^{1,2,a)}, T. R. Calhoun

^{3,4}, K. A. Nugent

^{1}and H. M. Quiney

^{1}

### Abstract

We present here the details of a phase retrieval technique that provides access to multidimensional modalities that are not currently available using existing interferometric techniques. The development of multidimensional optical spectroscopy has facilitated significant insights into electronic processes in physics, chemistry, and biology. The versatility and number of available techniques are, however, significantly limited by the requirement that the detection be interferometric. Many of these techniques are closely related to the vast range of multidimensional NMR spectroscopies, which revolutionized analytical chemistry more than 30 years ago. We focus here on the specific case of two-color multidimensional spectroscopy (analogous to heteronuclear NMR) and discuss the details of an iterative algorithm that recovers the relative phase relationships required to perform the Fourier transformation and find the unique solution for the 2D spectrum. A detailed guide is provided that describes the practical implementation of such algorithms. The effectiveness and accuracy of the phase retrieval process are assessed for simulated one- and two-color experiments. It is also compared with one-color experimental data for which the target phase information has been obtained independently by interferometry. In all the cases, the present algorithm yields results that compare well with the solutions obtained by other means. There are, however, some limitations and potential pitfalls that are identified and discussed. We conclude with a discussion of the potential applications and further advances that may be possible by adopting iterative phase retrieval algorithms of the type discussed here.

This work was supported by a Discovery Project grant from the Australian Research Council. H.Q. also acknowledges the support of the ARC Centre of Excellence for Coherent X-ray Science.

I. INTRODUCTION

II. FORMALISM

III. ASSESSMENT OF PHASE RETRIEVAL ALGORITHM

A. One-color simulations

B. Two-color simulations

C. Limiting cases and the use of additional information

IV. COMPARISON TO EXPERIMENTAL DATA

V. APPLICATIONS AND FURTHER ENHANCEMENTS

VI. CONCLUSIONS

### Key Topics

- Phase retrieval
- 44.0
- Fourier transforms
- 23.0
- Absorption spectra
- 13.0
- Interferometry
- 11.0
- Optical spectroscopy
- 9.0

## Figures

The flow diagram showing the basic phase retrieval algorithm, as described in the text and previously. (See Ref. 15.)

The flow diagram showing the basic phase retrieval algorithm, as described in the text and previously. (See Ref. 15.)

The simulated results for a three-level system, as described in the text is shown in part (a). The phase information was discarded and the phase retrieval algorithm was iterated using the data in the domain as pseudoexperimental data. The retrieved 2D data using just the basic error-reduction algorithm (b) reproduce the general features, but there are clearly problems with this recovery. Part (c) shows significant improvement in the recovered 2D spectrum with exponential decay outside the support and smooth initial guesses used. By incorporating several iterations of HIO every 100 iterations, the further improved 2D spectrum, shown in part (d), was recovered.

The simulated results for a three-level system, as described in the text is shown in part (a). The phase information was discarded and the phase retrieval algorithm was iterated using the data in the domain as pseudoexperimental data. The retrieved 2D data using just the basic error-reduction algorithm (b) reproduce the general features, but there are clearly problems with this recovery. Part (c) shows significant improvement in the recovered 2D spectrum with exponential decay outside the support and smooth initial guesses used. By incorporating several iterations of HIO every 100 iterations, the further improved 2D spectrum, shown in part (d), was recovered.

The inclusion of several HIO iterations can kick the algorithm out of local minima and into another solution with a lower (or higher) error value. The full red line shows the error metric as a function of iteration number with no HIO iterations. The dashed green line shows the error metric for three HIO iterations after 100 error-reduction iterations, followed by another 297 iterations of the error-reduction algorithm. The dash–dot blue line shows the error following 400 iterations, with three HIO iterations after the first and second 100 error-reduction iterations. The dotted black curve shows the error for 400 iterations, with three iterations of HIO every 100 iterations of error reduction. It can be seen that the first two applications of HIO lead to a lower error, but the final HIO application leads to a solution with a slightly higher error.

The inclusion of several HIO iterations can kick the algorithm out of local minima and into another solution with a lower (or higher) error value. The full red line shows the error metric as a function of iteration number with no HIO iterations. The dashed green line shows the error metric for three HIO iterations after 100 error-reduction iterations, followed by another 297 iterations of the error-reduction algorithm. The dash–dot blue line shows the error following 400 iterations, with three HIO iterations after the first and second 100 error-reduction iterations. The dotted black curve shows the error for 400 iterations, with three iterations of HIO every 100 iterations of error reduction. It can be seen that the first two applications of HIO lead to a lower error, but the final HIO application leads to a solution with a slightly higher error.

(a) The energy level diagram shows the three-level system used and the energy and interactions of the three incident pulses. (b) The double-sided diagram for the main off-diagonal peak under examination.

(a) The energy level diagram shows the three-level system used and the energy and interactions of the three incident pulses. (b) The double-sided diagram for the main off-diagonal peak under examination.

The 2D spectra for simulations with two-color excitation are shown together with the 2D spectra recovered from the intensity data alone. Parts (a) and (b) represent data that have been Fourier transformed with respect to the delay τ, for the case of (a) no inhomogeneous broadening and (b) 20 meV of inhomogeneous broadening. Parts (c) and (d) represent data that have been Fourier transformed with respect to the delay T, for the case of (c) no inhomogeneous broadening and (d) 20 meV of inhomogeneous broadening. In each case, the recovered spectrum agrees very well with the calculated 2D spectrum and different information is obtained in each case. The interpretation of such two-color correlation diagrams is somewhat different to their one-color counterparts, as discussed in the text.

The 2D spectra for simulations with two-color excitation are shown together with the 2D spectra recovered from the intensity data alone. Parts (a) and (b) represent data that have been Fourier transformed with respect to the delay τ, for the case of (a) no inhomogeneous broadening and (b) 20 meV of inhomogeneous broadening. Parts (c) and (d) represent data that have been Fourier transformed with respect to the delay T, for the case of (c) no inhomogeneous broadening and (d) 20 meV of inhomogeneous broadening. In each case, the recovered spectrum agrees very well with the calculated 2D spectrum and different information is obtained in each case. The interpretation of such two-color correlation diagrams is somewhat different to their one-color counterparts, as discussed in the text.

These two 2D spectra represent nearly homometric solutions for simulated data where it is nearly impossible to arbitrarily determine which solution is the correct one, as both satisfy the support constraints and give equally good fits to the measured intensities. In this case, knowledge about the system acquired from other complementary experiments may be able to assist in the determination of the correct solution. (The vertically elongated peaks seen here correspond to a nonresonant final interaction, as in Raman scattering, and reflect the pulse bandwidth.)

These two 2D spectra represent nearly homometric solutions for simulated data where it is nearly impossible to arbitrarily determine which solution is the correct one, as both satisfy the support constraints and give equally good fits to the measured intensities. In this case, knowledge about the system acquired from other complementary experiments may be able to assist in the determination of the correct solution. (The vertically elongated peaks seen here correspond to a nonresonant final interaction, as in Raman scattering, and reflect the pulse bandwidth.)

The laser spectrum, (a), and linear absorption spectrum, (b), for the carotenoid in LHCII used for the 2D experiments. Part (c) shows the spectrally resolved intensity acquired as a function of τ, with the phase information discarded, as used as the input for the phase retrieval algorithm. The 2D spectrum determined from interferometry (see Ref. 31), (d), compares well with the 2D spectrum determined using phase retrieval algorithm, with no prior knowledge of the phase information (e).

The laser spectrum, (a), and linear absorption spectrum, (b), for the carotenoid in LHCII used for the 2D experiments. Part (c) shows the spectrally resolved intensity acquired as a function of τ, with the phase information discarded, as used as the input for the phase retrieval algorithm. The 2D spectrum determined from interferometry (see Ref. 31), (d), compares well with the 2D spectrum determined using phase retrieval algorithm, with no prior knowledge of the phase information (e).

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