With GRAPES, the two-dimensional spectrum is physically mapped to an image with coherence time encoded along one spatial axis of the sample and frequency along the other, perpendicular axis, after passage through a dispersive element (grating or prism). Detection proceeds by using a 2D array detector (i.e. camera), which is generally very sensitive in the visible or NIR spectral regions. Eliminating the camera would give us the potential to (i) utilize single-element detectors in regions of the spectrum where cameras perform poorly and (ii) bypass uniform sampling requirements imposed by array detectors when such sampling is not optimal. Single-element detection would provide a route to ultrasensitive detection by employing, for instance, lock-in detection schemes or balanced heterodyne detection, which are currently very difficult to achieve with 2D array sensors.

*Particularly exciting would be the ability to perform 2D spectroscopy in other wavelength regions such as the mid-infrared, THz, X-ray, or ultraviolet regions* or using mixed-frequency methods. These spectral regions would report on other degrees-of-freedom (DOF) that uniquely identify the target system such as the identity of core electronic transitions, vibrational modes, collective excitations of extended systems, and high-lying valence electronic transitions. Recently, we introduced a novel method, which we call SPARSE that combines GRAPES with 2D spatial light modulation in order to encode the 2D spectrum onto a signal recorded by a single-element detector. We demonstrated SPARSE using two different encoding schemes discussed below:

**Hadamard Encoding:** The Hadamard transform matrix, , is analogous to the discrete Fourier transform matrix, but contains only binary (1 and -1) elements. Hadamard sampling can be expressed as the linear problem , where is a length- signal vector and is a length- measurement vector. The unknown signal can be reconstructed by performing the inverse Hadamard transform on . To implement experimentally, we use a digital micromirror device (DMD), which is a 2D array of electro-mechanical mirror elements whose surface normal angles can be controlled between two binary states: (“on” state, 1) and (“off” state 0). In this way, multiplication of the masks [each constructed from a different row of the Hadamard matrix ] with the unknown image occurs optically by reflecting the image formed at the spectrometer exit off of a spatial mask imprinted on the DMD. The reflected portions (i.e., pixels) of the image are summed to form the observation by focusing the reflected light with a lens onto a small active area photomultiplier tube (PMT) detector. The mask is generated by taking a single row of the Hadamard matrix and mapping it onto a two-dimensional region of the DMD.

Each row of the Hadamard matrix, therefore, corresponds to a single measurement value on the single-element detector. The signal is then recovered by simply multiplying the vector of measured amplitudes with the transpose of the Hadamard matrix (. The 1D signal is then wrapped (i.e. the inverse of the wrapping procedure earlier) to recover the desired 2D image. The raw signal as well as a reconstructed spatial spectral interferograms for the carbocyanine dye, IR-144, detected by 2D programmable Hadamard encoding and comparison to the directly detected signal from an sCMOS camera.

**Compressive Sensing Reconstruction:** As with the Fourier transform, Hadamard reconstruction requires Nyquist sampling to recover the signal. However, if the signal is sparse under a suitable unitary transform, then according to the paradigm of CS, an -element signal may be faithfully reconstructed from fewer than CS algorithms (e.g., convex optimization, basis pursuit, etc.) minimize the L1-norm of the recovered signal subject to the constraint , where is an observation matrix and is a length- measurement vector with . In this work, is a pseudo-randomly chosen subset of the rows of a Hadamard matrix , although other forms are possible such as a pseudo-random matrix of 0s and 1s. It is important to note that itself does not necessarily have to be sparse, but rather it should be sparse in a suitable basis representation.

To solve the constrained L1-norm minimization problem we used standard interior-point methods. Since heterodyne detection by a time-delayed reference field yields a sinusoidal spectral interference pattern, the discrete cosine transform (DCT) was the natural choice for sparsifying transform. To test the validity of this transform, we explicitly compared the DCT of the CS and Hadamard reconstructed interferograms. To perform the 1D DCT, the interferogram is first flattened to a vector. The DCT of the interferogram consists mostly of isolated groups of non-zero points. The Hadamard measurement confirms that the DCT is effective at making the signal sparse and also demonstrates good agreement with the CS reconstruction. Notable differences appear in the high frequency DCT coefficients where the relatively constant, low amplitude features in the Hadamard reconstruction contrast with the higher, but less frequent spikes in the CS reconstruction. These deviations are expected due to the inability to exactly reconstruct uncorrelated noise from a sub-Nyquist set of measurements.

SPARSE was used to collect 2DFT spectra of IR-144 and LH2 from spatial spectral interferograms using each of the three detection methods (camera, Hadamard, and CS) described above. The Hadamard and CS (10% of Nyquist for IR-144 and 35% for LH2) reconstructions are faithful to the camera-detected spectra with respect to peak positions, although peak shapes and relative amplitudes do differ somewhat, especially in the case of LH2. The fraction of Hadamard-encoded spatial mask measurements used for CS reconstruction of LH2 and IR-144 interferograms was chosen such that variations between 2DFT spectra generated with different pseudo-randomly chosen measurement subsets were below the 10% level. The higher signal-to-noise ratio of IR-144 measurements enabled greater undersampling than for LH2. Statistical variations in CS reconstruction were explored at a range of sampling percentages to ensure that CS reconstructed 2DFT spectra are reproducible and that they converge to the Hadamard reconstruction as sampling approaches 100%.

A.P. Spencer, B. Spokoyny, S. Ray, F. Sarvari, and E. Harel, Mapping Multidimensional electronic structure and ultrafast dynamics with single-element detection and compressive sensing, **Nat. Commun.**, 7, 10434 (2016). PDF