1. Metrology of time and length - a question of radio and optical frequencies
Josef Lazar
Královopolská 147
Brno
Czech Republic
www:

2. Metrology
Diagram shows relations among 7 fundamental quantities of the SI system:
The unit of length is dependent upon the unit of time
Their relation is through the constant of speed of light in vacuum „c“
Representation: time – cesium clock (rf oscilator), length – laser (interferometer)

3. Frequency, time and length
Unit of time is defined:
„The second is the duration of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.“
Etalon of time is highly precise and stable radiofrequency oscillator. Conversion of frequency into time is simple:
T = 1/f
Unit of length is defined:
„The metre is the length of the path travelled by light in vacuum during a time interval of 1/ of a second.“
Etalon of length is highly precise and stable laser. Conversion of optical frequency into length (wavelength) is:
l = c/n
Both etalons of time and length as well are oscillators operating in the radiofrequency domain, resp. optical spectral region.

4. Electronic oscillator
+US
Electronic oscillator (in this case Wien type oscillator) consists of a source of energy needed to cover losses, amplifier (usually broadband, able to amplify oscillations) and a selective feedback (here a combination of low-pass and high pass filters) C1 R1 A
UOUT R2 C2
transfer
low pass
high pass
Broadband amplifier with power supply – source of energy
Selective feedback f
f osc.

5. Laser – optical frequency oscillator
Laser can be viewed as a quantum amplifier of light with a highly selective feedback.
LASER – Light amplification by Stimulated Emission of Radiation
non-radiative
radiative transition
non-radiative
pumping
Example of multilevel system of quantum transitions. Source of energy populates higher energy level by a process called pumping (discharge, absorption of photons etc.) Collision of photon with excited atom can stimulate emission of another photon with identical phase.
semireflecting mirror
excited atoms
mirror
pumping
photons

6. Laser – spectral view amplifier gain profile gain quantum amplifier
resonator – selective feedback
transition frequency
frequency
line broadening
Gain profile of optical amplifier is broadened by various effects (Doppler, etc.) Linewidth of laser feedback – resonator (cavity) is given by its Q (quality) factor determined by reflectivity of mirrors, length and overall losses.
transmission
frequency
cavity linewidth

7. Stability of an oscillator
Oscillator as etalon of (radio-, optical-) frequency must be “precise”. It means stability of output frequency. This can be viewed as low frequency noise.
Amplitude and frequency noise go hand-in-hand. Shot noise is white, flicker and slower, random walk noise have a 1/f spectral distribution
frequency noise spectral density
random walk noise
flicker noise
shot noise level
frequency
Frequency stabilization of an oscillator can be interpreted as filtering frequency noise. Measuring of frequency variations (by counter) depends on integration time. Longer time averaging results in smaller deviations when the frequency fluctuations are pure random.

8. Allan variance – expression of freq. stability
Is defined as one half of the time average of the squares of the differences between successive readings of the frequency deviation sampled over the sampling period. The Allan variance depends on the time period between samples.
where t is a sample period
n is frequency
dn frequency error.
relative stability
Diagram of Allan variances in log-log scale represents relation between integration time and relative stability. Linear descending diagram shows presence of pure random frequency noise.
10-9
10-10
10-11
10-12
100
101
102
103
int. time [s]

9. Stabilization of frequency
Improvement of frequency stability may be done by an active frequency control of tunable oscillator. Error signal is derived from precise and stable frequency discriminator.
frequency discriminator
oscillator
feedback servo loop
frequency control
controller
discriminator curve V
noise
Quality of discriminator is given by “gain” of the discriminator curve, by its signal-to-noise ratio and bandwidth.
frequency DV Df

10. Bandwidth of the control loop
Ability of frequency control loop to reduce frequency noise of oscillator is expressed by frequency response of the controller.
frequency noise spectral density
frequency response of “I” controller
frequency noise response
shot noise level
frequency
Gain, slope of the frequency discriminator (limited by signal-to-noise ratio) and bandwidth of the control loop determines reduction of frequency noise on the frequency scale.

11. Laser – primary etalon of length
Metrology of length is more complex. The way from primary etalon (laser) and mecjanic meters has several steps.
Traceability of the etalons of length to the primary etalon:
High stability laser
Gauge block
Mechanic meters
Interferometer
Interferometer transforms the precise value of wavelength to practical measurements of length by counting of discrete wavelengths.
Its precision is given by:
Stability of optical frequency of laser
Interferometer
Masurement in vacuum, l = c / n
Stability of optical frequency of laser
Measurement in air,
l = c / (n . n)
Interferometer
Refractive index of air

12. Stabilized laser and interferometer
High stability laser – optical oscillator is a tunable laser, with an optical frequency controlled by a servo-loop to a constant value which is derived from some reference, most often an atomic transition detected as a narrow absorption line.
Tunable laser
Reference
Detection
Regulator
For measuring of length the Michelson interferometer is mostly used. The length is measured by counting of wavelengths when the length of the measuring arm is varied.
Motion
Mirror
Reference branch
Laser
Mirror
Semireflecting mirror
Measurement branch
Fotodetector
Counter

13. He-Ne laser, most common etalon of length
The He-Ne laser is a laser with an active media of a mixture of gasses helium and neon. He-Ne lasers at the 633 nm wavelength are suitable for metrological applications for their high coherence and relative simplicity.
anode
Glass discharge tube
cathode
mirror
Semi-reflecting mirror
Windows under Brewster angle
Tuning of the resonator by piezoelement
capilary

14. Longitudinal modes, creation of „holes“

15. Stabilization of frequency of two-mode laser
He-Ne laser with two longitudinal modes: polarizations of the modes are perpendicular.
Stabilization of frequency is possible by separation of modes by a polarizing beamsplitter and by deriving the control value from difference in power.

16. Derivation of discrimination profile
Stabilization of laser by narrow-linewidth absorption in a suitable medium is based most often on derivative spectroscopy.
Laser wavelength is modulated by a low-frequency with much higher frequency deviation.
During interaction with the absorption line the frequency modulation results in amplitude modulation.
With phase-sensitive detection a derivative of the absorption line is achieved. V f
useful region for frequency control

17. He-Ne-I2 laser, fundamental etalon of length
Stabilization of laser is derived from hyperfine components of the R(127)11-5 transition in molecular iodine. Absorption lines are very weak, absorption cell is placed inside the laser cavity.
Control loop uses third derivative of the absorption profile
Detection is based on phase-sensitive detection
Modulation a laser tuning is via piezoelectric elements
Overall control via PC computer

19. Gain profile and spectrum of the He-Ne-I2 laser
Gain profile of the He-Ne laser with iodine absorption cell inside of the cavity, its first and third derivative

20. Spectrum of transitions in molecular iodine
Iodine vapour is a suitable absorbing medium from the range from red to green part of the visible spectrum. They perform a rich set of narrow lines.

21. Coherent semiconductor lasers
Laser diodes are the most common lasers in these days. They are available in a wide variety of types, powers and wavelengths.

22. Semiconductor laser with external cavity (ECL)
LD with one facet (mirror) replaced by a selective reflector is a way of improvement of emission spectral properties.
Configuration “Littrow“, grating is a selective element, reflects the first-order beam back into laser diode. Grating angle sets the wavelength. Zero-order reflection is useful output.
Configuration “Littmann“ – LD beam is incident on the grating with large angle. Laser is tuned by mirror. Zero-order reflection is useful output.

23. Selection of single optical frequency in ECL
For a propper operation the front facet of the LD should be AR coated to suppress its internal resonator.
The lasing frequency is selected from the LD gain profile by a combination grating selectivity and one of the longitudinal modes of the extended cavity
To achieve a continuous tuning the cavity length and the grating selectivity must be synchronized

24. Experimental and compact versions of ECLs
ECSL designed for experiments and testing of antireflection coatings for laser diodes operated around the 635 nm wavelength
The compact ECSL, a high-stability laser for metrological applications, interferometry, spectroscopy, etc. Designed for the 633 nm wavelength, replacement for He-Ne lasers.

25. Comparison of stability of optical oscillators
Comparison of frequency stability of lasers is possible to perform with a high precision by recording of beat signal in the air (comparison of wavelengths by a differential interferometer would be complicated by fluctuations of the index of refraction of air)
Laser 1
Laser 2
Fotodetector
Counter
Recorder
Photodetector output is related to the intensity of the incident light which is related to the quadrate of the electric field.
Max. frequency difference between the lasers is limited by the bandwidth of the photodetector fD = 1/tD

26. Conversion of stable frequency
To get an (optical) frequency other than that of a stable etalon laser but with the same relative frequency stability the PLL (phase lock loop) with frequency divider or multiplier is used. Both oscillators (lasers) are running in phase transferring all frequency fluctuations from oscillator 1 to oscillator 2.
oscillator 1
phase comparator
freq. divider / multiplier
oscillator 2
feedback servo loop
frequency control
controller
output
Detection of radiofrequency beat signal between two optical frequencies is possible only if the two lasers operate at very close frequencies (no more than several GHz). A set of conversion PLL’s can bridge the frequency gap. Nonlinear crystals can generate higher harmonics of optical frequencies and act like frequency multpliers.

27. Traceability of the etalon of optical frequency to the rf etalon of time
Direct comparison of stability of the laser etalon of optical frequency in the visible spectral range with the time etalon – rf cesium clock was achieved by a chain of phase-locked oscillators generating higher harmonics.
This very complicated system realized only in several laboratories can be replaced by pulsed lasers.

28. Femtosecond pulsed lasers and their applications in metrology of optical frequencies
Generation of very short optical pulses became possible with introduction of lasers with active medium with broad spectral profile and with the invention of mode-lock operation. They are dye lasers and now predominantly Titanium-Sapphire lasers.
Laser with a gain profile covering several longitudinal modes of the cavity is able to generate several optical frequencies at once. Titanium-Sapphire laser is able to generate radiation on more than 104 longitudinal modes. Such a laser is able to cover a large frequency range.

29. Generation of the comb of optical frequencies by pulsed lasers
Df = 1/T
A single pulse of zero duration covers the whole spectrum of frequencies. The sequence of such pulses has a spectral representation of a comb of discrete frequencies with spacing of Df = 1/T.
Periodic pulses of finite length are represented by spectrally limited comb of frequencies. There is inverse proportion between the pulse length and spectral width.

30. Emerging of short pulses in a multimode laser
A single-frequency laser generates radiation of constant intensity
Two-frequency laser generates radiation modulated by an envelope of the beat frequency which = intermode frequency = 2L/c
Multimode laser with random varying phase of the discrete frequencies generates radiation with the amplitude of random noise character
Multimode laser in mode-lock regime generates short pulses arising by constructive interference with a maximum in the moment when f1 upto fn = 0

31. Arising of the mode-lock regime
Mode-lock regime can be created by periodic modulation of gain in the laser cavity. generating of periodic pulses is supported. Synchronisation of W with the cavity length is needed. It is so-called active mode-locking.
Passive mode-locking uses e.g. saturable absorber. Losses drop when the saturation threshold is crossed. Short and powerful pulses perform lower losses.
aperture
Kerr-lens modelocking arises by a non-linear Kerr effect when a non-linear crystal behaves under high intensity like a gradient lens. With an aperture losses are lower for strong pulses.

32. Comparison of frequencies of lasers by the comb of optical frequencies
Comparison of wavelength of laser with one of the frequencies of the optical com generator – the pulsed Ti:Sa laser is possible by selecting a single component by a selective element – grating from the “white continuum”.
This makes the comparison of lasers with distant frequencies possible
Intermode radiofrequency frequency (around 1 GHz) can be stabilized to the cesium atomic clock.
The pulsed broad spectrum lasers will in future allow unification of the etalon of length and etalon of time by bridging the frequency span from rf to optical frequencies.

33. Direct transfer of stability from rf to opitcal
difference between rf and optical
spectral region of comb generator
freq.
repetition frequency
Df = 1/T
Any component n the optical spectral range can be selected and PLL locked to stable laser
offset frequency
Repetition frequency can be controlled from a rf stable oscillator
Optical comb generator transfers the repetition frequency by a large number of multiples into the optical range. Order of each component (multiple) s an integer number. Relative stability of each optical spectral component s equal to the relative stability of the repetition frequency.
The comb generator can bridge the gap between rf and optical frequencies. It can operate as optical clock converting stable laser frequency into rf and vice versa.

34. Repetition and offset frequency
Difference between group and phase velocity in the cavity results in varying phase delay between carrier frequency of the pulses and envelope. This generates offset frequency which if uncontrolled fluctuates and causes additional shift of the whole spectrum of comb components.

35. Locking of the offset frequency
The optical comb generator can be viewed as a scale with two degrees of freedom: adjustable spacing of the components (repetition frequency) and a possibility to shift the whole scale (offset frequency). If the span of components covers more than 1 octave, offset frequency variations can be eliminated by self-locking.
Red-end frequency when doubled by nonlinear crystal and compared with octave blue-end frequency gives the offset frequency as a beat signal.

36. Schematics of the pulsed Titanium-Sapphire laser
Pumping of the Ti:Sa crystal is optical by a continuous Argon-ion or Nd:YAG laser. Length of the cavity adjustemnt influences the repetition frequency of pulses. Introduction of dispersive elements (prisms) allows to vary cavity length for different wavelengths and change the group delay. This makes possible the control of offset frequency

38. Cesium clock
A cloud of Cs atoms is created in a laser trap. Vertical lasers toss the cloud upward like fountain through a microwave-filled cavity. During the ca 1 second roundtrip, the atomic states of the atoms might or might not be altered as they interact with the microwave signal. Then another laser is pointed at the atoms. Those atoms whose atomic state were altered by the microwave signal emit light (fluorescence) which is measured by a detector. Frequency which alters the states of most of the cesium atoms and maximizes their fluorescence is the natural resonance frequency of the cesium atom (9,192,631,770 Hz) the frequency used to define the second.

39. Pound-Drever method of stabilization to F.-P. cavity
Locking of lasers to narrow but weak and noisy transitions cannot give robust lock to keep a noisy (i.e. semiconductor) laser stabilized. Multistage techniques such as PD method with frequency-modulation spectroscopy detection is the solution.
Passive Fabry-Perot cavity can be made with very high “Q” factor, very good stability (invar, zerodur) and gives excellent signal-to-noise ratio, over the whole dynamic range (transmission from 0 to nearly 100%). Stabilization to atomic transition only compensates long-term drift.

40. FM (Frequency Modulation) spectroscopy
Detection technique called FM spectroscopy uses F modulation of laser with mod. index close to 1.
Modulation frequency W must be a little higher, than linewidth of the detection transition.
Beat signals between carrier and sidebands are in opposite phase and cancel each other.
In case of attenuation and/or phase shift of one of sidebands due to interaction with absorption, resp. dispersion of the line the equilibrium is disturbed and signal detected.
After synchronous detection the frequency discrimination signal can be used for locking.

41. Optical clocks
The optical frequencies (hundreds of THz) gives chances to build oscillators with unprecedented relative stability.
optical transitions with natural linewidths around 1 Hz or less therefore offer potential Q-factors of order 1015 or higher.
A number of different candidates for optical frequency standards are currently being investigated in various laboratories, based on forbidden transitions in cold trapped ions or atoms, and over recent years there has been significant progress in both areas.
By laser cooling, the atom or ion can be confined to within a wavelength of light, ensuring that the transition is free from Doppler frequency shifts to all orders. Since there is only one ion held in the trap, in a vacuum, the transition is also free from frequency shifts caused by collisions. This results in very narrow transitions

42. The future
Transfer of relative frequency stability from the optical oscillator (laser) via comb generator into rf spectrum can bring optical clock into life.
This will lead to unification of the unit of length and time on the basis of a single precise oscillator.