Reconstruct a 2D spectrum from a complex record

Measure a weak field with a reference

A detector measures total intensity. Mix weak signal $E_s$ with strong local oscillator $E_L$ and subtract the oscillator-only background. The leading cross term is linear in the weak field, allowing sensitive detection.

One local-oscillator phase gives one projection of the signal. A quarter-turn phase shift supplies the orthogonal projection. Saving both gives a complex record. Those two calibrated projections retain the field phase needed for the reconstruction.

|E_L+E_s|^2-|E_L|^2=2\operatorname{Re}(E_L^*E_s)+|E_s|^2

With a strong reference and weak signal, the last term is usually small compared with the cross term. Background, phase drift and detector response still require treatment.

Reading: Tokmakoff · Park

Fix the positive directions first

Use excitation frequency horizontally and detection frequency vertically, with positive frequencies on both. Choose $R_{\mathrm R}\propto e^{+i\omega_at_1-i\omega_bt_3}$ and $R_{\mathrm{NR}}\propto e^{-i\omega_at_1-i\omega_bt_3}$ . Their first-interval phases are opposite. Opposite first-axis transform signs place the same transition at the same positive excitation frequency.

S_{\mathrm R}(\omega_1,\omega_3)=\int_0^\infty\!\int_0^\infty R_{\mathrm R}e^{-i\omega_1t_1+i\omega_3t_3}\,dt_1dt_3

The rephasing transform. Finite records replace the integral with weighted sums.

S_{\mathrm{NR}}(\omega_1,\omega_3)=\int_0^\infty\!\int_0^\infty R_{\mathrm{NR}}e^{+i\omega_1t_1+i\omega_3t_3}\,dt_1dt_3

The nonrephasing transform. Consistent time ordering, normalization and common phase let the real part of the sum form the selected absorptive spectrum.

Reading: Tokmakoff

Worked example for sample spacing and duration

Sample a rotating-frame envelope every $\Delta t=0.05\,\mathrm{ps}$ . The Nyquist half-range of frequency offsets is approximately $333.6\,\mathrm{cm}^{-1}$ . A $3.2\,\mathrm{ps}$ record has a characteristic frequency-grid scale of $10.4\,\mathrm{cm}^{-1}$ . The actual resolving power also depends on decay and window shape.

These samples describe a slowly varying complex envelope relative to a carrier. Direct sampling of the original mid-infrared field would require resolving that optical carrier. Store the rotating-frame convention so that the physical frequency axis can be restored.

|\delta\widetilde\nu|<\frac{1}{2c\Delta t},\qquad \delta\widetilde\nu_{\mathrm{grid}}\approx\frac{1}{cT_{\mathrm{window}}}

Sample spacing controls aliasing; duration sets the grid scale. Zero padding smooths the displayed transform without lengthening the observation.

Pulse spectra and windows reshape peaks

A finite pulse selects frequencies within its spectrum. Transitions far from the carrier may be weakly excited, so a raw edge-peak height cannot be compared directly with one at the centre. A finite time window can also truncate a persistent signal and introduce sidelobes.

A smooth window suppresses those sidelobes but broadens peaks. With noise, a longer record can eventually collect mostly noise. A waiting-time comparison needs shared windowing and phase treatment, with processing parameters saved.

Reading: Park

Save enough information to recalculate

A reproducible record includes both time axes, real and imaginary parts for both pathways, waiting time, carrier, units, pulse weights, window, random seed and model version. Reconstruct from that record. Changing a colour scale should leave the underlying samples unchanged.

Compare direct time-domain sums with an independent integral to check peak positions, signs and amplitudes. If changing the sampling step suddenly moves a peak, inspect aliasing and transform signs before assigning a molecular change.

Try the prediction

Read signed peaks from an exported record

Save a complex record and its spectrum, including time window and carrier convention.
Keep the sampling grid fixed and change coherence lifetime or noise. Predict which features blur first.
Explain changes in peak height separately from changes in frequency.

Compare a consistently processed waiting-time series

Keep bandwidth, window and noise fixed while inspecting different waits.
Export the series and check that its records share coordinates and processing parameters.

Exercises and explanations

Does four times denser zero padding resolve two previously overlapping narrow lines?

Read the explanation

Zero padding interpolates the transform of the existing finite record. It supplies no independent time information. Closer lines generally need a longer useful coherence record and adequate signal-to-noise ratio.

What happens if both pathways use the same first-axis transform sign?

Read the explanation

One pathway places the same positive excitation frequency at the opposite sign. Corresponding features no longer align for the intended absorptive combination.

Sources and calculation scope

Tokmakoff, Two-dimensional spectroscopy chapter (June 2011)

In the same deposited notes, the section whose printed page numbers restart at 1: pp. 4–5, Eqs. (1)–(3), connect response, Fourier transformation and heterodyne detection; pp. 10–11 discuss coupled-vibration peak pairs. No source figure is redistributed.

Park, Kwak and Fayer, Ultrafast 2D-IR vibrational echo spectroscopy (2007)

Laser Physics Letters 4, 704–718, DOI 10.1002/lapl.200710046. Experimental review: pulse sequence, local oscillator and absorptive phasing in Sections 2–3 and Fig. 3. Its center-line convention uses the inverse slope for the opposite slice direction; the course states its own axes before defining a slope.