Transcript Bez nadpisu - Hochschule Pforzheim

Metrology of time and length - a question
of radio and optical frequencies
Josef Lazar
Královopolská 147
612 00 Brno
Czech Republic
e-mail: [email protected]
www: http://www.isibrno.cz
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)
Frequency, time and length
Unit of time is defined:
„The second is the duration of 9 192 631 770 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 1/299 792 458 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.
Electronic oscillator
+US
C1
R1
A
R2
C2
UOU
T
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)
transfer
Selective feedback
Broadband amplifier
with power supply –
source of energy
low
pass
high
pass
f osc.
f
Laser – optical frequency oscillator
E
non-radiative
radiative
transition
non-radiative
Laser can be viewed as a
quantum amplifier of light with a
highly selective feedback.
LASER – Light amplification by
Stimulated Emission of Radiation
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.
mirror
pumping
photons
excited
atoms
semireflecting
mirror
Laser – spectral view
amplifier gain
profile
gain
quantum amplifier
resonator – selective feedback
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.
transition
frequency
frequency
transmission
frequency
cavity linewidth
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.
frequency
noise
spectral
density
random walk noise
flicker 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
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.
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]
Stabilization of frequency
frequency
discriminator
oscillator
frequency
control
V
controller
feedback
servo loop
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.
discriminator
curve
noise
frequency
DV
Df
Quality of discriminator is given
by “gain” of the discriminator
curve, by its signal-to-noise ratio
and bandwidth.
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.
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
Interferometer
Gauge
block
Mechanic
meters
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
Stability of optical
frequency of laser
Measurement in air,
Interferometer
Refractive index
of air
Masurement in vacuum,
l=c/n
l = c / (n . n)
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
Semireflecting
mirror
Mirror
Fotodetector
Counter
Measurement branch
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
Semireflecting
mirror
Tuning of the
resonator by
piezoelement
Windows under
Brewster angle
capilary
Longitudinal modes, creation of „holes“
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.
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
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 phasesensitive detection
Modulation a
laser tuning is via
piezoelectric
elements
Overall control
via PC computer
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
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.
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.
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 firstorder 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.
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
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.
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
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
oscillator 2
freq. divider
/ multiplier
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.
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 phaselocked oscillators generating
higher harmonics.
This very complicated system
realized only in several
laboratories can be replaced
by pulsed lasers.
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.
Generation of the comb of optical frequencies
by pulsed lasers
I
I
FT
t
T
f
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.
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
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 socalled 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 nonlinear crystal behaves under high
intensity like a gradient lens. With an
aperture losses are lower for strong
pulses.
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.
Direct transfer of stability from rf to opitcal
I
difference
between rf
and optical
repetition Df = 1/T
frequency
offset
frequency
Repetition frequency can be
controlled from a rf stable
oscillator
spectral region of
comb generator
freq.
Any component n the optical
spectral range can be selected
and PLL locked to stable laser
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.
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.
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 selflocking.
Red-end frequency when doubled by nonlinear crystal and compared with octave
blue-end frequency gives the offset frequency as a beat signal.
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
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.
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 frequencymodulation
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.
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.
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
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.