Gradients, Slice Selection & Spatial Encoding

From One Signal to a Map of Space

A spinning sample in a perfectly uniform field $B_0$ emits a single resonance at one frequency, $\omega_0 = \gamma B_0$. The receiver coil integrates contributions from every excited proton in the body and reports one number per instant in time. There is no information about where any of that signal came from. The entire enterprise of MR imaging is the trick of making frequency and phase depend on position, so that a measurement in time can be inverted into a measurement in space.

The instrument that accomplishes this is the gradient coil: a set of windings that superimpose a small, spatially linear field on top of $B_0$. By controlling the amplitude and timing of three orthogonal gradients, we select a slice, encode one in-plane axis by frequency, and encode the orthogonal axis by phase. This lesson derives each of those operations from the Larmor relation and shows how gradient area, not gradient amplitude alone, is the master variable that fixes field of view and resolution.

Linear Gradients and the Position-Dependent Larmor Frequency

A magnetic field gradient is a controlled spatial variation of the $z$-component of the main field. We define three gradient amplitudes, $G_x = \partial B_z / \partial x$, $G_y = \partial B_z / \partial y$, and $G_z = \partial B_z / \partial z$, collected into a vector $\mathbf{G} = (G_x, G_y, G_z)$ with units of millitesla per meter (mT/m). The total longitudinal field at position $\mathbf{r}$ is the base field plus the linear ramp.

Because the Larmor frequency is proportional to the local field, the resonance frequency now carries spatial information. Substituting Eq. 3.1 into $\omega = \gamma B$ gives the foundational equation of spatial encoding.

Slice Selection

Selective excitation confines the flip to a thin slab. We apply a slice-select gradient (call it $G_z$ for an axial slice) at the same time as a shaped radiofrequency pulse. The gradient spreads the Larmor frequency linearly along $z$, so each plane resonates at its own frequency. An RF pulse contains only a finite band of frequencies, $\Delta f$ (its transmit bandwidth), so it can only tip the spins whose Larmor frequency falls inside that band.

Set the bandwidth condition equal to the gradient-induced spread across the slice. Spins from $z_0$ to $z_0 + \Delta z$ span a frequency range $\Delta f = (\gamma / 2\pi)\,G_z\,\Delta z$. Solving for the slab thickness gives the master slice equation.

Two practical levers follow directly. To get a thinner slice, either narrow the RF bandwidth $\Delta f$ or increase the gradient $G_z$. To move the slice off isocenter to position $z_0$, shift the RF center frequency by $\Delta f_0 = (\gamma/2\pi)\,G_z\,z_0$, with no change to the gradient. Worked example: with $\gamma/2\pi = 42.58$ MHz/T, an RF bandwidth of 1000 Hz and $G_z = 10$ mT/m give $\Delta z = 1000 / (42.58 \times 10^6 \cdot 0.010) = 2.35 \times 10^{-3}$ m, about 2.3 mm.

Slice Profile and the Time-Bandwidth Product

The slice profile (signal versus $z$) is approximately the Fourier transform of the RF envelope. A rectangular profile would require an infinitely long sinc pulse, so real pulses are truncated sinc shapes that trade off pulse duration against profile sharpness. The governing dimensionless quantity is the time-bandwidth product $\mathrm{TBW} = \tau_{RF}\,\Delta f$, where $\tau_{RF}$ is the pulse duration. Typical excitation pulses use $\mathrm{TBW} \approx 2$ to $8$; higher TBW gives sharper slice edges at the cost of a longer pulse and more specific absorption rate.

The Slice-Refocusing Lobe

During the second half of a symmetric RF pulse, spins across the slice accrue different phases because $G_z$ is still imposing a position-dependent frequency. By the end of excitation the transverse magnetization is dephased across the slice, which would reduce the net signal. The fix is a negative gradient lobe applied immediately after, whose area is approximately half the area of the slice-select lobe (for a symmetric pulse, the effective excitation occurs near the pulse center). This refocusing lobe rewinds the through-slice phase to zero, restoring coherent signal. Forgetting it, or mis-setting its area, causes a global signal loss that is uniform across the image, an easy artifact to misread as low gain.

Frequency (Readout) Encoding

After exciting a slice we must localize signal within the slice plane. The first in-plane axis is encoded by switching on a readout gradient (say $G_x$) while the signal is being sampled. Each column of spins at position $x$ then precesses at its own offset frequency $\Delta\omega(x) = \gamma\,G_x\,x$. The received signal is a superposition of these tones, and a one-dimensional Fourier transform of the digitized echo separates them, mapping each frequency back to its position $x$.

Sampling sets the limits. If we acquire $N_x$ complex points at dwell time $\Delta t$ (so the total readout duration is $T_{acq} = N_x\,\Delta t$), the receiver bandwidth is $\mathrm{BW} = 1/\Delta t$. The Nyquist theorem requires that the highest spatial frequency in the object fit within this sampling band; frequencies outside it alias (wrap around) in the image. The range of resolvable spatial positions, the field of view along the readout, follows from equating the receiver bandwidth to the frequency spread across the FOV.

Phase Encoding

The orthogonal in-plane axis cannot use a second simultaneous readout gradient, because a single sampled time series cannot separate two frequency axes at once. Instead we use phase encoding. Before readout, a gradient $G_y$ is pulsed for a fixed duration $\tau$. While it is on, spins at position $y$ precess faster or slower and acquire a position-dependent phase $\phi(y) = \gamma\,G_y\,y\,\tau$. When $G_y$ is switched off, that phase is frozen into the transverse magnetization and persists through the readout.

A single phase-encode step gives one phase ramp across $y$, which is not enough to resolve the axis. We repeat the entire excitation-and-readout cycle many times, stepping the area of the phase-encode gradient on each repetition. Each step imposes a different spatial-frequency ramp along $y$, and after $N_y$ steps the orthogonal axis is fully sampled. The price is time: total scan time scales with the number of phase-encode steps, which is why phase encoding usually dominates the duration of a 2D acquisition.

Unifying the Picture: k-Space

Slice selection, frequency encoding, and phase encoding look like three different operations, but they are the same operation applied along different axes and bookkept by a single quantity: the spatial frequency $\mathbf{k}$. Define $\mathbf{k}$ as the time integral of the gradient. Its value at any instant is set purely by the accumulated gradient area up to that moment, regardless of whether that area was delivered as a tall brief pulse or a low long one.

In this language, the received signal is the Fourier transform of the spin density $\rho(\mathbf{r})$, sampled along whatever trajectory the gradients trace: $s(\mathbf{k}) = \int \rho(\mathbf{r})\, e^{-i 2\pi\, \mathbf{k}\cdot\mathbf{r}}\, d\mathbf{r}$. The readout gradient sweeps $k_x$ continuously during sampling (one horizontal line). Each phase-encode step jumps to a different $k_y$ (selects which line). The image is recovered by an inverse 2D Fourier transform of the filled k-space matrix.

From this single integral, the geometry of the image falls out as two reciprocal relationships. The extent of k-space you cover, $k_{max}$, sets the spatial resolution, and the spacing between k-space samples, $\Delta k$, sets the field of view.

| Image property | k-space quantity | Relationship | | --- | --- | --- | | Field of view (FOV) | sample spacing Δk | FOV = 1 / Δk | | Pixel size (resolution) | maximum extent k(max) | Δr = 1 / (2 · k(max)) | | Readout line direction | k(x) swept during sampling | filled in one TR | | Phase-encode direction | k(y) stepped between TRs | one line per TR |

Explore: Trajectories and Sampling in k-Space

The KSpaceExplorer below lets you change how k-space is sampled and watch the reconstructed image respond. Try shrinking the sampled extent ($k_{max}$) to see resolution blur, then widen the sample spacing ($\Delta k$) to provoke aliasing wrap. Note how the bright center governs overall contrast while the periphery sharpens edges.

Putting the Numbers Together

The three operations chain into a complete 2D acquisition. Slice select with $G_z$ plus a shaped RF pulse confines excitation to a slab of thickness $\Delta z = 2\pi\,\Delta f / (\gamma\,G_z)$. A phase-encode blip $G_y$ sets the $k_y$ line. The readout $G_x$ sweeps $k_x$ across one line while the receiver samples at bandwidth $\mathrm{BW}$, fixing $\mathrm{FOV}_x = 2\pi\,\mathrm{BW}/(\gamma\,G_x)$. Repeat over $N_y$ phase-encode steps to fill k-space, then inverse Fourier transform. A representative worked set of parameters appears below.

| Parameter | Symbol | Representative value | | --- | --- | --- | | Field strength | B0 | 1.5 T | | Slice-select gradient | Gz | 10 mT/m | | RF bandwidth | Δf | 1000 Hz → 2.3 mm slice | | Readout gradient | Gx | 8 mT/m | | Receiver bandwidth | BW | 32 kHz → ~94 mm FOV | | Matrix | Nx × Ny | 256 × 256 | | Max gradient amplitude | G(max) | 40 mT/m typical clinical |