### Abstract

We present a Fourier optical analysis of a typical femtosecond pulse shaping apparatus and derive analytic expressions for the space-time dependence of the emerging waveform after the pulse shaper and in the focal volume of an additional focusing element. For both geometries the results are verified experimentally. Hereafter, we analyze the influence of space-time coupling on nonlinear processes, specifically second harmonic generation, resonant interaction with an atomic three-level system, and resonant excitation of a diatomic molecule.

This work was supported by the National Center of Competence in Research: Quantum Photonics, Research Instrument of the Swiss National Science Foundation (SNSF).

I. INTRODUCTION

II. FOURIER OPTICAL ANALYSIS

A. Slowly varying electric field after the pulse shaper

B. Slowly varying electric field in focusing geometry

III. EXPERIMENTAL VERIFICATION OF SPACE-TIME COUPLING

A. Space-time coupling after the pulse shaper

B. Space-time coupling in focusing geometry

IV. EFFECTS OF SPACE-TIME COUPLING ON COHERENT CONTROL

A. Experimental verification of space-time coupling in second harmonic generation

B. Spatially averaged fields

C. Second harmonic generation

D. Resonant interaction with an atomic three-level system

E. Resonant interaction with a diatomic molecule

V. CONCLUSION

### Key Topics

- Phase modulation
- 22.0
- Coherent control
- 20.0
- Second harmonic generation
- 19.0
- Spacetime topology
- 18.0
- Diffraction gratings
- 16.0

## Figures

Typical pulse shaper in geometry using a pixelated SLM at the symmetry plane of the setup. In most experiments the shaper’s output waveform is focused by an additional focusing element to the sample of interest.

Typical pulse shaper in geometry using a pixelated SLM at the symmetry plane of the setup. In most experiments the shaper’s output waveform is focused by an additional focusing element to the sample of interest.

(a) Space-time coupling as a function of focal length given a constant frequency-to-space mapping of 6.807 THz/mm; the gratings have 300 lines/mm (dash-dotted curve), 600 lines/mm (dotted curve), 800 lines/mm (dashed curve), and 1200 lines/mm (solid curve), respectively. The arrows indicate the space-time coupling constant for the Littrow geometry, i.e., . (b) Angle of incidence and diffraction angle at the center frequency as a function of focal length.

(a) Space-time coupling as a function of focal length given a constant frequency-to-space mapping of 6.807 THz/mm; the gratings have 300 lines/mm (dash-dotted curve), 600 lines/mm (dotted curve), 800 lines/mm (dashed curve), and 1200 lines/mm (solid curve), respectively. The arrows indicate the space-time coupling constant for the Littrow geometry, i.e., . (b) Angle of incidence and diffraction angle at the center frequency as a function of focal length.

Top row and bottom row at five positions within the Rayleigh length of the focusing element.

Top row and bottom row at five positions within the Rayleigh length of the focusing element.

(a) Top row and bottom row at five positions within the Rayleigh length of the focusing element. (b) Spectral intensity and (c) temporal intensity as a function of for .

(a) Top row and bottom row at five positions within the Rayleigh length of the focusing element. (b) Spectral intensity and (c) temporal intensity as a function of for .

(a) Top row and bottom row at five positions within the Rayleigh length of the focusing element. (b) Spectral intensity and (c) temporal intensity as a function of for .

Spatial offset as a function of the delay time for (a) a linear phase and (b) a double-pulse modulation. The solid lines result from Eq. (12). (c) Measured and simulated beam profile for a sinusoidal phase.

Spatial offset as a function of the delay time for (a) a linear phase and (b) a double-pulse modulation. The solid lines result from Eq. (12). (c) Measured and simulated beam profile for a sinusoidal phase.

Top row: experimentally measured and bottom row: simulated space-frequency distributions as a function of transverse coordinate and frequency . [(a) and (e)] Unshaped pulse, [(b) and (f)] linear phase with , [(c) and (g)] double pulse with , and [(d) and (h)] sinusoidal phase with .

Top row: experimentally measured and bottom row: simulated space-frequency distributions as a function of transverse coordinate and frequency . [(a) and (e)] Unshaped pulse, [(b) and (f)] linear phase with , [(c) and (g)] double pulse with , and [(d) and (h)] sinusoidal phase with .

Spatial offset in the focus of a 400 mm lens as a function of the delay time for a (a) linear phase, (b) double-pulse modulation, and (c) sinusoidal phase. The solid lines are derived from Eq. (13).

Spatial offset in the focus of a 400 mm lens as a function of the delay time for a (a) linear phase, (b) double-pulse modulation, and (c) sinusoidal phase. The solid lines are derived from Eq. (13).

Top row: experimentally measured; bottom row: simulated space-frequency distributions as a function of transverse coordinate and frequency . [(a) and (e)] Unshaped pulse [(b) and (f)] linear phase with , [(c) and (g)] double pulse with , and [(d) and (h)] sinusoidal phase with .

Top row: experimentally measured; bottom row: simulated space-frequency distributions as a function of transverse coordinate and frequency . [(a) and (e)] Unshaped pulse [(b) and (f)] linear phase with , [(c) and (g)] double pulse with , and [(d) and (h)] sinusoidal phase with .

Measured (top row) and simulated (bottom row) spectra at the center of the focal plane, i.e., at , as a function of time delay and frequency . [(a) and (d)] Linear, [(b) and (e)] double pulse, and [(c) and (f)] sinusoidal phase modulation.

Measured (top row) and simulated (bottom row) spectra at the center of the focal plane, i.e., at , as a function of time delay and frequency . [(a) and (d)] Linear, [(b) and (e)] double pulse, and [(c) and (f)] sinusoidal phase modulation.

Spatially resolved second harmonic spectrum of a double pulse with a time delay of . (a) Measurement and (b) analytical solution, Eq. (25).

Spatially resolved second harmonic spectrum of a double pulse with a time delay of . (a) Measurement and (b) analytical solution, Eq. (25).

Spatially resolved second harmonic spectrum for a sinusoidal phase modulation. (a) Measurement and (b) analytical solution, Eq. (26).

Spatially resolved second harmonic spectrum for a sinusoidal phase modulation. (a) Measurement and (b) analytical solution, Eq. (26).

The three constants (a) , (b) , and (c) as a function of time delay in a double logarithmic plot. Spectral intensity for two different time delays, namely, (d) 400 fs and (e) 1400 fs.

The three constants (a) , (b) , and (c) as a function of time delay in a double logarithmic plot. Spectral intensity for two different time delays, namely, (d) 400 fs and (e) 1400 fs.

(a) Rubidium V-type three-level system. (b) Spatially resolved population distribution after interaction with a Gaussian-shaped bandwidth-limited pulse with a maximum fluence of . Ground state: black solid curve; lower excited state: red dotted curve; and upper excited state: blue dashed curve.

(a) Rubidium V-type three-level system. (b) Spatially resolved population distribution after interaction with a Gaussian-shaped bandwidth-limited pulse with a maximum fluence of . Ground state: black solid curve; lower excited state: red dotted curve; and upper excited state: blue dashed curve.

Spatially resolved final population of the Rb three-level system after excitation with (a) a downchirped pulse and (b) an upchirped pulse , respectively. (c) and (d) show the difference between the simulations in (a) and (b) and the corresponding ones without space-time coupling. Ground state: black solid curve; lower excited state: red dotted curve; and upper excited state: blue dashed curve.

Spatially resolved final population of the Rb three-level system after excitation with (a) a downchirped pulse and (b) an upchirped pulse , respectively. (c) and (d) show the difference between the simulations in (a) and (b) and the corresponding ones without space-time coupling. Ground state: black solid curve; lower excited state: red dotted curve; and upper excited state: blue dashed curve.

Spatially resolved final population of the Rb three-level system. The top row [(a) and (b)] shows simulations with space-time coupling and the center row [(c) and (d)] the difference with respect to simulations without space-time coupling. The bottom row [(e) and (f)] shows the spatially varying spectral intensity as a function of and the two horizontal dashed lines indicate the two resonance frequencies. For the left column the time delay is 51.7 fs and for the right column the time delay is 53.3 fs.

Spatially resolved final population of the Rb three-level system. The top row [(a) and (b)] shows simulations with space-time coupling and the center row [(c) and (d)] the difference with respect to simulations without space-time coupling. The bottom row [(e) and (f)] shows the spatially varying spectral intensity as a function of and the two horizontal dashed lines indicate the two resonance frequencies. For the left column the time delay is 51.7 fs and for the right column the time delay is 53.3 fs.

(a) Ground and excited state potential curves of . (b) Spatially resolved population in the ground (solid curve) and the excited state (dashed curve) after the interaction with a bandwidth-limited pulse.

(a) Ground and excited state potential curves of . (b) Spatially resolved population in the ground (solid curve) and the excited state (dashed curve) after the interaction with a bandwidth-limited pulse.

Spatially resolved population in the ground (solid curve) and in the excited state (dashed curve) of after interacting with a double pulse. (a) Time delay 720 fs and relative phase 0 and (b) time delay of 560 fs and relative phase of 2.3 rad. The differences with respect to simulations neglecting space-time coupling are shown in (c) and (d).

Spatially resolved population in the ground (solid curve) and in the excited state (dashed curve) of after interacting with a double pulse. (a) Time delay 720 fs and relative phase 0 and (b) time delay of 560 fs and relative phase of 2.3 rad. The differences with respect to simulations neglecting space-time coupling are shown in (c) and (d).

## Tables

Experimental parameters.

Experimental parameters.

Simulation parameters.

Simulation parameters.

Spatially averaged population for a pulse with no and with space-time coupling. Final populations calculated from spatially averaged fields with and with no space-time coupling. Populations for the maximum field at .

Spatially averaged population for a pulse with no and with space-time coupling. Final populations calculated from spatially averaged fields with and with no space-time coupling. Populations for the maximum field at .

Spatially averaged population for a pulse with no and with space-time coupling. Final populations calculated from spatially averaged fields with and with no space-time coupling.

Spatially averaged population for a pulse with no and with space-time coupling. Final populations calculated from spatially averaged fields with and with no space-time coupling.

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