Temporal Evolution of a Terahertz Electric Field

1. Optical delay line: displacement becomes time

In coherent THz time-domain detection, the THz pulse and an ultrashort optical probe pulse arrive at the detector with a controlled relative delay. A retroreflector mounted on a translation stage changes the probe path length. The detector does not directly measure an entire waveform at once; it measures one temporal sample for each delay-stage position.

Double-pass geometry

If the stage is displaced by \(x\), the optical path changes by \(2x\). The corresponding delay is

\[ \tau(x)=\frac{2x}{c}, \qquad \Delta \tau=\frac{2\Delta x}{c}. \tag{1} \]

Here \(c\) is the speed of light and \(\Delta x\) is the mechanical step. For a single-pass delay line, the factor of two is absent.

Position index

\[x_n=x_0+n\Delta x\]

Delay index

\[\tau_n=\tau_0+n\Delta\tau\]

Sampled signal

\[S_n=S(\tau_n)\]

The delay axis is a relative time coordinate. Its numerical zero is chosen by calibration; it does not have to coincide with the physical onset of the THz pulse.

2. Discrete temporal sampling and measurable bandwidth

For \(N_t\) equally spaced delay positions, the time interval, total observation window, frequency spacing, and Nyquist frequency are linked. These relations are fundamental: changing the delay step changes the maximum non-aliased frequency, while changing the number of samples changes the spectral resolution.

Temporal grid

\[ t_n=n\Delta t,\qquad n=0,\ldots,N_t-1. \tag{2} \]

Total sampled duration

\[ T=N_t\Delta t. \tag{3} \]

Frequency spacing

\[ \Delta f=\frac{1}{T}=\frac{1}{N_t\Delta t}. \tag{4} \]

Nyquist frequency

\[ f_{\mathrm{Nyq}}=\frac{1}{2\Delta t}. \tag{5} \]

A smaller delay step increases the Nyquist bandwidth but does not, by itself, improve the frequency resolution. A longer temporal window improves \(\Delta f\).

3. What the detector samples

In coherent detection, the measured detector output is proportional to a gated version of the THz electric field. In the ideal limit of an infinitely short probe pulse and a flat detector response, the signal is proportional to the instantaneous THz field.

\[ S(\tau)\propto E_{\mathrm{THz}}(\tau). \tag{6} \]

More generally, the detector records a convolution:

\[ S(\tau)=\int_{-\infty}^{\infty} E_{\mathrm{THz}}(t)\,h_{\mathrm{det}}(\tau-t)\,dt = \left(E_{\mathrm{THz}}*h_{\mathrm{det}}\right)(\tau), \tag{7} \]

where \(h_{\mathrm{det}}(t)\) contains the probe-pulse envelope, electro-optic crystal response, phase matching, absorption, analyzer response, and electronics.

\[ \widetilde S(\omega)=H_{\mathrm{det}}(\omega)\widetilde E_{\mathrm{THz}}(\omega). \tag{8} \]

4. From sequential frames to the spatiotemporal field cube

When a camera or two-dimensional detector records one image at every delay, the experiment produces a stack of frames. The \(n\)-th frame is a transverse electric-field map evaluated at \(t_n\).

\[ E_x^{(n)}(x,y)=E_x(x,y,t_n),\qquad E_y^{(n)}(x,y)=E_y(x,y,t_n). \tag{9} \]

Stacking the frames gives two three-dimensional arrays:

\[ \mathbf E(x,y,t_n)= \begin{bmatrix} E_x(x,y,t_n)\\[2mm] E_y(x,y,t_n) \end{bmatrix}, \qquad \text{array shape }(N_t,N_y,N_x). \tag{10} \]

Instantaneous amplitude

\[ A(x,y,t)=\sqrt{E_x^2(x,y,t)+E_y^2(x,y,t)}. \tag{11} \]

Instantaneous field direction

\[ \theta(x,y,t)=\operatorname{atan2}\!\left(E_y(x,y,t),E_x(x,y,t)\right). \tag{12} \]

Equation (12) gives a directed instantaneous vector angle. It is not identical to the polarization-ellipse azimuth obtained from frequency-domain Stokes parameters.

5. Step-by-step temporal evolution of a broadband THz pulse

A single-cycle THz transient is not a slowly varying intensity pulse. It is a signed electric field. Therefore, the field can be zero while electromagnetic energy is still present in the pulse envelope, and the vector direction reverses when the field changes sign.

Representative analytic field

A common theory model is the derivative of a Gaussian current or polarization response:

\[ E(t)=E_0\frac{t-t_0}{\sigma} \exp\!\left[-\frac{(t-t_0)^2}{2\sigma^2}\right]. \tag{13} \]

This waveform has two lobes with opposite signs and a zero crossing at \(t=t_0\).

Stage I
Pre-pulse

\[ |t-t_0|\gg \sigma,\qquad E_x\approx E_y\approx 0. \] The probe reaches the detector before temporal overlap with the THz transient. The measured signal is dominated by baseline and noise.

Stage II
Leading edge

The probe begins to overlap the first temporal lobe. The field amplitude grows: \[ \frac{\partial |E|}{\partial t}>0. \] The first nonzero spatial frame reveals where the pulse front arrives earliest.

Stage III
First extremum

At a temporal extremum, \[ \frac{\partial E}{\partial t}=0,\qquad |E|\ \text{is locally maximal}. \] The instantaneous field map has its strongest signed contrast for the first polarity.

Stage IV
Zero crossing

\[ E_x(x,y,t_z)\approx 0,\qquad E_y(x,y,t_z)\approx 0 \] for a scalar, uniformly delayed pulse. In a structured or dispersive beam, different pixels may cross zero at different times, so the global image need not vanish simultaneously.

Stage V
Polarity reversal

The electric field changes sign. For a fixed local polarization direction, \[ \mathbf E(x,y,t>t_z)\approx-\mathbf E(x,y,t

Stage VI
Second extremum

The opposite field polarity reaches maximum magnitude. Asymmetry between the two lobes signals spectral phase, detector transfer-function effects, material dispersion, or a non-ideal source current.

Stage VII
Trailing field

The main transient decays. Residual oscillations may arise from water-vapor resonances, etalon reflections, phonon resonances, waveguide modes, or detector ringing.

Theoretical time evolution

Move the delay slider. The curves are an analytic two-component THz pulse, not measured data.

\(t/\sigma=-2.00\)

\(E_x(t)\) \(E_y(t)\) current delay

6. Two-component temporal field and polarization trajectory

At one spatial coordinate, the time-dependent transverse field is \(\mathbf E_\perp(t)=E_x(t)\hat{\mathbf x}+E_y(t)\hat{\mathbf y}\). Its tip traces a trajectory in the \(E_x\)-\(E_y\) plane.

\[ \mathbf E_\perp(t)= \begin{bmatrix} E_x(t)\\ E_y(t) \end{bmatrix}. \tag{14} \]

For a monochromatic component,

\[ E_x(t)=A_x\cos(\omega t+\phi_x),\qquad E_y(t)=A_y\cos(\omega t+\phi_y). \tag{15} \]

Defining \(\delta=\phi_y-\phi_x\), elimination of time gives the polarization ellipse:

\[ \left(\frac{E_x}{A_x}\right)^2+ \left(\frac{E_y}{A_y}\right)^2 -2\frac{E_xE_y}{A_xA_y}\cos\delta =\sin^2\delta. \tag{16} \]

For a broadband pulse, the instantaneous vector trajectory is a coherent superposition of many frequency-dependent ellipses. A single time-domain angle does not fully specify the broadband polarization state.

7. Temporal baseline removal and windowing

Before the Fourier transform, a pixel-wise temporal mean is commonly removed. This suppresses the zero-frequency component caused by offsets.

\[ E_x'(x,y,t_n)=E_x(x,y,t_n) -\frac{1}{N_t}\sum_{n=0}^{N_t-1}E_x(x,y,t_n), \tag{17} \] \[ E_y'(x,y,t_n)=E_y(x,y,t_n) -\frac{1}{N_t}\sum_{n=0}^{N_t-1}E_y(x,y,t_n). \tag{18} \]

A window \(w_n\) may then be applied:

\[ E_{x,w}(x,y,t_n)=w_nE_x'(x,y,t_n). \tag{19} \]

Windowing reduces edge discontinuities but broadens spectral features. It changes the effective amplitude and spectral resolution, so it must be reported.

8. Pixel-wise Fourier transform

The temporal transform is performed independently at every spatial pixel. The output is a complex field, retaining both amplitude and phase.

\[ \widetilde E_x(x,y,f_k) = \sum_{n=0}^{N_t-1} E_{x,w}(x,y,t_n) \exp\!\left(-i2\pi\frac{kn}{N_t}\right), \tag{20} \] \[ f_k=\frac{k}{N_t\Delta t}, \qquad k=0,\ldots,\left\lfloor\frac{N_t}{2}\right\rfloor. \tag{21} \]

Spectral amplitude

\[ A_x(x,y,f)=|\widetilde E_x(x,y,f)|. \tag{22} \]

Spectral phase

\[ \phi_x(x,y,f)=\arg\widetilde E_x(x,y,f). \tag{23} \]

The same definitions apply to \(y\). The complex representation is

\[ \widetilde E_x=A_xe^{i\phi_x},\qquad \widetilde E_y=A_ye^{i\phi_y}. \tag{24} \]

9. Narrowband field reconstructed at a selected time

A frequency bin is not itself a time-domain snapshot. To reconstruct the real field contribution of frequency \(f_k\) at time \(t\), the complex spectral field must be multiplied by its temporal phase factor.

\[ E_{x,k}(x,y,t) = \Re\left\{ \widetilde E_x(x,y,f_k)e^{i2\pi f_kt} \right\}, \tag{25} \] \[ E_{y,k}(x,y,t) = \Re\left\{ \widetilde E_y(x,y,f_k)e^{i2\pi f_kt} \right\}. \tag{26} \]

The full real time-domain field is obtained by summing all positive- and negative-frequency contributions consistently:

\[ E_x(x,y,t)= \int_{-\infty}^{\infty} \widetilde E_x(x,y,f)e^{i2\pi ft}\,df. \tag{27} \]

Taking only \(\Re\{\widetilde E_x(f_k)\}\) corresponds to evaluating the selected frequency component at a chosen zero of time. It does not represent arbitrary \(t\) unless the factor \(e^{i2\pi f_kt}\) is included.

10. Spectral phase, delay, and pulse-front evolution

A linear spectral phase corresponds to a temporal delay. A spatially varying linear phase therefore maps the local pulse arrival time.

\[ \phi(x,y,\omega)=\phi_0(x,y)-\omega\tau_g(x,y)+\phi_{\mathrm{res}}(x,y,\omega). \tag{28} \]

\[ \tau_g(x,y,\omega) = -\frac{\partial\phi(x,y,\omega)}{\partial\omega}. \tag{29} \]

If \(\tau_g\) varies linearly across \(x\), the pulse front is tilted:

\[ \tau_g(x)=\tau_0+\alpha x, \qquad \alpha=\frac{\partial\tau_g}{\partial x}. \tag{30} \]

If it varies quadratically, the pulse front is curved:

\[ \tau_g(x,y)=\tau_0+a_xx^2+a_yy^2. \tag{31} \]

11. Frequency-resolved polarization and Stokes parameters

At a selected frequency, the complex Jones field is \(\widetilde{\mathbf E}=(\widetilde E_x,\widetilde E_y)^T\). The Stokes parameters are calculated from complex cross-products.

\[ S_0=|\widetilde E_x|^2+|\widetilde E_y|^2, \tag{32} \] \[ S_1=|\widetilde E_x|^2-|\widetilde E_y|^2, \tag{33} \] \[ S_2=2\Re\!\left(\widetilde E_x\widetilde E_y^*\right), \tag{34} \] \[ S_3=2\Im\!\left(\widetilde E_x\widetilde E_y^*\right). \tag{35} \]

The normalized parameters are \(s_i=S_i/S_0\). The polarization azimuth and ellipticity angle are

\[ \psi=\frac{1}{2}\operatorname{atan2}(S_2,S_1), \tag{36} \] \[ \chi=\frac{1}{2}\sin^{-1}\!\left(\frac{S_3}{S_0}\right). \tag{37} \]

For partially polarized data,

\[ \mathrm{DoP}= \frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}, \qquad \mathrm{DoLP}= \frac{\sqrt{S_1^2+S_2^2}}{S_0}. \tag{38} \]

Pixels with small \(S_0\) must be masked before normalization. Otherwise, division by noise produces unstable \(s_1,s_2,s_3,\psi,\chi\).

12. Spatial averaging and why the order matters

An ROI-averaged time trace is a coherent spatial average:

\[ \overline E_x(t)=\frac{1}{N_{\mathrm{ROI}}} \sum_{(x,y)\in\mathrm{ROI}}E_x(x,y,t). \tag{39} \]

Fourier transforming this average gives

\[ \mathcal F\{\overline E_x(t)\} = \frac{1}{N_{\mathrm{ROI}}} \sum_{(x,y)\in\mathrm{ROI}}\widetilde E_x(x,y,f). \tag{40} \]

This coherent average can cancel spatial regions with opposite phase. In contrast, averaging \(|\widetilde E_x|\) is an incoherent amplitude average and answers a different physical question.

13. Temporal resolution, timing jitter, and measured field

The experimentally reconstructed waveform is the true field filtered by the temporal instrument response and timing uncertainty.

\[ E_{\mathrm{meas}}(t)= E_{\mathrm{true}}(t)*g_{\mathrm{probe}}(t) *h_{\mathrm{crystal}}(t) *g_{\mathrm{jitter}}(t). \tag{41} \]

For Gaussian timing responses with RMS widths \(\sigma_i\),

\[ \sigma_{\mathrm{eff}}^2= \sigma_{\mathrm{probe}}^2+ \sigma_{\mathrm{jitter}}^2+ \sigma_{\mathrm{other}}^2. \tag{42} \]

A Gaussian timing blur attenuates high frequencies according to

\[ H_{\mathrm{jitter}}(\omega)= \exp\!\left(-\frac{\omega^2\sigma_{\mathrm{jitter}}^2}{2}\right). \tag{43} \]

14. Complete theoretical chain

Step	Mathematical object	Physical meaning
Delay stage	\(\tau_n=2x_n/c\)	Converts position into probe–THz delay
Frame acquisition	\(E_x(x,y,t_n),E_y(x,y,t_n)\)	Samples the instantaneous transverse field
Field cube	\(\mathbf E(x,y,t_n)\)	Stores temporal evolution at every pixel
Preprocessing	\(E'=E-\langle E\rangle_t\)	Removes temporal offset
Fourier transform	\(\widetilde E(x,y,f)\)	Produces complex spectral field
Amplitude and phase	\(\|\widetilde E\|,\arg\widetilde E\)	Separates spectral strength and timing information
Narrowband evolution	\(\Re\{\widetilde Ee^{i2\pi ft}\}\)	Shows a selected frequency evolving in time
Polarization	\(S_0,S_1,S_2,S_3\)	Describes the local polarization ellipse