Frequency Standards

A frequency standard is a measurement standard whose assigned quantity value is a frequency. In an atomic implementation, a quantum transition supplies a reproducible reference, an oscillator supplies continuous phase, an interrogation and feedback system connects the two, and synthesis or a frequency comb makes a usable output. The result is not merely a narrow spectral line. It is a specified frequency at a specified reference plane, with a correction model, an uncertainty, a stability characterization, and a documented route to a reference such as the SI second.

The distinction matters because different parts of the instrument answer different questions:

the reference determines what frequency should be realized;
the oscillator carries phase between observations of the reference;
the servo estimates and corrects their frequency difference;
the synthesis chain maps the disciplined oscillator to an output;
the comparison link connects that output to another standard;
the traceability record states what reference was used, when it was used, and what uncertainty every link contributed.

A standard can have excellent short-term stability but an incorrect assigned value. It can have a small systematic uncertainty but poor availability. It can be traceable yet unsuitable for a demanding application because its uncertainty is too large. These are separate properties, not synonyms.

Helpful background. Atomic clocks explains how atomic measurements steer an oscillator. Frequency combs explains how optical and radio frequencies are connected. This page follows the delivered signal through comparisons, time scales, and uncertainty estimates.

Oscillators, Standards, and Time Scales

Oscillator

An oscillator produces a signal with evolving phase. Write a nominal sinusoidal output as

V(t) = V_0\cos\!\left[ 2\pi\nu_{\rm nom}t+\phi(t) \right].

The instantaneous frequency is

\nu(t) = \nu_{\rm nom} + \frac{1}{2\pi}\frac{d\phi}{dt}.

Quartz oscillators, dielectric resonators, lasers, cryogenic resonators, and hydrogen masers can all serve as oscillators. An oscillator need not possess an accurate absolute frequency. Its central metrological asset is continuous phase with sufficiently predictable fluctuations over the interval between calibrations.

Reference

A frequency reference provides a reproducible condition that identifies a frequency. An atomic transition, an optical cavity mode, a molecular spectral line, or a signal received from another laboratory can act as a reference. A reference need not itself supply a continuous output.

An unperturbed transition frequency

\nu_0 = \frac{E_e-E_g}{h}

is a theoretical and operational specification. The laboratory observes a perturbed resonance, estimates shifts, and connects an oscillator to the specified unperturbed value. “The atom has frequency $\nu_0$ ” is shorthand for this measurement model, not a complete description of the apparatus.

Frequency standard

A frequency standard realizes, conserves, or reproduces an assigned frequency value. The term may refer to:

a primary or secondary atomic standard evaluated against the SI second;
a continuously operating reference oscillator with a calibration;
a working standard used to calibrate laboratory instruments;
a travelling standard transported between laboratories; or
a remote reference delivered through a calibrated transfer link.

The standard includes the relevant output plane and measurement procedure. Two nominally identical $10\ \mathrm{MHz}$ ports are not interchangeable if their distribution amplifiers, cables, or synthesizers contribute different offsets, phase steps, or noise.

Clock

A clock combines a frequency source with a procedure for assigning time or phase. If $C(t)$ is its reading and $t$ is a reference coordinate time, the time error is

x(t) = C(t)-t.

Frequency is the rate of clock time. A standard can provide frequency without publishing an epoch, while a clock must maintain both rate and a time reading. Conversely, a clock display can appear synchronized even if its oscillator has poor free-running frequency because it is corrected frequently.

Flywheel and holdover oscillator

A flywheel bridges intervals when a more accurate reference is unavailable. Hydrogen masers often serve this role between intermittent optical-clock evaluations. A flywheel is selected for predictable phase and frequency evolution over the relevant gap, not merely for the smallest one-second Allan deviation.

Holdover is operation without the external calibration input. The resulting uncertainty depends on the prior estimate of offset and drift, the oscillator noise model, the duration of the outage, and any environmental sensitivity. It is a prediction problem.

Time scale

A time scale is an algorithmic and operational procedure that combines clock data into a reference time. A schematic ensemble is

x_{\rm ens}(t) = \sum_{i=1}^{N} w_i \left[ x_i(t)+p_i(t) \right], \qquad \sum_{i=1}^{N}w_i=1,

where $x_i$ is a measured clock difference and $p_i$ is a prediction or steering term. Real algorithms estimate clock rates, drift, outliers, and future behavior; they also limit individual weights and preserve continuity. The computed ensemble must ultimately be represented by physical outputs.

The algorithm can be more stable than any typical member while remaining dependent on external standards for its long-term rate. An ensemble, a primary standard, and a disseminated time signal are therefore different objects.

A compact vocabulary table

Object	What it supplies	Typical weakness	Example
oscillator	continuous phase	unknown or drifting absolute frequency	quartz oscillator, laser
reference	reproducible frequency condition	may be intermittent or noncontinuous	atomic transition, cavity
standard	assigned frequency with procedure and uncertainty	limited uptime, calibration, or transfer	caesium fountain, calibrated maser
flywheel	predictable phase between calibrations	accumulates prediction error	hydrogen maser
clock	frequency plus epoch or time reading	steering can cause phase or rate artifacts	UTC( $k$ ) output
time scale	computed ensemble time	depends on data latency and steering policy	EAL, TAI, UTC
transfer link	comparison between locations	delay variation and calibration	GNSS, TWSTFT, optical fiber

Phase, Frequency, and Propagation Delay

Frequency metrology is unusually sensitive to sign, epoch, and reference plane. This page uses the following conventions.

Phase time and fractional frequency

Let $\phi(t)$ be the phase deviation of a signal from a nominal oscillator at frequency $\nu_{\rm nom}$ . Its phase-time deviation is

x(t) = \frac{\phi(t)}{2\pi\nu_{\rm nom}}.

Its fractional-frequency deviation is

y(t) = \frac{\nu(t)-\nu_{\rm nom}}{\nu_{\rm nom}} = \frac{dx}{dt}.

The interval-average fractional frequency from $t_1$ to $t_2$ is

\overline{y}_{[t_1,t_2]} = \frac{x(t_2)-x(t_1)}{t_2-t_1}.

This endpoint relation is exact for the interval average under the stated definition. It is also why a phase record is often more informative than a list of already averaged frequency values: phase steps, gaps, and time-tag errors remain visible.

For two clocks $A$ and $B$ , define

x_{A-B}(t)=x_A(t)-x_B(t), \qquad y_{A-B}(t)=y_A(t)-y_B(t).

A report must state its sign convention explicitly. BIPM products use expressions such as $[\mathrm{UTC}-\mathrm{UTC}(k)]$ ; silently reversing that sign reverses the inferred correction.

Frequency and angular frequency

Ordinary frequency is denoted $\nu$ or $f$ and measured in hertz. Angular frequency is $\omega=2\pi\nu$ . Mixing them introduces a factor of $2\pi$ in phase, linewidth, and servo calculations.

Reference planes

Every output is defined at a physical or logical reference plane:

a connector on a distribution amplifier;
an optical plane before or after an acousto-optic modulator;
the input of a counter;
the effective antenna phase center of a transfer receiver; or
a software epoch after a documented delay correction.

Let $d(t)>0$ be the propagation delay from the defined output plane to the counter. For an otherwise ideal source with phase $2\pi\nu_{\rm nom}t$ , the received phase is $2\pi\nu_{\rm nom}[t-d(t)]$ . Thus the received phase-time deviation and fractional-frequency shift are

x_{\rm delay}(t)=-d(t), \qquad y_{\rm delay}(t)=-\frac{dd}{dt}.

A longer path makes the signal arrive later: increasing delay lowers the received frequency. A timestamp observable defined as receive time minus send time instead contains $+d$ ; its sign must be translated before it is used as a source-minus-reference clock offset.

A constant delay affects time synchronization but not a long-term frequency average; a changing delay affects both. This distinction is central to frequency transfer.

Coordinate and proper frequency

Atomic standards operate along physical worldlines and realize proper-time rates. International comparisons refer them to specified coordinate-time conventions and gravitational reference potentials. To first order in a weak stationary field,

\frac{\Delta\nu}{\nu} \simeq \frac{\Delta U}{c^2},

with $\Delta U=U_A-U_B$ the Newtonian potential difference, increasing with height, and the frequency difference ordered $A-B$ . The gravitational correction and its uncertainty belong in a primary or secondary standard’s evaluation. A location label alone is not an adequate potential model.

From a Transition to a Usable Standard

The physical chain

In a passive standard, the oscillator probes the atoms; their response forms an error signal, and feedback corrects the oscillator frequency.

The atoms are observed intermittently or continuously, and the discriminator estimates an error proportional to the detuning near its operating point. The servo steers one or more oscillator actuators. A synthesis chain then produces the frequency needed by users or comparisons.

Four time stamps in a two-way exchange and the sign of the residual nonreciprocal delay — A two-way exchange compares two clock readings while cancelling a common propagation delay. Time increases downward. Clock $A$ records emission $t_1$ and reception $t_4$ ; clock $B$ records reception $t_2$ and emission $t_3$ . For nearly constant clock offsets over the exchange, $(t_2-t_1)=d_{AB}-x_{A-B}$ and $(t_4-t_3)=d_{BA}+x_{A-B}$ . Their reversed difference estimates $x_{A-B}$ , but retains half the directional delay difference. Local send/receive electronics are included in the corresponding delays.

Locking to a transition

Suppose an interrogation produces an observable $P(\delta)$ as a function of detuning

\delta = 2\pi(\nu_{\rm LO}-\nu_{\rm atom}).

Sampling on opposite sides of the resonance gives a signed error signal, for example

e_k = P_k(+\Delta)-P_k(-\Delta) \simeq K_d\,\delta_k+b_k,

where $K_d$ is the discriminator slope and $b_k$ collects offsets and asymmetries. A discrete controller updates an oscillator command:

u_{k+1} = u_k-g e_k -g_I\sum_{j\le k}e_j.

The sign of $gK_d$ must provide negative feedback. The loop bandwidth must be high enough to suppress useful oscillator noise but low enough to remain stable despite interrogation delay, sampling, and actuator dynamics.

This concise model hides the clock-specific physics. Ramsey phases, line shapes, projection noise, dead time, and the Dick effect are developed in Atomic Clocks and Ramsey Interferometry. Generic feedback design is developed in Laser Stabilization.

The realized frequency

The servo lock point is not automatically the unperturbed transition. A generic realization model is

\nu_{\rm std} = \nu_{\rm out} -\sum_i \Delta\nu_i -\Delta\nu_{\rm synth} -\Delta\nu_{\rm link},

or, in fractional form,

y_{\rm std} = y_{\rm out} -\sum_i c_i -c_{\rm synth} -c_{\rm link}.

All terms in this equation must be referred to the same output frequency. For example, dividing an atomic carrier by $M$ divides its shift in hertz by $M$ , while leaving its fractional shift unchanged. The $\Delta\nu_i$ are the reference shifts expressed on that output scale; synthesis and link terms describe further changes at the defined plane. A correction is an estimate with uncertainty, not a declaration that the effect has vanished.

For input quantities $q_i$ , a local linearized model gives

\Delta\nu_i \simeq k_i(q_i-q_{i,0}),

and the combined variance includes covariance:

u^2(\nu_{\rm std}) = \sum_{i,j} \frac{\partial\nu_{\rm std}}{\partial q_i} \frac{\partial\nu_{\rm std}}{\partial q_j} \operatorname{cov}(q_i,q_j).

The clock pages derive the detailed microwave and optical frequency shifts. The metrological point is that the assigned output value must include the correction model valid during the stated evaluation interval.

Active and passive standards

In a passive standard, an external oscillator interrogates a transition and is steered by the response. In an active standard, stimulated emission or masing produces the output field directly. A hydrogen maser is a standard active example.

“Active” does not mean free of calibration. Cavity pulling, state preparation, magnetic fields, wall interactions, and aging still affect the output. Active standards are often excellent flywheels, while passive primary standards can provide superior long-term accuracy.

Synthesis and division

A user rarely receives the transition frequency directly. Microwave standards use multipliers, dividers, phase-locked loops, and direct digital synthesis. Optical standards use a comb to connect hundreds of terahertz to radio frequency or to another optical carrier.

For a continuous-wave optical frequency measured against a comb,

\nu_{\rm cw} = n f_{\rm rep} +f_{\rm CEO} +s_b f_b +\sum_j s_j f_{{\rm AOM},j},

where $n$ is the tooth index and the signs $s_b,s_j=\pm1$ are determined from the optical layout and actuator conventions. The derivation, self-referencing conditions, and transfer-oscillator combinations are in Frequency Combs.

The synthesis uncertainty is not necessarily limited by nominal electronic resolution. Cycle slips, wrong tooth indices, untracked phase-lock offsets, counter dead time, distribution noise, and sign errors can dominate. A metrological implementation records every term and verifies the result out of loop.

Output phase continuity

A frequency correction can be applied in two qualitatively different ways:

a phase step changes the clock reading immediately;
a frequency steering changes the slope of phase and removes the offset gradually.

If phase continuity matters, a synthesizer update should preserve phase or the phase step must be measured and documented. A frequency standard used only for interval calibration may tolerate a known phase step; a time scale or coherent network often cannot.

Comparing Frequencies

Comparison is the observable

No instrument measures an isolated absolute frequency. It measures a phase or frequency difference, ratio, or beat against another oscillator. The basic comparison result is

y_{A-B} = \frac{\nu_A-\nu_B}{\nu_0},

with a clearly defined normalization $\nu_0$ . For very different carrier frequencies, a ratio is more natural:

R_{AB} = \frac{\nu_A}{\nu_B}.

An absolute frequency measurement means a comparison whose chain ends at a realization of the SI second. It does not mean a measurement without a reference. An optical ratio can be much more precise than either optical frequency’s absolute calibration because it avoids a microwave link and several common uncertainty terms.

Phase endpoints and mean frequency

If a comparison system records $x_{A-B}(t)$ , the interval-average fractional-frequency difference is

\overline{y}_{A-B} = \frac{x_{A-B}(t_2)-x_{A-B}(t_1)} {t_2-t_1}.

For a complete, continuous record this endpoint estimator is insensitive to bounded phase noise in the long-average limit. In practice, gaps, outliers, phase wraps, and changing delays require a documented estimator. A linear fit can use all samples, but its weighting and response to nonstationary noise must be stated.

A phase step $\Delta x$ inside an interval of duration $T$ biases a simple endpoint frequency estimate by

\Delta\overline{y} = \frac{\Delta x}{T}.

Thus a $1\ \mathrm{ns}$ unrecorded step biases a one-day average by about $1.16\times10^{-14}$ . Small timing discontinuities are not small at the fractional-frequency levels of modern standards.

Frequency counters

A counter estimates phase change over a gate. An ideal rectangular, dead-time-free average is

\overline{\nu}_k = \nu_{\rm nom} + \frac{\phi(t_k+\tau)-\phi(t_k)} {2\pi\tau}.

Here $\phi$ is the phase deviation defined above; using total phase would remove the explicit $\nu_{\rm nom}$ term. Real counters can use different estimators and weighting functions. Common categories include:

rectangular or $\Pi$ -type averaging;
triangular or $\Lambda$ -type averaging from overlapped estimates;
regression-based frequency estimates; and
timestamp or phase-recorder outputs.

Two data streams labelled “one-second frequency” need not have the same transfer function. Counter mode affects white-phase-noise rejection, adjacent-sample correlations, and which Allan statistic is appropriate. A report should state the instrument, mode, gate time, dead time, input bandwidth, trigger level, and common timebase.

Common timebases and false agreement

If two counters share a timebase derived from one of the standards, the result can contain useful common-mode rejection, but it can also hide errors. A ratio counter that references both channels to the same oscillator does not independently validate that oscillator. Likewise, two nominally independent comb paths can share an RF reference, fiber, synthesizer, or software sign convention.

An out-of-loop comparison should be independent in the mechanism relevant to the claim. Independence is architectural, not merely a second line on a plot.

Optical ratios and closure

For standards $A$ , $B$ , and $C$ , a ratio network should satisfy

R_{AB}R_{BC}R_{CA}=1.

Define a logarithmic closure residual

\epsilon_{\rm cl} = \ln R_{AB}+\ln R_{BC}+\ln R_{CA}.

For small errors, $\epsilon_{\rm cl}$ is approximately the sum of fractional ratio errors. A nonzero residual reveals inconsistency but does not by itself identify which comb or link is wrong. Any three fixed clock frequencies, including three independently biased ones, yield ratios whose product is one. Closure tests independently measured ratio links; it cannot determine the individual clocks’ systematic shifts.

Absolute frequency through a flywheel

An intermittent optical standard can calibrate a continuous maser, which is then compared with UTC( $k$ ) or TAI. A schematic decomposition is

y_{\rm opt-SI} = y_{\rm opt-H} +y_{\rm H-UTC(k)} +y_{\rm UTC(k)-TAI} +y_{\rm TAI-SI}.

These additive fractional offsets use a common nominal-frequency normalization (after any known synthesis ratios); equivalently they are first-order approximations to multiplicative ratios. Every term must refer to compatible intervals and sign conventions. Interpolating or extrapolating the maser over optical-clock dead time adds uncertainty that depends on the maser’s noise process and the placement of the measurement epochs. Equal uptime fractions can produce different dead-time uncertainties if the gaps are distributed differently.

Stability Statistics

Why ordinary variance is often unsuitable

Clock noise commonly includes flicker frequency noise, random-walk frequency noise, and drift. The classical variance of frequency samples can depend strongly on record length and need not converge for these processes. Frequency metrology therefore uses differenced statistics that are finite for important power-law noise classes and explicitly depend on averaging time $\tau$ .

Stability is a property of a process and measurement bandwidth. A single number without $\tau$ , estimator, data treatment, and confidence information is incomplete.

Allan variance

For contiguous interval averages $\overline y_k(\tau)$ , the two-sample Allan variance is

\sigma_y^2(\tau) = \frac{1}{2} \left\langle \left[ \overline y_{k+1}(\tau) -\overline y_k(\tau) \right]^2 \right\rangle.

In terms of phase-time samples $x_k=x(t_0+k\tau)$ ,

\sigma_y^2(\tau) = \frac{1}{2\tau^2} \left\langle \left( x_{k+2}-2x_{k+1}+x_k \right)^2 \right\rangle.

The Allan deviation is $\sigma_y(\tau)$ . It characterizes fluctuation amplitude after averaging for time $\tau$ ; it is not the systematic uncertainty of the standard.

For a finite record with $M$ nonoverlapping frequency averages,

\widehat{\sigma}_y^2(\tau) = \frac{1}{2(M-1)} \sum_{k=0}^{M-2} \left( \overline y_{k+1}-\overline y_k \right)^2.

Overlapping estimators reuse the phase data to improve statistical confidence, especially at long $\tau$ . The terms are then correlated, so the number of plotted points is not the number of independent degrees of freedom.

Reading slopes

For ideal power-law noises, log–log Allan-deviation slopes provide a useful diagnostic:

Dominant noise	Typical phase/frequency description	Allan-deviation scaling
white phase modulation	independent fast phase measurement noise	approximately $\tau^{-1}$ , bandwidth dependent
flicker phase modulation	$1/f$ phase noise	approximately $\tau^{-1}$ with logarithmic/bandwidth dependence
white frequency modulation	independent frequency averages	$\tau^{-1/2}$
flicker frequency modulation	$1/f$ frequency noise	$\tau^0$
random-walk frequency modulation	integrated white frequency increments	$\tau^{+1/2}$
deterministic linear frequency drift	predictable slope in $y(t)$	$\tau^{+1}$ if not removed

These are model signatures, not automatic identifications. Mixed noise, filtering, periodic environmental modulation, data gaps, and servo features can mimic slopes over a limited range.

Modified Allan deviation

Modified Allan variance adds averaging before the two-sample difference. It is especially useful for distinguishing white from flicker phase modulation, which ordinary Allan deviation does not reliably separate by slope. Its value depends on the base sampling interval and estimator definition.

For phase samples separated by $\tau_0$ and $\tau=m\tau_0$ , one common overlapping form is

\operatorname{mod}\sigma_y^2(\tau) = \frac{1}{2m^2\tau^2} \left\langle \left[ \sum_{i=0}^{m-1} \left( x_{k+2m+i}-2x_{k+m+i}+x_{k+i} \right) \right]^2 \right\rangle.

Software packages differ in accepted input type, overlap convention, and normalization. A publication should name the implementation or state the formula.

Hadamard deviation

Hadamard variance uses a second difference of interval-average frequency, equivalently a third difference of phase:

\sigma_H^2(\tau) = \frac{1}{6} \left\langle \left( \overline y_{k+2} -2\overline y_{k+1} +\overline y_k \right)^2 \right\rangle.

It is insensitive to linear frequency drift. That makes it useful for characterizing clocks whose drift is substantial or difficult to remove without introducing analyst choices. It does not make drift irrelevant to prediction or holdover.

Time deviation

Time deviation, TDEV, expresses the dispersion of timing error rather than fractional frequency. Under the standard modified-Allan convention,

\operatorname{TDEV}(\tau) = \frac{\tau}{\sqrt{3}}\, \operatorname{mod}\sigma_y(\tau).

This relation is convention dependent through the definition of modified Allan deviation. TDEV is useful for synchronization and time-transfer applications, while Allan deviation is usually more direct for frequency standards.

Stability does not equal uncertainty of a mean

It is tempting to infer

u_A(\overline y) \stackrel{?}{=} \frac{\sigma_y(\tau_0)}{\sqrt{T/\tau_0}}.

That relation is valid only under assumptions close to independent white frequency noise. Flicker floors, random walk, drift, dead time, correlations, and estimator filtering change the uncertainty of the average. A defensible Type A evaluation uses the observed noise model, the actual sampling pattern, and sensitivity to analysis choices.

Three-cornered comparisons

A beat between $A$ and $B$ measures the sum of their noise contributions and any link noise. If three standards are mutually compared and their noises are uncorrelated, one may estimate

\sigma_A^2 = \frac{1}{2} \left( \sigma_{AB}^2 +\sigma_{AC}^2 -\sigma_{BC}^2 \right),

with cyclic expressions for $B$ and $C$ . Correlations, common reference noise, shared links, or estimator differences invalidate the simple decomposition and can even produce negative variance estimates. A three-cornered hat is a model-based inference, not a direct single-clock measurement.

Dead time and gaps

Dead time changes the statistical question. For white frequency noise, separated samples merely reduce the effective sample count. For flicker or random-walk frequency noise, unobserved evolution during gaps can dominate. The uncertainty depends on:

noise type and fitted coefficients;
gap lengths and locations;
interpolation or extrapolation algorithm;
calibration interval and endpoint constraints;
whether a continuously observed flywheel bridges the gaps; and
correlations between the flywheel and the comparison link.

Monte Carlo simulation from a validated noise model is often appropriate, but its result is only as trustworthy as the model, parameter estimates, and gap pattern supplied to it.

Primary, Secondary, and Working Standards

The SI Second

The SI second is defined by fixing the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom:

\Delta\nu_{\rm Cs} = 9\,192\,631\,770\ \mathrm{Hz}.

The definition refers to an unperturbed atom, and its practical realization requires corrections for environmental, motional, and relativistic effects. Optical standards can realize secondary representations with uncertainties below those of the best caesium fountains, but this does not make an optical transition the present defining transition.

Primary frequency standard

A primary frequency standard realizes the current definition of the second without calibration against another standard of the same quantity. For the present definition, operational primary frequency standards are caesium standards whose systematic effects have been evaluated to the standards required for primary operation.

Not every caesium clock is primary. Commercial beam clocks, local caesium ensembles, and caesium devices used as working standards generally do not become primary merely because they use the defining species.

Secondary representation and secondary frequency standard

The CIPM maintains recommended values for selected transitions that may be used as secondary representations of the second (SRS). The list includes optical transitions and the $^{87}\mathrm{Rb}$ ground-state hyperfine transition; recommended values and representation uncertainties are periodically adjusted from the international body of frequency data.

A secondary frequency standard (SFS) is an operational standard based on such a secondary representation and evaluated for use in metrology. When an SFS reports a calibration of TAI, its uncertainty includes both the standard’s realization uncertainty and the uncertainty assigned to the recommended representation.

The abbreviations describe different logical levels:

SRS is the recognized transition value and uncertainty;
SFS is an operating instrument realizing an SRS;
PFS is an operating instrument realizing the defining caesium transition as a primary standard.

Calling an optical transition “a secondary standard” without distinguishing the recommended representation from a particular apparatus obscures this structure.

Reference, transfer, and working standards

Laboratories also use a hierarchy based on function:

a reference standard calibrates other standards at a location;
a working standard is used routinely for instrument calibration;
a transfer standard or device carries a comparison between locations or systems; and
a travelling standard is physically transported for comparisons.

These roles do not map one-to-one onto PFS and SFS status. A hydrogen maser can be a laboratory reference or transfer oscillator without being a PFS or SFS. An optical SFS can calibrate a maser that then serves as the continuous working output.

For developments toward an optical definition, see optical-clock frontiers and the BIPM redefinition portal.

Metrological Traceability

What traceability means

The International Vocabulary of Metrology defines metrological traceability as a property of a measurement result: the result can be related to a reference through a documented, unbroken chain of calibrations, each contributing to the measurement uncertainty.

Four consequences are easy to miss:

traceability belongs to a result, not permanently to an instrument;
the reference, calibration procedure, epoch, and validity interval must be identified;
every link contributes uncertainty; and
traceability does not guarantee that the uncertainty is small enough for an application or that no mistake occurred.

A serial number and an old calibration sticker are not a traceability argument. The result must fall within the calibration’s stated conditions and interval, with drift, transport, adjustment, and measurement uncertainty accounted for.

Comparison versus calibration

A comparison estimates a difference or ratio between standards. It becomes a calibration when the comparison is used to establish the relation between an indication and a reference value, together with the associated uncertainty, and that relation is then used for a result or correction.

For a device under test (DUT) compared with a reference,

y_{\rm DUT-SI} = y_{\rm DUT-ref} +y_{\rm ref-SI}.

The combined standard uncertainty has the schematic form

u_c^2 ={} u_{\rm DUT-ref}^2 +u_{\rm ref}^2 +u_{\rm transfer}^2 +u_{\rm dead}^2 +u_{\rm drift}^2 + 2\sum_{i<j} \operatorname{cov}(y_i,y_j).

The quadrature-only expression is justified only when the terms are independent. Common synthesizers, gravitational models, transfer receivers, or reference calibrations produce covariance.

A local traceability chain

A practical laboratory chain might be

\begin{aligned} \text{device under test} &\longleftrightarrow \text{working standard} \\ &\longleftrightarrow \text{laboratory reference} \\ &\longleftrightarrow \mathrm{UTC}(k) \\ &\longleftrightarrow \mathrm{UTC} \\ &\longleftrightarrow \text{SI second}. \end{aligned}

The route need not be purely hierarchical. Optical ratios, multiple GNSS receivers, fiber comparisons, and independent PFS/SFS evaluations can form a network. Each branch relevant to the measurement model must be traceable, including environmental sensors when their uncertainties materially affect frequency corrections.

For institutes participating in the CIPM Mutual Recognition Arrangement, the continuous key comparison CCTF-K001.UTC publishes degrees of equivalence for UTC( $k$ ) realizations. Other recognized routes can pass through a participating NMI or designated institute with validated comparison results and published calibration and measurement capabilities.

Calibration interval

A calibration applies over the interval for which drift and environmental changes are adequately characterized. Suppose the fractional frequency immediately after calibration is $y_0$ and the frequency drift is $D=dy/dt$ . A simple deterministic prediction is

y(t) = y_0+Dt.

After holdover time $T$ , the accumulated time error is

\Delta x(T) = y_0T+\frac{1}{2}DT^2.

If $y_0$ and $D$ are uncertain and correlated,

u_x^2(T) ={} T^2u^2(y_0) +\frac{T^4}{4}u^2(D) + T^3\operatorname{cov}(y_0,D) +u_{\rm stoch}^2(T),

where $u_{\rm stoch}(T)$ represents stochastic oscillator noise and unmodelled influences. The calibration interval is chosen so that the result remains within the target uncertainty with suitable confidence.

Recalibration after repair, transport, loss of environmental control, firmware changes, or unexplained phase steps may be required even before the nominal interval expires.

Traceability through time data

For a continuous time comparison, frequency calibration can be inferred from the slope of a time-difference record. If Circular T gives $[\mathrm{UTC}-\mathrm{UTC}(k)]$ at epochs $t_1$ and $t_2$ , the mean fractional-frequency relation over the interval is approximately

\overline y_{\mathrm{UTC}-\mathrm{UTC}(k)} = \frac{ x_{\mathrm{UTC}-\mathrm{UTC}(k)}(t_2) - x_{\mathrm{UTC}-\mathrm{UTC}(k)}(t_1) }{t_2-t_1}.

The uncertainty is not obtained by dividing one endpoint’s time uncertainty by the interval. Endpoint uncertainties can be correlated, link noise can average nontrivially, and calibration changes can introduce discontinuities. The covariance of the published values and the transfer method must be considered.

International Atomic Time and UTC

Local real-time realizations

The BIPM computes international time scales from data submitted by laboratories. It does not distribute a single physical UTC cable signal to the world. National metrology institutes and observatories maintain real-time local realizations denoted UTC( $k$ ), where $k$ identifies the laboratory.

A UTC( $k$ ) system commonly combines:

an ensemble of continuously operating caesium clocks and masers;
a local time-scale algorithm;
one or more physical output generators;
steering toward UTC estimates;
GNSS and/or TWSTFT comparison equipment;
calibrated distribution systems; and
monitoring for phase steps, frequency changes, environmental events, and equipment faults.

The local scale exists in real time. The best estimate of its offset from UTC arrives later through BIPM products.

EAL

The Free Atomic Time Scale, EAL, is computed from an ensemble of clock data submitted by participating laboratories. Its clock weighting and prediction procedures seek long-term stability and robustness. EAL is “free” in the sense that its rate is not, at that stage, forced to agree with primary and secondary standards.

The ensemble contains continuously operating clocks useful for timekeeping. Intermittent state-of-the-art optical standards need not themselves provide continuous phase to improve the long-term accuracy of the international scale: they can calibrate its rate.

TAI

International Atomic Time, TAI, is obtained by steering the rate of EAL using evaluations from approved primary and secondary frequency standards. The steering is designed to make the scale interval of TAI conform to the SI second while preserving the stability and continuity of the ensemble.

Schematically,

y_{\rm TAI} = y_{\rm EAL}+d_{\rm steer},

where $d_{\rm steer}$ is derived from the available PFS/SFS evaluations. The actual BIPM algorithm uses time-distributed data, uncertainty weighting, and operational constraints; the equation only expresses the logical role of steering.

TAI is a postprocessed international reference scale. Laboratories do not wait for the monthly computation to generate local time; they predict and steer UTC( $k$ ), then compare it with the published solution.

UTC

Coordinated Universal Time, UTC, has the same rate as TAI and differs from it by an integral number of seconds. Leap seconds maintain the prescribed relation between UTC and Earth-rotation time according to the applicable international procedure.

Thus:

TAI supplies continuous atomic time;
UTC supplies the civil reference timescale with leap seconds;
UTC( $k$ ) supplies a physical local approximation in real time; and
a transmitted signal supplies a delayed and noisy realization of a local scale.

UTC is not the same thing as a GNSS system time, a computer’s local clock, or a timestamp received from an internet server.

Circular T

BIPM Circular T is the monthly publication that provides the official traceability of UTC( $k$ ) realizations to UTC. It reports

\left[ \mathrm{UTC}-\mathrm{UTC}(k) \right]

at five-day epochs for participating laboratories, together with time-link and frequency-standard information used in the computation.

For participating CIPM MRA institutes, the continuous key comparison CCTF-K001.UTC expresses the degree of equivalence as

D_k = \left[ \mathrm{UTC}-\mathrm{UTC}(k) \right]

with associated expanded uncertainty. Values are available at Modified Julian Dates ending in 4 or 9. The key comparison documents international equivalence; it is not a live synchronization service.

Rapid UTC

UTCr is a rapid BIPM solution published weekly with daily values for a subset of participating laboratories. It helps laboratories monitor and steer their local realizations with lower latency. BIPM states that UTCr is typically consistent with Circular T at the few-nanosecond level, but Circular T remains the unique source of traceability to UTC.

The practical distinction is:

use UTCr for timely operational guidance;
use Circular T for the final traceability statement.

Replacing the final data silently with rapid predictions understates the status and uncertainty of the result.

TT(BIPM)

TT(BIPM) is a deferred realization of Terrestrial Time computed annually from a weighted average of PFS and SFS evaluations of TAI. It is designed for scientific applications requiring excellent long-term frequency stability and accuracy.

TT(BIPM) is not an operational real-time signal and should not be confused with UTC. It can improve retrospective scientific analyses, such as pulsar timing, because later frequency-standard information can be incorporated.

Time scale and disseminated signal

The hierarchy from computation to a user can be written

\mathrm{UTC} \longrightarrow \mathrm{UTC}(k) \longrightarrow \text{laboratory output} \longrightarrow \text{transfer service} \longrightarrow \text{user}.

Each arrow adds latency, uncertainty, and possible discontinuities. A user receiving an internet timestamp is traceable only if the service, network model, local hardware, software corrections, and uncertainty evaluation support the claimed result. Naming UTC is not enough.

Reporting Primary and Secondary Standards to TAI

Evaluation intervals

Current BIPM guidance asks PFS/SFS evaluations intended for TAI steering to cover a multiple of five days and begin at an MJD ending in 4 or 9. The Circular T reporting interval is usually 30 days but can be 25 or 35 days. Submission timing matters because late data may not enter immediate TAI steering.

These requirements align local measurements with the international computation. They do not imply that five days is a universal optimum for every standard’s instability or dead-time uncertainty.

What the reported frequency represents

The reported local comparison is typically between the PFS/SFS and an intermediate reference such as UTC( $k$ ) or a clock contributing to TAI. To cancel that reference consistently in the later TAI calculation, the BIPM guidance uses an interval frequency based on endpoint phase:

\overline y = \frac{x_{\rm end}-x_{\rm begin}}{T}.

When operation is intermittent, a linear fit to individual frequency data may be used, but asymmetric sampling and dead time must be evaluated. An intermediate reference adjustment, time-link jump, or new link calibration during the interval must be avoided, measured, or explicitly corrected and reported.

Uncertainty terms

A useful decomposition for a PFS/SFS evaluation is

u_{\rm eval}^2 ={} u_A^2 +u_B^2 +u_{A/{\rm lab}}^2 +u_{B/{\rm lab}}^2 +u_{\rm TAI}^2 + \delta_{\rm SFS}\,u_{\rm SRep}^2 +2\sum_{i<j}\operatorname{cov}_{ij}.

Here:

$u_A$ describes statistical uncertainty from the standard’s instability;
$u_B$ combines systematic uncertainties of the standard, including the gravitational redshift correction;
$u_{A/{\rm lab}}$ includes statistical local-link uncertainty and dead-time uncertainty;
$u_{B/{\rm lab}}$ describes systematic local-link effects;
$u_{\rm TAI}$ describes the link from UTC( $k$ ) to TAI for the interval;
$u_{\rm SRep}$ is the recommended-representation uncertainty for an SFS; and
$\delta_{\rm SFS}$ is one for an SFS and zero for a PFS.

The names used in specific BIPM products can vary slightly, so an actual report should follow the current format rather than reconstruct it from this schematic equation. The covariance term is shown explicitly because blind quadrature is not valid for shared effects.

Uptime and Dead-Time Uncertainty

Uptime is the fraction of the full evaluation interval during which the standard-to-reference measurement actually contributes. Maintenance, parameter measurements, outages, and unusable data count as dead time.

A trustworthy report includes:

total evaluation interval and operating epochs;
uptime percentage;
a map of gaps, not only their total duration;
the flywheel noise model and its validation;
the method used to compute dead-time uncertainty;
sensitivity to plausible alternative noise coefficients; and
any steering or phase discontinuities in the intermediate reference.

Two reports with 50% uptime can have very different dead-time uncertainties: one might operate every other hour, while the other operates only during the first half of the month.

Gravity is part of the standard

A PFS/SFS report must refer the standard’s frequency to the adopted relativistic reference convention. If the gravitational potential at the clock is uncertain by $u(U)$ , the fractional uncertainty contribution is

u_{\rm grav} = \frac{u(U)}{c^2}.

Near Earth’s surface, a height-equivalent uncertainty $u(h)$ gives the rough estimate

u_{\rm grav} \simeq \frac{g\,u(h)}{c^2},

but precision work requires a potential model, reference system, tide and loading conventions, and survey uncertainty. A geometric height with an unspecified datum is not enough.

Frequency and Time Transfer

A One-Way Delay Model

Let $C_A(t)=t+x_A$ and $C_B(t)=t+x_B$ , with offsets nearly constant during propagation. A signal sent at coordinate time $t$ reaches $B$ at $t+d_{AB}$ . If the comparison subtracts the receiving clock reading from the transmitted timestamp, it measures

m_B=C_A(t)-C_B(t+d_{AB}) =x_A-x_B-d_{AB}.

Here $d_{AB}$ includes positive transmit, propagation, and receive delays. Clock offset and delay cannot be separated from this one observation. Slowly varying delays give, to first order,

\frac{dm_B}{dt}\simeq y_{A-B}-\frac{d d_{AB}}{dt}.

The opposite observable, receive timestamp minus send timestamp, reverses both signs. This is why a delay convention must be stated alongside a clock-offset convention.

One-way dissemination

One-way methods include broadcast radio, GNSS timing, and network services. They model the propagation delay using orbital, atmospheric, geometric, and hardware information. Their advantages are scalability and real-time availability. Their limits include:

transmitter and receiver hardware calibration;
antenna and cable delays;
orbit and satellite-clock errors;
ionospheric and tropospheric delay;
multipath and interference;
local coordinate and time-tag errors; and
dependence on the integrity of the transmitting system.

GNSS system time is an operational scale maintained by the constellation. The broadcast relation to UTC or a UTC( $k$ ) realization is another calibration layer.

Common-view comparison

In common view, stations $A$ and $B$ observe the same satellite or source over overlapping epochs. Here define each observation as the local receive timestamp minus satellite send timestamp, the opposite ordering to $m_B$ in the preceding example:

\begin{aligned} m_A &= x_A-x_S+d_{S\to A}+b_A+\epsilon_A,\\ m_B &= x_B-x_S+d_{S\to B}+b_B+\epsilon_B. \end{aligned}

Subtracting cancels the satellite-clock term:

m_A-m_B = x_A-x_B + (d_{S\to A}-d_{S\to B}) + (b_A-b_B) + (\epsilon_A-\epsilon_B).

Cancellation is strongest when observations are simultaneous and path errors are correlated. Differential propagation, receiver calibration, and site geometry remain. “Same satellite” does not mean “zero link uncertainty.”

All-in-view methods combine multiple satellites and can use precise orbit and clock products in postprocessing. They improve sampling and robustness but introduce data-processing and calibration dependencies.

Two-Way Transfer

Let $t_1$ be the timestamp of emission at $A$ , $t_2$ of reception at $B$ , $t_3$ of return emission at $B$ , and $t_4$ of return reception at $A$ . For offsets nearly constant across the exchange,

\begin{aligned} t_2-t_1&=d_{AB}-x_{A-B},\\ t_4-t_3&=d_{BA}+x_{A-B}. \end{aligned}

Assuming reciprocal delays leads to the estimator

\widehat{x}_{A-B} =\frac{(t_4-t_3)-(t_2-t_1)}{2} =x_{A-B}+\frac{d_{BA}-d_{AB}}{2}.

Thus reciprocal delay cancels, while the residual bias has the sign $(d_{BA}-d_{AB})/2$ for this clock-offset ordering. Unequal local hardware delays, uplink/downlink frequencies, motion, and asynchronous sampling require additional corrections. Two-way satellite time and frequency transfer (TWSTFT) applies this idea with a detailed satellite and station model.

Optical fiber links

An optical carrier sent through fiber accumulates phase noise from length fluctuations:

\phi_{\rm link}(t) = -\frac{2\pi\nu}{c} n_{\rm eff}\,\delta L(t) +\cdots.

This is the received phase deviation in the $\cos(2\pi\nu t+\phi)$ convention; the omitted terms include changes in refractive index. Actively stabilized links use a round-trip phase measurement to estimate and cancel one-way noise under a reciprocity model. Fiber links can support comparisons approaching optical-clock performance, but their uncertainty still depends on:

reciprocity and propagation delay;
uncompensated fiber at both ends;
interferometer offsets and thermal sensitivity;
optical amplifiers and regeneration stations;
cycle slips and loss of lock;
reference-plane definition; and
independent end-to-end verification.

The round trip samples the path at different times. Finite propagation delay limits cancellation bandwidth, and nonreciprocal effects do not cancel.

Free-space optical links

Free-space optical transfer can connect sites without fiber and can support moving or satellite platforms. Atmospheric turbulence, path interruption, pointing, terminal motion, asynchronous sampling, and relativistic coordinate transformations become central. A high carrier frequency does not by itself guarantee a small transfer uncertainty.

Internet and packet timing

Protocols such as NTP and PTP can provide valuable synchronization at application-dependent levels. Their performance is set by network-delay asymmetry, hardware timestamping, topology, queuing, path changes, oscillator holdover, and calibration. They should not be assigned the uncertainty of the upstream atomic standard without propagating the network and endpoint contributions.

For high-accuracy claims, “traceable to UTC” must include the realized uncertainty at the user’s reference plane, not merely the identity of the server.

Transfer calibration and closure

Hardware delays should be calibrated with a method appropriate to the system:

common-clock tests place two systems on the same source;
travelling receivers compare remote hardware against a calibrated unit;
loopback and round-trip tests probe reciprocity;
redundant links expose jumps and long-term drift; and
link closure around a network tests consistency.

A common-clock test removes true clock difference, so the measured residual estimates link and equipment contributions. It does not validate effects that are absent in the common-clock geometry, such as long-baseline atmospheric differences.

Steering, Holdover, and Availability

Steering is estimation plus control

A local scale estimates its offset and rate relative to a reference and applies corrections. A simple state model is

\mathbf z_k = \begin{pmatrix} x_k\\ y_k\\ D_k \end{pmatrix}, \qquad \mathbf z_{k+1} = \begin{pmatrix} 1&T&T^2/2\\ 0&1&T\\ 0&0&1 \end{pmatrix} \mathbf z_k + \mathbf w_k.

Measurements of time difference update the state, and a controller converts the estimate into phase or frequency commands. The process-noise model $\mathbf w_k$ embodies oscillator fluctuations and model uncertainty.

Aggressive steering follows a noisy reference and can degrade short-term stability. Weak steering preserves the flywheel but allows larger offsets. The optimum depends on the reference latency, link noise, oscillator noise, continuity requirements, and target application.

Phase steps and frequency steps

A phase step changes $x$ discontinuously. A frequency step changes its slope. Both can be legitimate controls, but the record must distinguish them. Hidden phase steps corrupt endpoint frequency estimates and can invalidate coherent applications.

When a local scale must remain continuous, an offset is often removed by a bounded frequency correction:

y_{\rm steer} \simeq -\frac{x_{\rm target}}{T_{\rm steer}}.

The chosen steering interval trades correction speed against frequency disturbance.

Holdover budget

A holdover statement should specify:

epoch and source of the last calibration;
estimated frequency offset and drift with covariance;
stochastic noise model and environmental sensitivities;
outage duration;
prediction algorithm;
phase-continuity status; and
confidence interval for both time and frequency error.

“The oscillator is stable to $10^{-13}$ ” is not a holdover budget. It omits averaging time, drift, calibration uncertainty, and how the statistic maps to a future prediction.

Availability is metrological

Availability affects more than convenience. Intermittent operation changes dead-time uncertainty; restarts can introduce phase ambiguity; relocking can change the servo operating point; and environmental settling can alter systematic corrections.

Useful availability metrics include:

fraction of valid operation;
mean and maximum outage duration;
phase continuity through outages;
relock time and post-relock validation;
fraction of time within a stated uncertainty; and
latency of the calibration reference.

A highly accurate standard that runs briefly can still be scientifically decisive, provided a characterized flywheel and comparison chain bridge the gaps.

Common Pitfalls

A delay sign is part of the observable. A longer cable delays the received phase and lowers its frequency while the delay grows. A timestamp defined as receive time minus send time has a positive delay term instead. Write the actual events before assigning a correction sign.

Allan deviation is neither an absolute calibration nor automatically an uncertainty of the mean. A constant offset cancels from the Allan difference. Colored noise, gaps, and drift determine how uncertainty in an interval average differs from a simple white-noise extrapolation.

A computed time scale and its physical realization differ. UTC is computed internationally; UTC( $k$ ) is maintained locally in real time. The received signal also includes the distribution link, so its uncertainty cannot be inferred just from the upstream clock.

Duty fraction is not a complete sampling model. Concentrating six hours of observations at one end of a day requires more extrapolation than distributing them across the day when the flywheel drifts. Use the actual measurement epochs and a tested oscillator model.

Further Connections

Atomic clocks develops the interrogation loop, oscillator aliasing, and frequency-shift evaluation.
Optical clocks compares ion and lattice standards and explains relativistic frequency comparisons.
Atomic clocks in quantum estimation treats oscillator tracking and quantum resources before calibration.
Frequency combs derives the signed frequency combinations used for optical comparison and division.
Variation of constants searches interprets long frequency-ratio records as tests of dimensionless parameters.

References

D. W. Allan, “Statistics of atomic frequency standards,” Proceedings of the IEEE 54, 221–230 (1966), doi:10.1109/PROC.1966.4634.
Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition, version 4.01 (June 2026), doi:10.59161/AUEZ1291.
Bureau International des Poids et Mesures, “Time Metrology,” official descriptions of UTC, UTCr, TT(BIPM), UTC( $k$ ), and BIPM time products (reviewed 25 July 2026).
Bureau International des Poids et Mesures, Circular T, monthly source of traceability from UTC( $k$ ) to UTC (reviewed 25 July 2026).
Bureau International des Poids et Mesures, “Rapid UTC (UTCr),” weekly rapid solution with daily values (reviewed 25 July 2026).
Bureau International des Poids et Mesures, “TT(BIPM),” deferred realization of Terrestrial Time (reviewed 25 July 2026).
Bureau International des Poids et Mesures, CCTF-K001.UTC key-comparison record, measurand $[\mathrm{UTC}-\mathrm{UTC}(k)]$ (reviewed 25 July 2026).
Bureau International des Poids et Mesures, Guidelines for Reporting Primary (PFS) or Secondary (SFS) Frequency Standards Data for TAI Calibration, CCTF-WGPSFS guidance, September 2024, official PDF.
Bureau International des Poids et Mesures, “Recommended Values of Standard Frequencies,” including secondary representations of the second and the 2025 adjustment (reviewed 25 July 2026), doi:10.59161/StdFreq2026.
Bureau International des Poids et Mesures, “Roadmap to the Redefinition of the Second,” current process and possible 2030 schedule (reviewed 25 July 2026).
CCTF Working Group on the CIPM MRA, CCTF Criteria for Obtaining Traceability in Time and Frequency, Guideline 9 (June 2017), official PDF.
Consultative Committee for Time and Frequency, “Recommendations for Operating, Comparing and Reporting Frequency Standards as Secondary Representations of the Second,” Recommendation 1 (2017), doi:10.59161/CCTF2017REC1E.
Consultative Committee for Time and Frequency, “Steering of International Atomic Time,” Recommendation 3 (2004), doi:10.59161/CCTF2004REC3E.
N. Dimarcq et al., “Roadmap towards the redefinition of the second,” Metrologia 61, 012001 (2024), doi:10.1088/1681-7575/ad17d2.
Joint Committee for Guides in Metrology, International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM), JCGM 200:2012, doi:10.59161/JCGM200-2012.
Joint Committee for Guides in Metrology, Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement, JCGM 100:2008, doi:10.59161/JCGM100-2008E.
Joint Committee for Guides in Metrology, Guide to the Expression of Uncertainty in Measurement—Part 1: Introduction, JCGM GUM-1:2023, doi:10.59161/JCGMGUM-1-2023.
J. Levine, “Introduction to time and frequency metrology,” Review of Scientific Instruments 70, 2567–2596 (1999), doi:10.1063/1.1149844.
J. Levine, “A review of time and frequency transfer methods,” Metrologia 45, S162–S174 (2008), doi:10.1088/0026-1394/45/6/S22.
J. Levine, “Distributing time and frequency information,” in Position, Navigation, and Timing Technologies in the 21st Century (Wiley-IEEE Press, 2021), doi:10.1002/9781119458449.ch29.
A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Reviews of Modern Physics 87, 637–701 (2015), doi:10.1103/RevModPhys.87.637.
G. Panfilo and F. Arias, “The Coordinated Universal Time (UTC),” Metrologia 56, 042001 (2019), doi:10.1088/1681-7575/ab1e68.
F. Riehle, Frequency Standards: Basics and Applications (Wiley-VCH, 2004), doi:10.1002/3527605991.
W. J. Riley and D. A. Howe, Handbook of Frequency Stability Analysis, NIST Special Publication 1065 (2008), doi:10.6028/NIST.SP.1065.
T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002), doi:10.1038/416233a.

Exercises

1. Classify a frequency laboratory

A laboratory contains:

a quartz oscillator that supplies a continuous $10\ \mathrm{MHz}$ ;
a caesium fountain operated for ten days each month;
a hydrogen maser continuously compared with the fountain;
a synthesizer generating the laboratory output;
a GNSS receiver comparing that output with UTC( $k$ ); and
software combining several local clocks into a time scale.

Classify the roles of these components. Explain why neither “the caesium clock is the output” nor “the GNSS receiver is the standard” is generally correct.

Solution

The quartz oscillator is an oscillator and might be a working source, but its role depends on whether it is disciplined. The caesium fountain is an intermittent atomic reference and may be a PFS only if it realizes the SI definition with the required systematic evaluation and operational acceptance. The hydrogen maser is the continuous flywheel and can be the laboratory reference oscillator. The synthesizer maps the flywheel to the defined output plane and is part of the standard’s realization.

The GNSS receiver is a transfer device. It compares the local output with a remote time reference through a propagation and hardware-delay model. It is not by itself the remote standard and does not inherit that standard’s uncertainty without link calibration.

The software realizes an ensemble time scale from clock-difference data, prediction, weighting, and steering. Its physical output still requires an oscillator and generator.

The statement “the caesium clock is the output” ignores intermittent operation, the flywheel, synthesis, and output plane. “The GNSS receiver is the standard” ignores the remote scale, path, receiver delays, and calibration chain.

2. Mean frequency from phase endpoints

At the beginning and end of a $5$ -day comparison, the measured phase-time difference $x_{A-B}$ is

x_{A-B}(t_1)=12.4\ \mathrm{ns}, \qquad x_{A-B}(t_2)=17.8\ \mathrm{ns}.

Find the mean fractional-frequency difference $\overline y_{A-B}$ .
Halfway through the interval, an unrecorded $0.80\ \mathrm{ns}$ phase step occurred in channel $A$ . What was the physical mean frequency difference before that measurement artifact?
Why does the step epoch not matter for the endpoint estimator but matter for gap handling and diagnostic fits?

Solution

The interval is

T = 5(86\,400\ \mathrm{s}) = 432\,000\ \mathrm{s}.

The observed endpoint change is

\Delta x_{\rm obs} = 5.4\ \mathrm{ns}.

Therefore

\overline y_{A-B}^{\rm obs} = \frac{5.4\times10^{-9}}{432\,000} = 1.25\times10^{-14}.

The positive phase step contributed $0.80\ \mathrm{ns}$ to the endpoint change. Removing it gives

\Delta x_{\rm phys} = 4.6\ \mathrm{ns},

and hence

\overline y_{A-B}^{\rm phys} = \frac{4.6\times10^{-9}}{432\,000} \simeq 1.06\times10^{-14}.

Any step inside the endpoint interval changes the final minus initial phase by the same amount, independent of its epoch. Its epoch matters for local frequency estimates, linear fits, Allan statistics, interpolation across gaps, and attribution to equipment events.

3. Allan deviation from a short record

Four contiguous $100\ \mathrm{s}$ fractional-frequency averages are

\overline y_k = (1,\ 3,\ 2,\ 6)\times10^{-13}.

Compute the nonoverlapping Allan deviation at $\tau=100\ \mathrm{s}$ . State what the result does and does not tell you.

Solution

The adjacent differences in units of $10^{-13}$ are

2,\ -1,\ 4.

The finite-sample Allan variance is

\begin{aligned} \widehat{\sigma}_y^2 &= \frac{1}{2(4-1)} \left[ 2^2+(-1)^2+4^2 \right]10^{-26} \\ &= \frac{21}{6}\times10^{-26} = 3.5\times10^{-26}. \end{aligned}

Thus

\widehat{\sigma}_y(100\ \mathrm{s}) = \sqrt{3.5}\times10^{-13} \simeq 1.87\times10^{-13}.

This estimates instability at one averaging time for this estimator and very short record. It does not determine the standard’s systematic uncertainty, identify the noise process, establish long-term behavior, or provide a precise confidence interval. Three squared differences provide few effective degrees of freedom.

4. Holdover with offset, drift, and covariance

A flywheel begins a holdover interval with

y_0=(2.0\pm0.5)\times10^{-14}

and drift

D=(1.0\pm0.3)\times10^{-20}\ \mathrm{s}^{-1}.

The correlation coefficient between $y_0$ and $D$ is $\rho=-0.60$ . Ignore additional stochastic noise.

Find the predicted time error after $T=10$ days.
Find its standard uncertainty including covariance.
Compare with the result obtained by ignoring covariance.

Solution

The duration is

T = 864\,000\ \mathrm{s}.

The predicted time error is

\begin{aligned} \Delta x &= y_0T+\frac{1}{2}DT^2 \\ &= (2.0\times10^{-14})(864\,000) + \frac{1}{2}(1.0\times10^{-20})(864\,000)^2 \\ &\simeq 1.728\times10^{-8} + 3.732\times10^{-9}\ \mathrm{s} \\ &\simeq 21.0\ \mathrm{ns}. \end{aligned}

Let

u(y_0)=0.5\times10^{-14}, \qquad u(D)=0.3\times10^{-20},

\operatorname{cov}(y_0,D) = \rho\,u(y_0)u(D) = -9.0\times10^{-36}\ \mathrm{s}^{-1}.

Propagation gives

u_x^2 ={} T^2u^2(y_0) +\frac{T^4}{4}u^2(D) + T^3\operatorname{cov}(y_0,D).

Numerically, the three terms are approximately

(4.32\ \mathrm{ns})^2, \qquad (1.12\ \mathrm{ns})^2, \qquad -5.80\ \mathrm{ns}^2.

Therefore

u_x \simeq \sqrt{14.1115}\ \mathrm{ns} \simeq 3.76\ \mathrm{ns}.

Ignoring covariance would give

u_x^{(\rho=0)} \simeq \sqrt{18.6624+1.2538}\ \mathrm{ns} \simeq 4.46\ \mathrm{ns}.

The negative covariance reduces the prediction uncertainty. In a real holdover budget, stochastic maser noise and environmental model uncertainty must also be included.

5. Combining PFS and SFS Uncertainties

An optical SFS evaluation reports the following independent standard uncertainties, in units of $10^{-16}$ :

Contribution	Value
statistical standard uncertainty $u_A$	1.2
standard systematic uncertainty $u_B$	1.5
local statistical and dead-time term $u_{A/{\rm lab}}$	2.0
local systematic link term $u_{B/{\rm lab}}$	0.8
link-to-TAI term $u_{\rm TAI}$	2.5
SRS representation term $u_{\rm SRep}$	1.8

Compute the combined standard uncertainty.
What would change if the apparatus were a caesium PFS with the same other terms?
Suppose $u_{A/{\rm lab}}$ and $u_{\rm TAI}$ have correlation $\rho=0.30$ . Recompute the SFS uncertainty.

Solution

For independent terms,

\begin{aligned} u_{\rm SFS} &= \sqrt{ 1.2^2+1.5^2+2.0^2+0.8^2+2.5^2+1.8^2 }\times10^{-16} \\ &= \sqrt{17.82}\times10^{-16} \\ &\simeq 4.22\times10^{-16}. \end{aligned}

For a PFS, there is no SRS representation term because the standard directly realizes the defining caesium transition:

u_{\rm PFS} = \sqrt{17.82-1.8^2}\times10^{-16} \simeq 3.82\times10^{-16}.

With correlation, add

2\rho\,u_{A/{\rm lab}}u_{\rm TAI} = 2(0.30)(2.0)(2.5) = 3.0

in squared units. Thus

u_{\rm SFS,corr} = \sqrt{20.82}\times10^{-16} \simeq 4.56\times10^{-16}.

The example is schematic: actual BIPM reporting categories and covariance must follow the current guidance and the physical comparison chain.

6. Signed comb synthesis

An optical standard is measured by a comb with

f_{\rm rep}=250.000\,000\ \mathrm{MHz}, \qquad f_{\rm CEO}=20.000\,000\ \mathrm{MHz}.

The nearest tooth has $n=1\,716\,000$ . The optical beat is $f_b=35.000\,000\ \mathrm{MHz}$ and increases when the optical frequency increases, so $s_b=+1$ . An acousto-optic modulator between the atoms and the comb shifts the light seen by the atoms upward by $f_{\rm AOM}=80.000\,000\ \mathrm{MHz}$ relative to the comb-side light.

Find the comb-side optical frequency.
Find the atomic-reference frequency.
Show the error caused by choosing the wrong beat sign.

Solution

The comb-side continuous-wave frequency is

\begin{aligned} \nu_{\rm cw} &= n f_{\rm rep}+f_{\rm CEO}+f_b \\ &= (1\,716\,000)(250.000\,000\ \mathrm{MHz}) +20.000\,000\ \mathrm{MHz} +35.000\,000\ \mathrm{MHz} \\ &= 429\,000\,055.000\,000\ \mathrm{MHz} \\ &= 429.000\,055\,000\,000\ \mathrm{THz}. \end{aligned}

Because the atoms see an upward AOM shift,

\nu_{\rm atom} = \nu_{\rm cw}+f_{\rm AOM} = 429.000\,135\,000\,000\ \mathrm{THz}.

Using $s_b=-1$ would change the inferred comb-side frequency by

\Delta\nu_{\rm sign} = -2f_b = -70\ \mathrm{MHz},

a conspicuous absolute error but one that software can propagate consistently unless the sign is independently tested by changing the optical frequency.

7. Common-view and two-way residuals

Two laboratories observe one satellite in common view. Their modelled one-way residual delays are

\delta d_A=1.8\ \mathrm{ns}, \qquad \delta d_B=0.7\ \mathrm{ns},

and their receiver-delay calibration errors are

\delta b_A=0.4\ \mathrm{ns}, \qquad \delta b_B=-0.2\ \mathrm{ns}.

What residual bias enters the common-view clock difference $x_A-x_B$ ?
In a two-way exchange, the residual path delays are $\delta d_{A\to B}=1.1\ \mathrm{ns}$ and $\delta d_{B\to A}=0.5\ \mathrm{ns}$ . What path bias remains?
Explain the assumptions behind both cancellations.

Solution

For common view, the satellite-clock term cancels, but differential path and receiver delays remain:

\begin{aligned} \delta x_{\rm CV} &= (\delta d_A-\delta d_B) + (\delta b_A-\delta b_B) \\ &= (1.8-0.7+0.4-(-0.2))\ \mathrm{ns} \\ &= 1.7\ \mathrm{ns}. \end{aligned}

For the simplified two-way estimator, the path bias is half the nonreciprocity:

\delta x_{\rm 2w} = \frac{1}{2} (0.5-1.1)\ \mathrm{ns} = -0.30\ \mathrm{ns}.

Common view assumes sufficiently simultaneous observation and correlated source/path errors so that the common satellite term and shared errors cancel. Differential geometry, atmosphere, and hardware remain.

Two-way cancellation assumes reciprocal propagation and consistent timestamp epochs. Direction-dependent hardware delay, different uplink and downlink paths or frequencies, motion, and nonreciprocal media remain.

8. Design a traceable calibration

A research group must calibrate a $100\ \mathrm{MHz}$ synthesizer output with target standard uncertainty $5\times10^{-14}$ over a one-day interval. It has access to:

a local maser;
an optical SFS that runs for six hours;
a calibrated GNSS receiver linked to UTC( $k$ );
a phase recorder; and
two independent distribution paths.

Design the measurement and reporting plan. Identify the measurand, measurement equation, dead-time problem, traceability route, validation tests, and measurements needed to decide whether the uncertainty target is achieved.

Solution

One defensible measurand is the one-day average fractional-frequency difference of the synthesizer’s defined $100\ \mathrm{MHz}$ output plane from the SI second:

\overline y_{\rm synth-SI} = \overline y_{\rm synth-H} + \overline y_{\rm H-UTC(k)} + \overline y_{\rm UTC(k)-UTC} + \overline y_{\rm UTC-SI}.

The optical SFS supplies an additional calibration of the maser:

\overline y_{\rm H-SI} = -\overline y_{\rm opt-H} +\overline y_{\rm opt-SI},

over its six-hour operating epochs. A noise model for the maser then bridges the remaining 18 hours. The final estimator should combine the optical and UTC( $k$ ) routes with covariance rather than treating them as automatically independent.

The plan should:

define the output connector, cable state, sign convention, UTC day, and endpoint estimator;
record continuous phase from the synthesizer and maser on both distribution paths with dead-time-free timestamping;
operate the optical SFS in epochs distributed across the day if possible, reducing extrapolation compared with one six-hour block;
validate the maser noise model over comparable intervals and compute dead-time uncertainty using the actual optical operating schedule;
obtain final Circular T data for the UTC( $k$ )-to-UTC relation rather than substituting UTCr as the final traceability source;
include GNSS receiver, antenna, cable, and hardware-delay calibration uncertainties;
include SFS systematic, statistical, local-link, SRS representation, and gravitational terms;
compare the two distribution paths, search for phase steps, and perform a common-source counter test;
verify synthesizer ratio and phase continuity independently; and
propagate all terms and covariances before checking whether $5\times10^{-14}$ is achieved.

The report should include raw phase sampling, actual SFS uptime, gap map, counter mode, all corrections, noise-model checks, calibration certificates, Circular T issue, link data, anomalies, and the complete uncertainty budget. If the combined result exceeds the target, traceability still exists, but the intended uncertainty claim is not supported.