Pound-Drever-Hall technique
Encyclopedia
The Pound–Drever–Hall technique is an application of FM spectroscopy to detect the resonance
s of the optical cavity
by demodulating the beam reflected from the cavity. The PDH technique is often part of a control system
which regulates the frequency of a laser
to match the resonance condition of a cavity.
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 used for microwave cavities.
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.
Phase modulated
light, consisting of a carrier frequency and two side bands (well outside the FWHM of the cavity), 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.
(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 ωres of the cavity, but also whether ω is greater or less than ωres. Additionally, it 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
can be represented mathematically by its electric field, E0eiωt. If this light is then phase-modulated by βsin(ωmt), the resulting field Ei is
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 and , respectively, called the sideband
s.
In general, the light Eout reflected out of a Fabry–Pérot two-mirror cavity is related to the light Ein incident on the cavity by the following transfer function
:
where , 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.
Applying this transfer function to the phase-modulated light Ei gives the reflected light Er:
The power of the reflected light, which is proportional to photodetector voltage, is given by
After some algebraic manipulation, this becomes
Here P0 = |E0|2 is the power of the light incident on the Fabry–Pérot cavity. The terms of Pr which involve ωm are of interest because they oscillate with the same frequency as that of the incident laser beam, giving the derivative.
The reflected light is incident upon a photodetector
, which produces a voltage Vr proportional to Pr. The terms of Vr involving ωm are isolated by first using a mixer
to multiply Vr by a reference sinusoid at the same modulation frequency ωm and with phase φ:
With the use of some trigonometric identities, it can be shown that
where
Finally, Vr′ is passed through a low-pass filter
to remove the oscillating terms. This gives the final the error signal voltage V:
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
s of the optical cavity
Optical cavity
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric...
by demodulating the beam reflected from the cavity. The PDH technique is often part of a control system
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
which regulates the frequency of a laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
to match the resonance condition of a cavity.
Named after R. V. Pound, Ronald Drever
Ronald Drever
Ron Drever is a Scottish physicist. He is currently a Professor Emeritus at the California Institute of Technology.Among other accomplishments he co-founded the LIGO project, and was a co-inventor of the Pound-Drever-Hall technique for laser stabilisation....
, and John L. Hall
John L. Hall
John Lewis "Jan" Hall is an American physicist, and Nobel laureate in physics. He shared one half of the 2005 Nobel Prize in Physics with Theodor W. Hänsch for his work in precision spectroscopy.-Biography:...
, the technique was described in 1983 by Drever, Hall and others working at the University of Glasgow
University of Glasgow
The University of Glasgow is the fourth-oldest university in the English-speaking world and one of Scotland's four ancient universities. Located in Glasgow, the university was founded in 1451 and is presently one of seventeen British higher education institutions ranked amongst the top 100 of the...
and the U. S. National Bureau of Standards, and had similarities to an older frequency-modulation technique used 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 activelyFeedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
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
Interferometry
Interferometry refers to a family of techniques in which electromagnetic waves are superimposed in order to extract information about the waves. An instrument used to interfere waves is called an interferometer. Interferometry is an important investigative technique in the fields of astronomy,...
gravitational wave
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...
detection depends critically on enhanced sensitivity afforded by optical cavities.
Overview of technique
Phase modulation
Phase modulation is a form of modulation that represents information as variations in the instantaneous phase of a carrier wave.Unlike its more popular counterpart, frequency modulation , PM is not very widely used for radio transmissions...
light, consisting of a carrier frequency and two side bands (well outside the FWHM of the cavity), is directed onto a two mirror cavity. Light reflected off the cavity is measured using a high speed photodetector
Photodetector
Photosensors or photodetectors are sensors of light or other electromagnetic energy. There are several varieties:*Active pixel sensors are image sensors consisting of an integrated circuit that contains an array of pixel sensors, each pixel containing a both a light sensor and an active amplifier...
, 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
Frequency mixer
In electronics a mixer or frequency mixer is a nonlinear electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals at frequencies f1 and f2 are applied to a mixer, and it produces new signals at the sum f1 + f2 and difference f1 -...
down with a phase shifted local oscillator
Local oscillator
A local oscillator is an electronic device used to generate a signal normally for the purpose of converting a signal of interest to a different frequency using a mixer. This process of frequency conversion, also referred to as heterodyning, produces the sum and difference frequencies of the...
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 functionTransfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
(which is symmetric and even
Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series...
) with respect to frequency, it is an odd
Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series...
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 ωres of the cavity, but also whether ω is greater or less than ωres. Additionally, it 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
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
can be represented mathematically by its electric field, E0eiωt. If this light is then phase-modulated by βsin(ωmt), the resulting field Ei is
This field may be regarded as the superposition
Superposition
Superposition can refer to:* The superposition principle in physics, mathematics, and engineering, describes the overlapping of waves. Particular applications include :** Quantum superposition, in quantum physics** Superposition theorem, in electronics....
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 and , respectively, called the sideband
Sideband
In radio communications, a sideband is a band of frequencies higher than or lower than the carrier frequency, containing power as a result of the modulation process. The sidebands consist of all the Fourier components of the modulated signal except the carrier...
s.
In general, the light Eout reflected out of a Fabry–Pérot two-mirror cavity is related to the light Ein incident on the cavity by the following transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
:
where , and where r1 and r2 are the reflection coefficients
Reflectivity
In optics and photometry, reflectivity is the fraction of incident radiation reflected by a surface. In general it must be treated as a directional property that is a function of the reflected direction, the incident direction, and the incident wavelength...
of mirrors 1 and 2 of the cavity, and t1 and t2 are the transmission coefficients
Transmissivity
Transmissivity may refer to:* Transmissivity , the rate which groundwater flows horizontally through an aquifer* Transmittance, in optics...
of the mirrors.
Applying this transfer function to the phase-modulated light Ei gives the reflected light Er:
The power of the reflected light, which is proportional to photodetector voltage, is given by
After some algebraic manipulation, this becomes
Here P0 = |E0|2 is the power of the light incident on the Fabry–Pérot cavity. The terms of Pr which involve ωm are of interest because they oscillate with the same frequency as that of the incident laser beam, giving the derivative.
The reflected light is incident upon a photodetector
Photodetector
Photosensors or photodetectors are sensors of light or other electromagnetic energy. There are several varieties:*Active pixel sensors are image sensors consisting of an integrated circuit that contains an array of pixel sensors, each pixel containing a both a light sensor and an active amplifier...
, which produces a voltage Vr proportional to Pr. The terms of Vr involving ωm are isolated by first using a mixer
Electronic mixer
An electronic mixer is a device that combines two or more electrical or electronic signals into one or two composite output signals. There are two basic circuits that both use the term mixer, but they are very different types of circuits: additive mixers and multiplying mixers...
to multiply Vr by a reference sinusoid at the same modulation frequency ωm and with phase φ:
With the use of some trigonometric identities, it can be shown that
where
Finally, Vr′ is passed through a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
to remove the oscillating terms. This gives the final the error signal voltage V: