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I. H. McLaren coauthored with W. C. Wiley the 1955 Bendix study of time-of-flight mass spectrometry with improved mass resolving power. A notable part of their analysis was time-lag focusing: waiting between ion formation and application of an accelerating pulse lets initial velocities become position differences that can compensate some arrival-time spread.^[ref]

A first-order calculation exposes the mechanism. Let $T(v,s)$ be arrival time after immediate extraction for initial axial velocity $v$ and position $s$. During a field-free delay $\tau$, an ion moves to $s+v\tau$. Apart from the common delay added to all times,The partial derivatives are evaluated at $v=0$ and the reference position. First-order sensitivity to $v$ vanishes ifA positive delay requires opposite signs of these sensitivities. For example, $T_v=2\times 10^{-9}{\mathrm{s}}^2{\mathrm{m}}^{-1}$ and $T_s=-10^{-3}\mathrm{s}{\mathrm{m}}^{-1}$ give $\tau =2\mu \mathrm{s}$. These values illustrate the cancellation, rather than reproducing an instrument setting from the paper.

This also reveals the tradeoff: the cancellation needs a nonzero position sensitivity, whereas ideal first-order space focusing sets $T_s=0$. The paper's delayed-extraction condition and measured spectra therefore address competing sources of broadening, not a universal elimination of every initial-condition effect.^[ref]

Space focusing in time-of-flight mass spectrometry reduces differences in arrival time caused by the initial positions of otherwise equivalent ions. An ion starting farther from the exit of an accelerating region spends longer there but can acquire greater kinetic energy, allowing it to catch up during the drift. Electrode geometry and electric fields are chosen so that these effects compensate at the detector.^[ref]

A simple idealized derivation makes the compensation explicit. Consider positive ions of mass $m$ and electric charge $q$, initially at rest, with distance $a$ remaining in a uniform accelerating electric field $E$. Let $L$ be the field-free distance beyond its exit. Their total flight time isThe first term is acceleration time; the second is drift time. Setting the derivative $dt/da$ to zero at a reference position $a_0$ givesFor example, $a_0=1\mathrm{cm}$ gives a first-order space focus at $L=2\mathrm{cm}$ in this simplified single-field model. Nearby position errors then affect arrival time only at higher order in the Taylor expansion. Practical two-field sources allow the focus to be placed at more useful drift distances. Initial velocity spread is a separate contribution to peak width, motivating time-lag focusing.

Time-lag focusing in time-of-flight mass spectrometry uses a controlled delay between creation of an ion packet and application of its extraction electric field. During this delay, different initial velocities produce different positions. The later extraction pulse can then give initially faster ions less accelerating work than initially slower ones, reducing their arrival-time spread at a chosen detector plane.^[ref]

Suppose two equal-mass, equal-electric charge ions differ in their initial forward velocity by $\Delta u$. A field-free delay $\tau$ creates a position differenceFor example, $\Delta u=100\mathrm{m}{\mathrm{s}}^{-1}$ and $\tau =1\mu \mathrm{s}$ give $\Delta x=100\mu \mathrm{m}$. In a subsequently applied uniform field, an ion farther toward the exit experiences a smaller remaining accelerating electric potential difference. Choosing the delay and fields appropriately allows the slower-starting ion to catch up later.

The delay creates a useful correlation between position and velocity; waiting alone does not focus the packet. Space focusing compensates an initial position distribution, whereas time-lag focusing exploits motion during the delay to address an initial velocity distribution. An instrument's focusing condition must state the geometry, fields, reference mass-to-charge ratio and initial-distribution assumptions.

A time-of-flight mass spectrometer is a mass spectrometer whose analyzer infers mass-to-charge ratio from ion arrival times. A simple arrangement uses an ion source, pulsed acceleration, a drift region under vacuum and a mass spectrometer detector. The 1955 Wiley–McLaren instrument connected such an apparatus to calculations of how initial position and velocity spread affect resolution.^[ref]

In an ideal drift, $t_{\mathrm{drift}}=L\sqrt{m/(2qU)}$ for flight length $L$, positive charge $q$ and accelerating electric potential difference $U$. Recorded arrival time can also include a time-of-flight timing offset $t_0$. A useful ideal instrument calibration model iswhere $x$ is the numerical mass-to-charge coordinate in a declared unit and $k$ absorbs the instrument geometry and unit conversion.

For original invented standards, let $x=25$ arrive at $12\mu \mathrm{s}$ and $x=100$ at $22\mu \mathrm{s}$. Since the square roots are 5 and 10, $k=2\mu \mathrm{s}$ and $t_0=2\mu \mathrm{s}$. An unknown arriving at $17\mu \mathrm{s}$ then has $x=((17-2)/2{)}^2=56.25$. Ignoring the offset and scaling from the first standard alone would instead give $25(17/12{)}^2\approx 50.17$. A timing offset becomes a mass-assignment error when omitted from the model.

Real peaks have finite widths. Space focusing and time-lag focusing address specified sources of that spread; neither is simply a longer wait before detection. Calibration standards, acquisition timing and focusing conditions therefore belong in the interpretation of a measured spectrum.

A time-of-flight timing offset, $t_0$, is an additive time-origin parameter in a model connecting recorded ion arrival time with physical flight time. In a simple time-of-flight mass spectrometer calibration,where $x$ is the numerical mass-to-charge ratio in declared units. A 2017 instrument-analysis paper uses this equation and determines instrument-dependent constants from identified isotope peaks.^[ref]

An original error calculation shows why the offset matters. Suppose the physical flight contribution is $20\mu \mathrm{s}$ and $t_0=3\mu \mathrm{s}$, so the recorded time is $23\mu \mathrm{s}$. With $k_0$ held fixed, ignoring the offset multiplies the inferred coordinate bya $32.25\%$ overestimate. This deliberately large example isolates the effect; it does not describe a measured instrument error.

With $k_0$ fixed, a small error in the fitted offset propagates approximately asAn offset uncertainty of $0.05\mu \mathrm{s}$ at a corrected flight time of $20\mu \mathrm{s}$ therefore contributes about $0.5\%$ relative uncertainty in this approximation. The same absolute offset error has greater relative impact on earlier arrivals.

This parameter concerns timing, whereas time-lag focusing is a physical delayed-extraction technique affecting the ion packet. Fitting an offset does not by itself sharpen a peak. More detailed instruments can require additional calibration terms; the stated equation is a model, not a universal transfer function.

W. C. Wiley worked on time-of-flight mass spectrometry at Bendix Aviation Corporation Research Laboratories. With I. H. McLaren, Wiley published a 1955 instrument study addressing the arrival-time spread of ions with the same mass-to-charge ratio. Their apparatus combined an ion-producing source, accelerating electric fields, an evacuated drift region and a collector; its analysis separated broadening from initial position and initial motion.^[ref]

The basic timing principle can be derived without the focusing details. An initially stationary ion of positive electric charge $q$ accelerated through an electric potential difference $V$ acquires kinetic energy $qV=m{v}^2/2$. Over a field-free drift length $L$,For a fixed charge state and instrument setting, differentiating $m\propto t^2$ gives the approximate mass resolving powerFor illustration, a $20\mu \mathrm{s}$ arrival time with a $20\mathrm{ns}$ timing width corresponds to $R\simeq 500$, if that width is the relevant small peak-width measure.

The focusing problem addressed by Wiley and I. H. McLaren is therefore substantive: a spread of initial conditions can broaden a peak even when the detector timing is precise. Their work on space focusing and time-lag focusing changes the ion trajectories to reduce that broadening, rather than merely measuring the same broad peak with a faster clock.