laser’s emission frequency also affect its intensity. A typical application of the PDH technique is a control system in which the resonances of a Fabry-Pérot optical cavity are detected and used to feed back a correction signal to the laser.

Named after R. V. Pound, Ronald Drever, and John L. Hall, the technique was described in 1983 by Drever, Hall and others working at the University of Glasgow and the U. S. National Bureau of Standards, and had similarities to an older frequency-modulation technique developed by Pound for microwave cavities.

Laser stabilization

All lasers demonstrate some kind of frequency wander – this is an artifact of temperature variations, which change laser cavity lengths, laser driver current and voltage fluctuations, atomic transition widths, and many other factors. PDH locking offers one possible solution to this problem by actively tuning the laser to match the resonance condition of a stable reference cavity.

In recent years the Pound–Drever–Hall technique has become a mainstay of modern laser frequency stabilization and is the basis of many of types of precision laser stabilization. Prominently, the field of interferometric gravitational wave detection depends critically on enhanced sensitivity afforded by optical cavities.

Overview of technique

Schematic of PDH servo loop to lock a the frequency of a laser to a Fabry–Perot cavity. Light from the laser is sent through a phase-modulator and is then directed upon the cavity. The isolator is not involved in the PDH setup; it is present only to ensure that light from various optical components does not reflect back into the laser. The polarizing beamsplitter and λ/4 plate act in combination to discriminate between the two directions of light travel: light traveling in the direction 1→2 passes straight through and on to the cavity, while light traveling in the direction 2→1 (i.e., from the cavity) is diverted toward the photodetector.

The phase modulator is driven with a sinusoidal signal from the oscillator; this impresses sidebands onto the laser light. As described in the section on the PDH readout function, the reflected light is directed upon a photodetector, and the resulting voltage is demodulated (that is, passed through the mixer and the low-pass filter) to produce an error signal that is fed back into the laser's frequency control port.

Phase modulated light, consisting of a carrier frequency and two side bands, is directed onto a two mirror cavity. Light reflected off the cavity is measured using a high speed photodetector, the reflected signal consists of the two unaltered side bands along with a phase shifted carrier component. This light is then measured using a photodetector and is mixed down with a phase shifted local oscillator and low pass filtered. The resulting electronic readout signal gives a measure of how far the laser carrier is off resonance with the cavity.

PDH readout function

The PDH readout function gives a measure of the resonance condition of a cavity. By taking the derivative of the cavity transfer function (which is symmetric and even) with respect to frequency, it is an odd function of frequency and hence indicates not only whether there is a mismatch between the output frequency ω of the laser and the resonant frequency ω_res of the cavity, but also whether ω is greater or less than ω_res. The zero-crossing of the readout function is sensitive only to intensity fluctuations due to the frequency of light in the cavity and insensitive to intensity fluctuations from the laser itself.

Light of frequency f = ω/2π can be represented mathematically by its electric field, E₀e^iωt. If this light is then phase-modulated by βsin(ω_mt), the resulting field E_i is

${\begin{aligned}E_{\text{i}}&=E_{0}e^{i(\omega t+\beta \sin(\omega _{\mathrm {m} }t))}\\&\approx E_{0}e^{i\omega t}[1+i\beta \sin(\omega _{\mathrm {m} }t)]\\&=E_{0}e^{i\omega t}\left[1+{\frac {\beta }{2}}e^{i\omega _{\mathrm {m} }t}-{\frac {\beta }{2}}e^{-i\omega _{\mathrm {m} }t}\right].\end{aligned}}$

This field may be regarded as the superposition of three components. The first component is an electric field of angular frequency ω, known as the carrier, and the second and third components are fields of angular frequency ω + ω_m and ω − ω_m, respectively, called the sidebands.

In general, the light E_out reflected out of a Fabry–Pérot two-mirror cavity is related to the light E_in incident on the cavity by the following transfer function:

$R(\omega )={\frac {E_{\text{out}}}{E_{\text{in}}}}={\frac {r_{1}-(r_{1}^{2}+t_{1}^{2})r_{2}e^{i2\alpha }}{1-r_{1}r_{2}e^{i2\alpha }}},$

where α = ωL/c, and where r₁ and r₂ are the reflection coefficients of mirrors 1 and 2 of the cavity, and t₁ and t₂ are the transmission coefficients of the mirrors.

Simulated plots of a two-mirror Fabry–Perot cavity reflection transfer function and a PDH readout signal. Top: Square magnitude R^*R of reflection transfer function; i.e., the reflected power. Middle: Phase arctan[Im(R)/Re(R)] of reflection transfer function. Bottom: PDH readout function V, with demodulation phase φ = π/2. The mirrors of the simulated cavity were chosen to have amplitude reflectivities r₁ = 0.99 and r₂ = 0.98, and the cavity length was L = 1 m. The phase modulation frequency of the light was chosen to be f_m = 23 MHz (f_m = ω_m/2π). The portion of the PDH readout function that is useful as a servo error signal is the linear region near f_res.

The reflected power and the PDH readout function are often monitored in real time as traces on an oscilloscope in order to assess the state of an optical cavity and its servo loop.

Applying this transfer function to the phase-modulated light E_i gives the reflected light E_r:

$E_{\text{r}}=E_{0}\left[R(\omega )e^{i\omega t}+R(\omega +\omega _{\mathrm {m} }){\frac {\beta }{2}}e^{i(\omega +\omega _{\mathrm {m} })t}-R(\omega -\omega _{\mathrm {m} }){\frac {\beta }{2}}e^{i(\omega -\omega _{\mathrm {m} })t}\right].$

The power P_r of the reflected light is proportional to the square magnitude of the electric field, E_r^* E_r, which after some algebraic manipulation can be shown to be

${\begin{aligned}P_{\text{r}}=&\ P_{0}\left|R(\omega )\right|^{2}+P_{0}{\frac {\beta ^{2}}{4}}{\Big \{}\left|R(\omega +\omega _{\mathrm {m} })\right|^{2}+\left|R(\omega -\omega _{\mathrm {m} })\right|^{2}{\Big \}}\\&+P_{0}{\frac {\beta }{2}}{\Big \{}{\textrm {Re}}[\chi (\omega )]\cos {\omega _{\mathrm {m} }t}+{\textrm {Im}}[\chi (\omega )]\sin {\omega _{\mathrm {m} }t}{\Big \}}+({\text{terms in }}2\omega _{\mathrm {m} }).\end{aligned}}$

Here P₀ ∝ |E₀|² is the power of the light incident on the Fabry–Pérot cavity, and χ is defined by

$\chi (\omega )=R(\omega )R^{*}(\omega +\omega _{\mathrm {m} })-R^{*}(\omega )R(\omega -\omega _{\mathrm {m} }).$

This χ is the ultimate quantity of interest; it is an antisymmetric function of ω − ω_res. It can be extracted from P_r by demodulation. First, the reflected beam is directed onto a photodiode, which produces a voltage V_r that is proportional to P_r. Next, this voltage is mixed with a phase-delayed version of the original modulation voltage to produce V′_r:

${\begin{aligned}V_{\text{r}}'&=V_{\text{r}}\cos(\omega _{\mathrm {m} }t+\varphi )\propto P_{\text{r}}\cos(\omega _{\mathrm {m} }t+\varphi ).\\\end{aligned}}$

Finally, V′_r is sent through a low-pass filter to remove any sinusoidally oscillating terms. This combination of mixing and low-pass filtering produces a voltage V that contains only the terms involving χ:

$V(\omega )\propto {\textrm {Re}}[\chi (\omega )]\cos \varphi +{\textrm {Im}}[\chi (\omega )]\sin \varphi .$

In theory, χ can be completely extracted by setting up two demodulation paths, one with φ = 0 and another with φ = π/2. In practice, by judicious choice of ω_m it is possible to make χ almost entirely real or almost entirely imaginary, so that only one demodulation path is necessary. V(ω), with appropriately chosen φ, is the PDH readout signal.