1. Physical Concept of a Spatio-Temporal THz Field
The complete description of a broadband terahertz pulse is not simply its temporal waveform $E(t)$, spectrum $E(f)$, or transverse beam profile $E(x,y)$, but the full spatio-temporal electric field $E(x,y,t)$. This quantity describes the instantaneous THz electric field at every transverse position $(x,y)$ and at every time $t$. A broadband THz pulse should therefore be regarded as a multidimensional electromagnetic object whose spatial distribution, temporal waveform, spectral phase, polarization and propagation characteristics are interconnected. At a fixed observation plane, $E(x,y,t)$ can be visualized as a sequence of two-dimensional electric-field maps recorded at successive time delays. Each map $E(x,y,t_0)$ is an instantaneous snapshot of the electromagnetic field rather than an intensity image. Because the electric field is a signed quantity, positive and negative regions correspond to opposite instantaneous electric-field directions. During the oscillation of a THz pulse, the spatial map may therefore reverse polarity, deform, rotate or reorganize depending on the spatial phase and spectral content of the field. At one particular pixel, the dataset gives a conventional time-domain waveform $E(x_0,y_0,t)$, while at one particular time it provides the complete transverse field distribution $E(x,y,t_0)$. This makes the measured THz data cube a direct representation of electromagnetic evolution in space and time.
The connection between the temporal and spectral representations is established through the Fourier transform, $E(x,y,\omega)=\int_{-\infty}^{\infty}E(x,y,t)e^{i\omega t}dt$, with inverse transformation $E(x,y,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}E(x,y,\omega)e^{-i\omega t}d\omega$. The complex spectral field can be written as $E(x,y,\omega)=A(x,y,\omega)e^{i\Phi(x,y,\omega)}$, where $A(x,y,\omega)$ describes the spatial amplitude of each spectral component and $\Phi(x,y,\omega)$ contains its phase. The observed temporal pulse results from coherent interference between all these frequency components. A single-cycle THz pulse is therefore not a single oscillation existing independently in time; it is created by the coherent addition of a broad spectrum of frequencies possessing a specific relative phase. Changing the spectral amplitude changes the pulse shape, while changing the spectral phase changes the temporal organization of those frequencies. A temporal delay $\tau$ introduces a linear spectral phase according to $E(\omega)\rightarrow E(\omega)e^{-i\omega\tau}$, so that $\Phi(\omega)=-\omega\tau$. Consequently, the slope of spectral phase contains direct information about pulse arrival time, and spatial variations of this slope indicate that different regions of the beam arrive at different times.
The simplest possible broadband beam is approximately space-time separable and may be written as $E(x,y,t)=U(x,y)T(t)$, where $U(x,y)$ represents the transverse spatial mode and $T(t)$ represents the common temporal waveform. In such a beam every spatial point experiences essentially the same normalized waveform, differing only in amplitude. Real THz beams are often not perfectly separable. Diffraction, chromatic focusing, material dispersion, waveguide propagation, angular dispersion, source geometry and structured optical elements can make the spatial distribution depend strongly on frequency. The general field must then be written as $E(x,y,\omega)$ rather than as a product of independent spatial and spectral terms. This condition is known as spatio-temporal coupling. Examples include pulse-front tilt, spatial chirp, frequency-dependent beam size, frequency-dependent wavefront curvature, angular dispersion and frequency-dependent spatial modes. Pulse-front tilt can be represented approximately as $E(x,t)=E_0(t-\alpha x)$, indicating that the pulse reaches different transverse positions at different times. Spatial chirp occurs when the local spectral distribution changes across the beam, so that different transverse positions contain different dominant frequencies. Both effects demonstrate why a broadband THz pulse cannot always be understood by separately studying its space and time dependence.
The phase of the field contains equally important spatial information. For an individual frequency, $\Phi(x,y,\omega)$ describes the electromagnetic wavefront. An approximately constant phase indicates a planar wavefront, while a quadratic spatial phase is associated with converging or diverging wavefronts. A helical phase indicates a vortex field, while more complicated distributions can contain several singularities or spatially dependent phase gradients. The group delay can be obtained from the frequency derivative of the spectral phase, commonly written as $\tau_g=-d\Phi/d\omega$ depending on the adopted Fourier convention. Performing this calculation spatially produces a group-delay map $\tau_g(x,y,\omega)$ that can reveal pulse-front curvature, local temporal delays and dispersive behavior across the THz beam. Spatio-temporal characterization therefore goes beyond simply observing a pulse movie; it provides access to the physical relationship between wavefront, spectrum, propagation and arrival time.
2. Propagation, Diffraction, Dispersion and Temporal Evolution
The spatio-temporal structure of THz radiation evolves naturally during propagation because diffraction depends on frequency. The wavelength is $\lambda=c/f$, so low-frequency THz components have longer wavelengths and generally diffract more strongly than higher-frequency components. For a Gaussian beam, the Rayleigh range is $z_R=\pi w_0^2/\lambda$, demonstrating directly that propagation behavior depends on wavelength. Consequently, a broadband THz beam may possess different beam sizes, divergence angles and wavefront curvatures at different frequencies. When all spectral components are recombined in time, the instantaneous spatial profile can change throughout the waveform. The early portion of a pulse may therefore possess a different spatial distribution from the central peak or trailing oscillations. This is an important characteristic of broadband THz radiation and becomes particularly significant close to a focus or after propagation through dispersive optical elements.
Propagation can be treated efficiently in the frequency domain. The transverse field $E(x,y,\omega)$ is first transformed into transverse spatial-frequency coordinates $(k_x,k_y)$ to obtain $\tilde{E}(k_x,k_y,\omega)$. Propagation over a distance $z$ introduces the phase factor $e^{ik_z z}$, where $k_z=\sqrt{k^2-k_x^2-k_y^2}$ and $k=n(\omega)\omega/c$. Thus, $\tilde{E}(k_x,k_y,\omega;z)=\tilde{E}(k_x,k_y,\omega;0)e^{ik_z z}$. Performing the inverse spatial Fourier transform gives the propagated field at each frequency, and a final inverse temporal Fourier transform reconstructs $E(x,y,z,t)$. This four-dimensional quantity describes the complete scalar electric-field evolution in two transverse spatial dimensions, propagation distance and time. For vector fields, the more complete description becomes $\mathbf{E}(x,y,z,t)=E_x\hat{x}+E_y\hat{y}+E_z\hat{z}$.
Material dispersion adds another level of temporal evolution. In a dispersive medium, the propagation constant is $\beta(\omega)=n(\omega)\omega/c$. Expanding around a central angular frequency $\omega_0$ gives $\beta(\omega)=\beta_0+\beta_1(\omega-\omega_0)+\frac{1}{2}\beta_2(\omega-\omega_0)^2+\cdots$. The coefficient $\beta_1$ controls group delay, while $\beta_2$ describes group-velocity dispersion. A linear spectral phase mainly shifts the pulse in time, whereas nonlinear spectral phase changes its shape. Quadratic spectral phase can broaden a pulse and generate chirp, while higher-order dispersion becomes increasingly important for extremely broadband THz radiation. A short single-cycle pulse can therefore evolve into a long multi-cycle waveform after propagation through a dispersive structure.
A useful way to inspect this behavior is through time-frequency analysis. The short-time Fourier transform may be expressed as $\mathrm{STFT}(t_0,\omega)=\int E(t)w(t-t_0)e^{i\omega t}dt$, where $w(t-t_0)$ is a localized temporal window. The resulting spectrogram reveals when particular frequency components occur within the pulse. If high-frequency components occur earlier than low-frequency components, the pulse exhibits one sign of chirp; if low frequencies occur earlier, the chirp is reversed. Such frequency-time slopes are particularly useful for studying dispersive THz waveguides, crystals, lenses and resonant structures. Applying the same analysis at multiple spatial positions produces a spatio-temporal-frequency description of the field.
Focusing also strongly affects THz spatio-temporal evolution. As a broadband beam approaches focus, its spatial width decreases, wavefront curvature changes and the Gouy phase evolves. For a fundamental Gaussian mode, the Gouy phase is $\psi_G(z)=\tan^{-1}(z/z_R)$. Since $z_R$ depends on wavelength, broadband components can experience different Gouy-phase variations. Chromatic optical elements may additionally focus different frequencies at different longitudinal positions. As a result, the local waveform before focus, at focus and after focus may not be identical. Broadband focusing must therefore be understood as a spatio-temporal propagation problem rather than simply as geometric focusing.
Waveguides provide an especially clear example of coupling between spatial and temporal evolution. A broadband field inside a waveguide may be expressed as $E(x,y,\omega,z)=\sum_m C_m(\omega)u_m(x,y,\omega)e^{i\beta_m(\omega)z}$, where $u_m$ represents the spatial profile of mode $m$, $C_m$ its excitation coefficient and $\beta_m$ its propagation constant. Each mode can possess a different cutoff frequency, group velocity and dispersion. After sufficient propagation distance, different spectral and modal components can arrive at different times. A pulse that initially contains a single-cycle waveform can therefore become temporally extended and multi-cycle, while its spatial pattern changes throughout the pulse. An early temporal region may be dominated by one mode and a later region by another. This is a direct manifestation of spatio-temporal modal evolution.
3. Structured THz Light: Vortices, Vector Beams and Frequency-Resolved Reconstruction
Structured THz light provides one of the most striking demonstrations of spatio-temporal electric-field evolution. A monochromatic vortex beam with topological charge $\ell$ contains the spatial phase factor $e^{i\ell\phi}$ and may be written as $E(r,\phi,\omega)=A(r,\omega)e^{i\ell\phi}$. Its real electric field varies approximately as $E(r,\phi,t)=A(r)\cos(\ell\phi-\omega t)$. At a fixed time, regions of positive and negative electric field occur at different azimuthal positions. As time evolves, the constant-phase pattern moves around the beam axis because $\ell\phi-\omega t=\mathrm{constant}$. This produces an apparent rotation of the instantaneous electric-field lobes. The rotation of the signed electric-field map should not be confused with physical rotation of the time-averaged beam intensity. It is instead a direct manifestation of the combination of temporal oscillation and helical spatial phase.
At the center of an ideal vortex, the azimuthal coordinate $\phi$ is undefined and the phase therefore becomes singular. The amplitude approaches zero at this position, forming a phase singularity. Around any closed loop enclosing the singularity, the accumulated phase change is $\oint\nabla\Phi\cdot d\mathbf{l}=2\pi\ell$. The integer $\ell$ is the topological charge. For $\ell=1$, the phase changes by $2\pi$ around the beam; for $\ell=2$, it changes by $4\pi$. Such helical phase structures are associated with orbital angular momentum. A measured vortex field can be decomposed into azimuthal modes using coefficients such as $C_\ell=\frac{1}{2\pi}\int_0^{2\pi}E(\phi)e^{-i\ell\phi}d\phi$, while a more complete analysis may use Laguerre-Gaussian modes that include both azimuthal and radial dependence. Repeating this decomposition across frequency reveals whether the topological charge and modal purity are preserved throughout a broadband THz spectrum.
Broadband vortices are considerably more complex than monochromatic ones because the spatial amplitude, radial profile, wavefront curvature and even modal composition may depend on frequency. A general field may be written as $E(r,\phi,t)=\int A(r,\omega)e^{i[\ell(\omega)\phi+\Phi_r(r,\omega)-\omega t]}d\omega$. If the same vortex structure is maintained over the entire spectrum, the temporal field retains a robust topology. If different frequencies contain different vortex content, the instantaneous structure can deform strongly throughout the pulse. Frequency-resolved reconstruction is therefore essential for understanding broadband THz vortex beams.
The same principle applies to vector beams. A complete transverse THz field contains two orthogonal electric-field components, $\mathbf{E}(x,y,t)=E_x(x,y,t)\hat{x}+E_y(x,y,t)\hat{y}$. Fourier transformation gives the complex fields $E_x(x,y,\omega)$ and $E_y(x,y,\omega)$. Their relative amplitudes and phases determine the local polarization state. The frequency-domain Stokes parameters are $S_0=|E_x|^2+|E_y|^2$, $S_1=|E_x|^2-|E_y|^2$, $S_2=2\mathrm{Re}(E_xE_y^)$, and $S_3=2\mathrm{Im}(E_xE_y^)$. Mapping these quantities across $(x,y)$ reveals the spatial polarization topology. Repeating the calculation at several frequencies reveals whether the polarization pattern is broadband or frequency dependent.
In an ideal radially polarized THz beam, the local electric-field axis points away from or toward the beam center, while in an azimuthally polarized beam it follows the tangential direction around the center. Because the THz field oscillates in time, every local electric vector reverses direction every half-cycle, but its polarization axis can remain radial or azimuthal. If the relative spectral phase between $E_x$ and $E_y$ changes with frequency, however, the local polarization may evolve from linear to elliptical or circular during the pulse. The field then possesses simultaneous spatial, temporal, spectral and polarization structure.
An experimentally measured data cube $E(x,y,t)$ allows direct frequency-resolved reconstruction. Fourier transformation along the temporal dimension produces $E(x,y,f)$. Selecting one frequency $f_0$ gives the complex spatial field $E(x,y,f_0)$, whose amplitude is $|E(x,y,f_0)|$ and whose phase is $\arg[E(x,y,f_0)]$. Repeating this process across frequency reveals how the spatial structure changes spectrally. This procedure can accurately be described as frequency-resolved spatial field reconstruction, spectral field decomposition, or frequency-resolved THz field reconstruction. Selecting a single Fourier frequency produces a monochromatic spatial field, whereas selecting a finite spectral band and performing an inverse transform produces a band-limited time-domain field, $E_{\mathrm{band}}(x,y,t)=\mathcal{F}^{-1}[E(x,y,f)W(f)]$, where $W(f)$ defines the retained spectral region.
Single-frequency reconstruction is especially useful for visualizing vortex evolution. A selected frequency oscillates continuously according to $E(x,y,t)=A(x,y)\cos[\omega_0t+\Phi(x,y)]$. If $\Phi(x,y)$ contains a helical phase, the positive and negative field regions move azimuthally during the oscillation. A finite-band reconstruction is more representative of a physical pulse because it maintains temporal localization. By reconstructing low-, intermediate- and high-frequency spectral bands independently, one can directly compare how different portions of a broadband THz field contribute to its complete temporal behavior.
4. Experimental Mapping, Data Analysis and Physical Interpretation
Spatio-temporal THz field evolution can be measured using raster-scanned THz time-domain spectroscopy, near-field techniques or two-dimensional electro-optic sampling. In 2D electro-optic sampling, a THz field induces transient birefringence in an electro-optic crystal such as ZnTe or GaP. A synchronized optical probe samples this birefringence, and an imaging system maps the probe onto a camera. Repeating the measurement for successive probe delays creates the complete field dataset $E(x,y,t)$. If two orthogonal THz polarization components are measured, the vector field $\mathbf{E}(x,y,t)$ can also be reconstructed.
The experimental data are commonly stored as an array with dimensions $(N_t,N_y,N_x)$. Each temporal index contains one spatial electric-field frame, while each pixel contains one complete THz waveform. From the same dataset, $E(x,y,t_0)$ gives an instantaneous spatial field, $E(x_0,y_0,t)$ gives a local temporal waveform, $E(x,y,f_0)$ gives a frequency-resolved complex spatial field and $E(x_0,y_0,f)$ provides a local spectrum. These are not separate measurements but different representations of the same electromagnetic dataset.
The temporal Fourier transform is one of the most important processing steps. The frequency resolution is approximately $\Delta f=1/T_{\mathrm{window}}$, where $T_{\mathrm{window}}$ is the total measured time interval. Increasing the acquisition window improves frequency resolution, whereas reducing the temporal sampling interval increases the Nyquist frequency. These two quantities serve different purposes and should not be confused. The resulting Fourier field is complex and contains both amplitude and phase. The phase becomes unreliable wherever the spectral amplitude is very small, so phase maps are commonly masked below a chosen amplitude threshold. Such masking should only remove regions without meaningful signal-to-noise ratio and should not be used to hide genuine features such as the low-intensity center of a vortex.
Spectral phase is generally obtained modulo $2\pi$, producing values within a limited interval such as $-\pi$ to $\pi$. Phase unwrapping removes artificial $2\pi$ discontinuities when the physical phase changes continuously. This is valuable for measuring phase slopes, wavefront curvature and accumulated delay. True phase singularities, however, require special care because the discontinuity around a singular point is physical rather than numerical. Similarly, the temporal origin must remain consistent between measurements. A temporal shift of the waveform automatically introduces a linear spectral phase, so different choices of time zero can produce different phase slopes even when the underlying pulse shape is identical.
It is also essential to distinguish between temporal evolution and propagation evolution. A movie of $E(x,y,t)$ recorded at one fixed detection plane shows how the electromagnetic field oscillates as the THz pulse passes that plane. It does not directly show the beam physically moving along the propagation direction $z$. Actual longitudinal evolution requires either measurements at several $z$ positions or numerical propagation of the measured complex field. Once $E(x,y,f)$ is known, every spectral component can be propagated using an angular-spectrum or Fresnel method and then recombined through an inverse temporal Fourier transform to reconstruct $E(x,y,z,t)$. This produces a genuine spatio-temporal propagation map.
The ability to measure the electric field itself rather than only intensity is one of the major advantages of coherent THz techniques. Intensity measurements proportional to $|E|^2$ remove field polarity and much of the phase information. Time-domain THz measurements preserve the sign and phase of the electromagnetic waveform, allowing direct determination of wavefront curvature, spectral phase, group delay, polarization, chirp, modal structure and phase singularities. For a complete electromagnetic description, the magnetic field and Poynting vector $\mathbf{S}=\mathbf{E}\times\mathbf{H}$ would also describe local energy flow, but the electric field alone already provides exceptionally rich information in most THz experiments.
The central physical picture is therefore that a broadband THz pulse is not simply a spatial beam carrying a temporal waveform. It is a coherent superposition of frequency-dependent spatial modes whose amplitudes and phases evolve together. Diffraction determines how different frequencies spread, dispersion determines when they arrive, wavefront phase determines how they propagate, polarization determines the orientation of the electric field, and modal structure determines the topology of the beam. Spatio-temporal THz field mapping unifies all of these properties within a single experimentally measurable quantity. The complete field $E(x,y,z,t)$, or its vector equivalent $\mathbf{E}(x,y,z,t)$, should therefore be regarded as the natural description of broadband structured THz light. Once this field is known, the same dataset can reveal temporal waveforms, spectra, wavefronts, group delays, chirp, polarization states, vortex phase, orbital angular momentum, modal purity and propagation dynamics. In this sense, spatio-temporal THz electric-field characterization does not simply show where a THz beam exists or when a THz pulse arrives; it reveals how the electromagnetic field itself evolves through space, time, frequency and polarization.
Suggested Reading
- Yun-Shik Lee, Principles of Terahertz Science and Technology. A strong foundation for THz generation, coherent detection, pulse propagation and spectroscopy.
- Xi-Cheng Zhang and Jingzhou Xu, Introduction to THz Wave Photonics. Particularly useful for understanding THz time-domain methods, ultrafast generation, detection and propagation.
- A. E. Siegman, Lasers. A fundamental reference for Gaussian beams, diffraction, wavefront evolution, higher-order modes and Gouy phase, all of which extend naturally to structured THz radiation.